Abstract
On the reference tetrahedron \(K\), we construct, for each \(k \in {\mathbb {N}}_0\), a right inverse for the trace operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\partial K}\). The operator is stable as a mapping from the trace space of \(W^{s, p}(K)\) to \(W^{s, p}(K)\) for all \(p \in (1, \infty )\) and \(s \in (k+1/p, \infty )\). Moreover, if the data is the trace of a polynomial of degree \(N \in {\mathbb {N}}_0\), then the resulting lifting is a polynomial of degree N. One consequence of the analysis is a novel characterization for the range of the trace operator.
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1 Introduction
The numerical analysis of high-order finite element and spectral element methods heavily rely on the existence of stable polynomial liftings—bounded operators mapping suitable piecewise polynomials on the boundary of the element to polynomials defined over the entire element. A number of operators have been constructed on the reference triangle and square, beginning with the pioneering work of Babuška et al. [14, 15]. Their lifting maps \(H^{\frac{1}{2}}(\partial E)\) boundedly into \(H^1(E)\), where E is either a suitable reference triangle or square, with the additional property that if the datum is continuous and its restriction to each edge is a polynomial of degree \(N \ge 0\), then the lifting is a polynomial of degree N. Other constructions for continuous piecewise polynomials on \(\partial E\) are stable from a discrete trace space to \(L^2(E)\) [4], from \(L^2(\partial E)\) to \(H^{\frac{1}{2}}(E)\) [3], from \(W^{1-\frac{1}{p}, p}(\partial E)\) to \(W^{1, p}(E)\) for \(1< p < \infty \) [45], and from \(W^{s-\frac{1}{p}, p}(\partial E)\) to \(W^{s, p}(E)\) for \(s \ge 1\) and \(1< p < \infty \) [48]. Liftings for other types of traces are also available; e.g. lifting the normal trace of \(H(\textrm{div}; E)\) [3], lifting the trace and normal derivative simultaneously into \(H^2(E)\) [2], and lifting an arbitrary number of normal derivatives simultaneously into \(W^{s, p}(E)\) [48].
Many of the above results have been extended to three space dimensions. Muñoz-Sola [46] generalized the construction of Babuška et al. [14, 15] to the tetrahedron, while Belgacem [16] gave a different construction for the cube using orthogonal polynomials. Commuting lifting operators for the spaces appearing in the de Rham complex on tetrahedra [30,31,32] and hexahedra [27] have also been constructed. These operators, among others, have been used extensively in a priori error analysis [7, 15, 36, 39, 45, 46], a posteriori error analysis [22, 25, 26, 37], the analysis of preconditioners [4, 5, 8, 9, 11, 14, 49], the analysis of sprectral element methods, particularly in weighted Sobolev spaces [18,19,20,21], and in the stability analysis of mixed finite element methods [6, 13, 28, 29, 33, 43]. Nevertheless, two types of operators are notably missing from the currently available results in three dimensions: (i) lifting operators stable in \(L^p\)-based Sobolev spaces, crucial in the analysis of high-order finite element methods for nonlinear problems; and (ii) lifting operators for the simultaneous lifting of the trace and normal derivative (and higher-order normal derivatives) which appear in the analysis of fourth-order (and higher-order) problems and in the analysis of mixed finite element methods for problems in electromagnetism and incompressible flow.
We address both of the above problems; namely, for each \(k \in {\mathbb {N}}_0\), we construct a right inverse for the trace operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\partial K}\) on the reference tetrahedron \(K\) that is stable from the trace of \(W^{s, p}(K)\) to \(W^{s, p}(K)\) for all \(p \in (1, \infty )\) and \(s \in (k+1/p, \infty )\). Additionally, if the data is the trace of a polynomial of degree \(N \in {\mathbb {N}}_0\), then the resulting lifting is a polynomial of degree N. A precise statement appears at the end of Sect. 2, which also contains a characterization for the trace space that appears to be novel and some potential applications of the results. These results generalize our construction on the reference triangle [48] to the reference tetrahedron and to Sobolev spaces with minimal regularity.
The remainder of the manuscript is organized as follows. In Sect. 3, we detail an explicit construction of the lifting operator in a sequence of four steps, each consisting of an intermediate single-face lifting operator. The remainder of the manuscript is devoted to the analysis of the intermediate single-face operators: Sects. 4 and 5 characterize the continuity of a related operator defined on all of \({\mathbb {R}}^3\), while Sect. 6 concludes with the proofs of the continuity properties of the intermediate operators.
2 The Traces of \(W^{s, q}(K)\) Functions and Statement of Main Result
We begin by reviewing the regularity properties of the traces of a function u defined on a tetrahedron. Here, we will work in the setting of Sobolev spaces defined on an open Lipschitz domain \({\mathcal {O}} \subseteq {\mathbb {R}}^d\). Let \(s = m + \sigma \) be a nonnegative real number with \(m \in {\mathbb {N}}_0\) and \(\sigma \in [0, 1)\). We denote by \(W^{s, p}({\mathcal {O}})\), \(p \in [1, \infty )\), the standard fractional Sobolev(-Slobodeckij) space [1] equipped with norm defined by
where the integer-valued seminorms and fractional seminorms are given by
with the usual modification for \(p = \infty \). When \(s = 0\), the Sobolev space \(W^{0, p}({\mathcal {O}})\) coincides with the standard Lebesque space \(L^p({\mathcal {O}})\), and we denote the norm by \(\Vert \cdot \Vert _{p, {\mathcal {O}}}\). We also require fractional Sobolev spaces defined on domain boundaries. Given a \(C^{k, 1}\), \(k \in {\mathbb {N}}_0\), \((d-1)\)-dimensional manifold \(\Gamma \subseteq \partial {\mathcal {O}}\), the surface gradient \(D_{\Gamma }\) is well-defined a.e. on \(\Gamma \), and we define \(W^{s, p}(\Gamma )\), \(0 \le s \le k+1\), analogously (see e.g. [47, §2.5.2]) with the norm
where the sums are over multi-indices \(\beta \in {\mathbb {N}}_0^{d-1}\). The seminorms \(|\cdot |_{s, p, \Gamma }\) are defined similarly.
2.1 Elementary Trace Results
Reference a tetrahedron and b triangle, where \(\textbf{e}_i\) are the standard unit vectors. Note that the label for \(\Gamma _4 = \{ (x, y, z) \in {\bar{K}}: x + y + z = 1 \}\) is omitted in (a)
When the domain is the reference tetrahedron \(K:= \{(x, y, z) \in {\mathbb {R}}^3: 0< x, y, z, x + y + z < 1\}\) depicted in Fig. 1a, the space \(W^{r, p}(\partial K)\), \(0 \le r < 1\), may be equipped with an equivalent norm that is more amenable to the analysis of traces. Let \(\Gamma _i\) and \(\Gamma _j\), \(1 \le i < j \le 4\), be two faces of \(K\) and let \(\gamma _{ij} = \gamma _{ji}\) denote the shared edge with vertices \(\varvec{a}\) and \(\varvec{b}\). Then, the vertices of \(\Gamma _i\) are denoted by \(\varvec{a}\), \(\varvec{b}\), and \(\varvec{c}_i\), while the vertices of \(\Gamma _j\) are denoted by \(\varvec{a}\), \(\varvec{b}\), and \(\varvec{c}_j\). Since \(\Gamma _i\) and \(\Gamma _j\) are both triangles, there exist unique affine mappings \(\varvec{F}_{ij}: T\rightarrow \Gamma _i\) and \(\varvec{F}_{ji}: T\rightarrow \Gamma _j\) from the reference triangle \(T:= \{ (x, y) \in {\mathbb {R}}^2: 0< x, y, x+ y < 1 \}\), labeled as in Fig. 1b, satisfying
and we define the following norm:
where \(f_i\) denotes the restriction of f to \(\Gamma _i\) and \({\mathcal {I}}_{i j}^p(f, g)\) is defined by the rule
Thanks to Corollary B.2, \({\left| \hspace{-1.0625pt}\left| \hspace{-1.0625pt}\left| \cdot \right| \hspace{-1.0625pt}\right| \hspace{-1.0625pt}\right| }_{r, p, \partial K}\) is an equivalent norm on \(W^{r, p}(\partial K)\); i.e.
and we shall use the two norms interchangeably with the common notation \(\Vert \cdot \Vert _{r, p, \partial K}\). Here, and in what follows, we use the standard notation \(a \lesssim _c b\) to indicate \(a \le C b\) where C is a constant depending only on c, and \(a \approx _c b\) if \(a \lesssim _c b\) and \(b \lesssim _c a\).
Now let \(u \in W^{s, p}(K)\), \(1< p < \infty \), \(s = m + \sigma > 1/p\) with \(m \in {\mathbb {N}}_0\) and \(\sigma \in [0, 1)\) (so that the trace operator is well-defined), be a function defined on the reference tetrahedron. The presence of edges and corners on the boundary of \(K\) limits the regularity of the trace of u. Nevertheless, we can iteratively apply the standard \(W^{s, p}(K)\) trace theorem (e.g. [41, Theorem 3.1] or [42, p. 208 Theorem 1]): \(W^{s, p}(K)\) embeds continuously into \(W^{s-\frac{1}{p}, p}(\partial K)\) for \(1/p< s < 1 + 1/p\). In particular, for \(k \in {\mathbb {N}}_0\), the kth-order derivative tensor given by
satisfies \(D^{k} u \in W^{s-k, p}(K) \subset W^{1+\sigma , p}(K)\), \(0 \le k \le m-1\), and \(D^m u \in W^{\sigma , p}(K)\); thus, the traces satisfy
Additionally, the trace of \(W^{m+\frac{1}{2}, 2}({\mathbb {R}}^3)\), \(m \ge 1\), on the plane \({\mathbb {R}}^2 \times \{0\}\) belongs to \(W^{m, 2}({\mathbb {R}}^2)\) (see e.g. [1, Chapter 7] or [42, p. 20 Theorem 4]), and so standard arguments show that the trace of \(W^{m+\frac{1}{2}, 2}(K)\) on the face \(\Gamma _i\), \(1 \le i \le 4\), belongs to \(W^{m, 2}(\Gamma _i)\). Thanks to the norm-equivalence (2.3), we arrive at the following conditions:
Remark 2.1
The case \(\sigma p = 1\) for \(p \ne 2\), which is not included in conditions (2.4), is beyond the scope of this paper since the trace of a \(W^{m+\frac{1}{p}, p}({\mathbb {R}}^3)\), \(m \in {\mathbb {N}}\), function on the plane \({\mathbb {R}}^2 \times \{0\}\) belongs to a Besov space, which cannot be identified with an integer-order Sobolev space [42, p. 20 Theorem 4].
When \(s > 2/p\), we obtain additional conditions since the trace of a \(W^{s, p}(K)\) function on the edge \(\gamma _{ij}\), \(1 \le i < j \le 4\) is well-defined. This can be seen from standard arguments owing to the fact that the trace of \(W^{s, p}({\mathbb {R}}^3)\) on the line \({\mathbb {R}} \times \{0\}^2\) is well-defined. In particular, the traces of the k-th derivative tensor, \(0 \le k < s-2/p\), on \(\Gamma _i\) and \(\Gamma _j\), \(1 \le i < j \le 4\), must agree on the shared edge \(\gamma _{ij}\):
where (2.5) is to be interpreted in the trace sense.
2.2 Trace Operators
We now turn to the consequences of (2.4) and (2.5) for various trace operators.
2.2.1 Zeroth-Order Operator
First consider the 0th-order “boundary-derivative” operator \({\mathfrak {D}}_i^0\) on \(\Gamma _i\), \(1 \le i \le 4\), defined formally by the rule
Then, (2.4) and (2.5) show that for \(u \in W^{s, p}(K)\), \((s, q) \in {\mathcal {A}}_0\), where
the trace \(f = u|_{\partial K}\) satisfies the following conditions:
-
1.
\(W^{s-\frac{1}{p}, p}\) regularity on each face:
$$\begin{aligned} {\mathfrak {D}}_i^0(f) \in W^{s-\frac{1}{p}, p}(\Gamma _i), \qquad 1 \le i \le 4. \end{aligned}$$(2.8) -
2.
Compatible traces along edges: For \(1 \le i < j \le 4\), there holds
$$\begin{aligned} {\mathfrak {D}}_{i}^0(f)|_{\gamma _{ij}} - {\mathfrak {D}}_{j}^0(f)|_{\gamma _{ij}}&= 0 \qquad{} & {} \text {if } sp > 2, \end{aligned}$$(2.9a)$$\begin{aligned} {\mathcal {I}}_{ij}^{p}({\mathfrak {D}}_{i}^0(f), {\mathfrak {D}}_{j}^0(f))&< \infty \qquad{} & {} \text {if } sp = 2. \end{aligned}$$(2.9b)
If \((s-n)p = 2\) for some \(n \in {\mathbb {N}}\), then we obtain an additional condition since the n-th derivative tensor satisfies \({\mathcal {I}}_{ij}^p(D^n u, D^n u) < \infty \) for \(1 \le i < j \le 4\). To describe this condition we define the following notation for a d-dimensional tensor S and vector \(v \in {\mathbb {R}}^3\):
In particular, for \(1 \le i < j \le 4\), denoting by \(\textbf{t}_{ij}\) a unit vector tangent to \(\gamma _{ij}\), we can differentiate \({\mathfrak {D}}_i^0(u)\) and \({\mathfrak {D}}_j^0(u)\) in the direction \(\textbf{t}_{ij}\) to obtain the following identity.
Consequently, the trace \(f = u|_{\partial K}\) also satisfies the following property:
-
3.
Compatible tangential derivatives: For \(1 \le i < j \le 4\) and \(n \in {\mathbb {N}}\), there holds
$$\begin{aligned} {\mathcal {I}}_{ij}^{p}\left( \frac{\partial ^n {\mathfrak {D}}_i^0(u)}{\partial \textbf{t}_{ij}^n}, \frac{\partial ^n {\mathfrak {D}}_j^0(u)}{\partial \textbf{t}_{ij}^n}\right) < \infty \qquad \text {if } (s-n)p = 2. \end{aligned}$$(2.10)
2.2.2 First-Order Operator
For \((s, p) \in {\mathcal {A}}_1\), we turn to the regularity of the trace of the gradient of \(u \in W^{s, p}(K)\). To this end, on each face \(\Gamma _i\), \(1 \le i \le 4\), let \(\{ \varvec{\tau }_{i, 1}, \varvec{\tau }_{i, 2} \}\) be orthonormal vectors tangent to \(\Gamma _i\) and let \(\textbf{n}_i\) denote the outward unit normal vector on \(\Gamma _i\). We define the 1st-order “boundary-derivative” operator \({\mathfrak {D}}_i^1\) on \(\Gamma _i\), \(1 \le i \le 4\), by the rule
so that \({\mathfrak {D}}_i^1(u, \partial _{\textbf{n}} u) = Du|_{\Gamma _i}\). Again applying (2.4) and (2.5), we obtain analogues of (2.8) and (2.9) stated below in (2.13) and (2.14) with \(k=0\). However, if \((s-2)p > 2\), then the second derivative tensor has matching traces on edges (i.e. (2.5) holds with \(k=2\)). In particular, for \(1 \le i < j \le 4\), we define the vectors
where we recall that \(\textbf{t}_{ij}\) is a unit vector tangent to \(\gamma _{ij}\), so that on \(\gamma _{ij}\), there holds
in the sense of traces. As a consequence, the operator \({\mathfrak {D}}_i^1\) satisfies the additional condition (2.15) below with \(n=0\) thanks to (2.4) and (2.5). Finally, we can differentiate in the direction tangent to each edge to obtain the analogue of (2.10) stated in (2.14b) and (2.15b) below. To summarize, the traces \(f = u|_{\partial K}\) and \(g = \partial _{\textbf{n}} u|_{\partial K}\) satisfy the following for all \((s, p) \in {\mathcal {A}}_1\):
-
1.
\(W^{s-1-\frac{1}{p}, p}\) regularity on each face:
$$\begin{aligned} {\mathfrak {D}}_i^1(f, g) \in W^{s-1-\frac{1}{p}, p}(\Gamma _i), \qquad 1 \le i \le 4. \end{aligned}$$(2.13) -
2.
Compatible traces along edges: For \(1 \le i < j \le 4\) and \(n \in {\mathbb {N}}_0\), there holds
$$\begin{aligned} {\mathfrak {D}}_{i}^1(f, g)|_{\gamma _{ij}} - {\mathfrak {D}}_{j}^1(f, g)|_{\gamma _{ij}}&= 0 \qquad{} & {} \text {if } (s-1)p > 2, \end{aligned}$$(2.14a)$$\begin{aligned} {\mathcal {I}}_{ij}^{p}\left( \frac{\partial ^n {\mathfrak {D}}_{i}^1(f, g)}{ \partial \textbf{t}_{ij}^n }, \frac{\partial ^n {\mathfrak {D}}_{j}^1(f, g)}{\partial \textbf{t}_{ij}^n} \right)&< \infty \qquad{} & {} \text {if } (s-n-1)p = 2. \end{aligned}$$(2.14b) -
3.
Compatible traces of higher derivatives along edges: For \(1 \le i < j \le 4\) and \(n \in {\mathbb {N}}_0\), there holds
$$\begin{aligned} \left. \textbf{b}_{ji} \cdot \frac{\partial {\mathfrak {D}}^1_{i}(f, g)}{\partial \textbf{b}_{ij}} \right| _{\gamma _{ij}} - \left. \textbf{b}_{ij} \cdot \frac{\partial {\mathfrak {D}}^1_{j}(f, g)}{\partial \textbf{b}_{ji}}\right| _{\gamma _{ij}}&= 0 \qquad{} & {} \text {if } (s-2)p > 2, \end{aligned}$$(2.15a)$$\begin{aligned} {\mathcal {I}}_{ij}^{p}\left( \textbf{b}_{ji} \cdot \frac{\partial ^{n+1} {\mathfrak {D}}^1_{i}(f, g)}{ \partial \textbf{t}_{ij}^{n} \partial \textbf{b}_{ij}}, \textbf{b}_{ij} \cdot \frac{\partial ^{n+1} {\mathfrak {D}}^1_{j}(f, g)}{ \partial \textbf{t}_{ij}^n \partial \textbf{b}_{ji}} \right)&< \infty \qquad{} & {} \text {if } (s-n-2)p = 2. \end{aligned}$$(2.15b)
Remark 2.2
For smooth enough functions, conditions (2.14a) and (2.15a) may be interpreted as the application of the vertex compatibility conditions for traces on the triangle (see e.g. [48, eqs. (2.11a) and (2.12a)]) at every point on the edge \(\gamma _{ij}\).
2.2.3 mth-Order Operator
We now turn to the general case of the trace of the m-th derivative tensor of a function \(u \in W^{s, p}(K)\), where \(m \ge 2\) and \((s, p) \in {\mathcal {A}}_m\). Given a collection of functions \(F = (f^0, f^1, \ldots , f^m)\) defined on \(\partial K\), we define the m-th order “boundary-derivative” operator \({\mathfrak {D}}_i^m\) on \(\Gamma _i\), \(1 \le i \le 4\), by the rule
where the set \({\mathfrak {M}}_{i}(\alpha )\) consists of the following mappings.
where we recall that \(\{ \varvec{\tau }_{i, 1}, \varvec{\tau }_{i, 2} \}\) are orthonormal vectors tangent to \(\Gamma _i\). For notational convenience, we set
Then, one may readily verify that \({\mathfrak {D}}^m_i(u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^m u) = D^m u\) on \(\Gamma _i\). Let \(f^l = \partial _{\textbf{n}}^l u\) on \(\partial K\). As before, we obtain \(W^{s-m-\frac{1}{p}, p}\) regularity of \({\mathfrak {D}}_i^m(F)\) (2.17) below on each face thanks to (2.4) and the edge compatibility conditions (2.18) below with \(l = 0\) from (2.5).
As was the case with the first-order operator, there are additional edge compatibility conditions. In particular, if \((s-m-l)p > 0\) for some \(1 \le l \le m\), then (2.5) shows that the \((m+l)\)th derivative tensor has matching traces on edges. Some components of the \((m+l)\)th derivative tensor can be expressed in terms of \({\mathfrak {D}}_l^{m}(F)\). In particular, on the edge \(\gamma _{ij}\), \(1 \le i < j \le 4\), there holds
in the sense of traces, where we used symmetry of the derivative tensor \(D^{m+l} u\). We can also differentiate in the direction tangent to each edge to obtain similar conditions. Consequently, \({\mathfrak {D}}_i^m(F)\) satisfies (2.18) below. In summary, for \(m \in {\mathbb {N}}_0\), the traces \(F = (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^{m} u)\) satisfy the following for all \((s, p) \in {\mathcal {A}}_m\):
-
1.
\(W^{s-m-\frac{1}{p}, p}\) regularity on each face:
$$\begin{aligned} {\mathfrak {D}}_i^m(F) \in W^{s-m-\frac{1}{p}, p}(\Gamma _i), \qquad 1 \le i \le 4, \end{aligned}$$(2.17)where \({\mathfrak {D}}_i^0\), \({\mathfrak {D}}_i^1\), and \({\mathfrak {D}}_i^l\), \(l \ge 2\), are defined in (2.6), (2.11), and (2.16).
-
2.
Compatible traces along edges: For \(1 \le i < j \le 4\) and \(0 \le l \le m\) and \(n \in {\mathbb {N}}_0\), there holds
$$\begin{aligned} \left. \textbf{b}_{ji}^{\otimes l} \cdot \frac{\partial ^l {\mathfrak {D}}_i^{m}(F)}{\partial \textbf{b}_{ij}^{l}} \right| _{\gamma _{ij}} - \left. \textbf{b}_{ij}^{\otimes l} \cdot \frac{\partial ^l {\mathfrak {D}}_j^{m}(F)}{\partial \textbf{b}_{ji}^{l}} \right| _{\gamma _{ij}}&= 0 \qquad{} & {} \text {if } (s-m-l)p > 2, \end{aligned}$$(2.18a)$$\begin{aligned} {\mathcal {I}}_{ij}^{p}\left( \textbf{b}_{ji}^{\otimes l} \cdot \frac{\partial ^{l+n} {\mathfrak {D}}_i^{m}(F)}{\partial \textbf{t}_{ij}^{n} \partial \textbf{b}_{ij}^{l}}, \textbf{b}_{ij}^{\otimes l} \cdot \frac{\partial ^{l+n} {\mathfrak {D}}_j^{m}(F)}{\partial \textbf{t}_{ij}^{n} \partial \textbf{b}_{ji}^{l}} \right)&< \infty \qquad{} & {} \text {if } (s-m-l-n)p = 2. \end{aligned}$$(2.18b)
Remark 2.3
As was the case in Remark 2.2, condition (2.18a) is simply the application of the vertex compatibility conditions for traces on the triangle [48, eq. (7.2)] at every point on the edge \(\gamma _{ij}\), provided that u is smooth enough.
2.3 The Trace Theorem on a Tetrahedron
Motivated by the conditions derived in the previous section, we define trace spaces as follows. Given a set of indices \({\mathcal {S}} \subseteq \{1,2,3,4\}\) with \(|{\mathcal {S}}| \ge 1\), let \(\Gamma _S:= \cup _{i \in {\mathcal {S}}} \Gamma _i\). We define the trace space on part of the boundary \({{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\) for \(k \in {\mathbb {N}}_0\) and \((s, p) \in {\mathcal {A}}_k\) as follows.
equipped with the norm
Note that the sum in the definition contains only finitely many nonzero terms, and hence is well defined. When \({\mathcal {S}} = \{1,2,3,4\}\), we set \({{\,\textrm{Tr}\,}}^{s,p}_k(\partial K):= {{\,\textrm{Tr}\,}}^{s,p}_k(\Gamma _{{\mathcal {S}}})\) and \(\Vert \cdot \Vert _{{{\,\textrm{Tr}\,}}_k^{s, p}, \partial K}:= \Vert \cdot \Vert _{{{\,\textrm{Tr}\,}}_k^{s, p}, \Gamma _{{\mathcal {S}}}}\). The following trace theorem is a consequence of the discussion in the previous section.
Theorem 2.4
Let \({\mathcal {S}} \subseteq \{1,2,3,4\}\), \(k \in {\mathbb {N}}_0\), and \((s, p) \in {\mathcal {A}}_k\) be given. Then, for every \(u \in W^{s, p}(K)\), the traces satisfy \((u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^{k} u)|_{\Gamma _{{\mathcal {S}}}} \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\) and
2.4 The Trace of Polynomials
Given \(N \in {\mathbb {N}}_0\), let \({\mathcal {P}}_N(K)\) denote the set of all polynomials of total degree at most N, while \({\mathcal {P}}_{-M}:= \{0\}\) for \(M > 0\). If \(u \in {\mathcal {P}}_N(K)\), then \(u \in W^{s, p}(K)\) for all \(s \ge 0\) and \(p \ge 1\). Consequently, for each \(k \in {\mathbb {N}}_0\) and \({\mathcal {S}} \subseteq \{1,2,3,4\}\), the traces \(F = (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\Gamma _{{\mathcal {S}}}} \in {{\,\textrm{Tr}\,}}^{s, p}_k(\Gamma _{{\mathcal {S}}})\) for all \((s, p) \in {\mathcal {A}}_k\). In particular, s may be taken to be arbitrarily large in (2.18a). Thus, the traces satisfy
Note that we have not included the integral condition (2.18b) in the list (2.20) above. The following lemma shows that if a tuple of functions defined on \(\partial K\) satisfy (2.20), then (2.18b) is automatically satisfied.
Lemma 2.5
Let \({\mathcal {S}} \subseteq \{1,2,3,4\}\) and \(k \in {\mathbb {N}}_0\). If \(F: \Gamma _{{\mathcal {S}}} \rightarrow {\mathbb {R}}^{k+1}\) satisfies (2.20), then \(F \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\) for all \((s, p) \in {\mathcal {A}}_k\).
Proof
Let \((s, p) \in {\mathcal {A}}_k\), \(0 \le l \le m \le k\), be given. Thanks to (2.20b) and (2.20c), the difference
vanishes on the edge \(\gamma _2\) of the reference triangle \(T\) and \(H_{ij}\) has entries \({\mathcal {P}}_{N-m-l}(T)\). Thus, \(H_{ij} = x_2 G_{ij}\), where \(G_{ij}\) has entries in \({\mathcal {P}}_{N-m-l-1}(T)\). Consequently, for all \(n \in {\mathbb {N}}_0\), there holds
which is finite since \(G_{ij}\) has polynomial entries. The inclusion \(F \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\) now follows from (2.20). \(\square \)
2.5 Statement of the Main Result
The aim of the current work is to construct a right inverse \({\mathcal {L}}_k\) of the operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^{k} u)|_{\partial K}\) for each \(k \in {\mathbb {N}}_0\) that is bounded from \({{\,\textrm{Tr}\,}}_k^{s, p}(\partial K)\) into \(W^{s, p}(K)\) for all \((s, p) \in {\mathcal {A}}_k\) and preserves polynomials in the following sense: if \(F = (f^0, f^1, \ldots , f^k)\) is the trace of some degree N polynomial, then \({\mathcal {L}}_k (F)\) is a polynomial of degree N. In particular, the main result is as follows.
Theorem 2.6
Let \(k \in {\mathbb {N}}_0\). There exists a linear operator
satisfying the following properties: for all \((s, p) \in {\mathcal {A}}_{k}\) and \(F = (f^0, f^1, \ldots , f^k) \in {{\,\textrm{Tr}\,}}_k^{s, p}(\partial K)\), \({\mathcal {L}}_k(F) \in W^{s, p}(K)\),
Moreover, if F is a piecewise polynomial of degree \(N \in {\mathbb {N}}_0\) satisfying (2.20) with \({\mathcal {S}} = \{1,2,3,4\}\), then \({\mathcal {L}}_k(F) \in {\mathcal {P}}_N(K)\).
The construction of the lifting operator \({\mathcal {L}}_k\) in Theorem 2.6 is the focus of the next section, and the proof of Theorem 2.6 appears in Sect. 3.5. An immediate consequence is the following characterization of the range of the trace operator.
Corollary 2.7
For each \(k \in {\mathbb {N}}_0\), the operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^{k} u)|_{\partial K}\) is surjective from \(W^{s, p}(K)\) onto \({{\,\textrm{Tr}\,}}_k^{s, p}(\partial K)\) for all \((s, p) \in {\mathcal {A}}_k\).
2.6 Potential Applications
Theorem 2.6 has many potential applications, particularly in the analysis of high-order finite element methods. For brevity, we discuss three applications. Firstly, the extension operator may be used analogously to the constructions in [7, 15, 45] to establish optimal (with respect to mesh size and polynomial degree) a priori error estimates for \(W^{s, p}\)-conforming finite element spaces for all \(p \in (1,\infty )\) and \(s > 1/p\). Secondly, the lifting operator will be crucial to obtain bounds explicit in polynomial degree for preconditioners for high-order finite element discretizations of fourth-order (and higher-order) elliptic problems similar to \(H^1\)-stable extensions for second-order problems in 2D and 3D [14, 49] and \(H^2\)-stable extensions for fourth-order problems in 2D [8]. Finally, in a similar vein to [6, 10, 13, 29, 43], the extension operator may be helpful in constructing a polynomial-preserving right inverse of the curl operator that preserves some trace properties (e.g. vanishing tangential trace, vanishing trace, etc.) and in proving discrete Friedrichs inequalities. These results have applications to the stability, convergence theory, and preconditioning of high-order discretizations of mixed and parameter-dependent problems (see also e.g. [11, 28, 33]).
3 Construction of the Lifting Operator
The construction of the lifting operator \({\mathcal {L}}_k\), \(k \in {\mathbb {N}}_0\), proceeds face-by-face using similar techniques to [46, 48]. The main idea is to perform a sequence of liftings and corrections using a fundamental convolution operator (see e.g. [12, eq. (4.2)], [14, 18, 19, p. 56, eq. (2.1)], [47, §2.5.5]) and subsequent modifications to it. Given a nonnegative integer \(k \in {\mathbb {N}}_0\), a smooth compactly supported function \(b \in C_c^{\infty }(T)\), and a function \(f: T\rightarrow {\mathbb {R}}\), we define the operator \({\mathcal {E}}_k^{[1]}\) formally by the rule
and we use the notation \({\mathcal {E}}_k^{[1]}[b]\) when we want to make the dependence on b explicit. Note that for \((\varvec{x}, z) \in K\) and \(\varvec{y} \in T\), there holds \(\varvec{x} + z \varvec{y} \in T\), and so (3.1) is well-defined for e.g. \(f \in C^{\infty }({\bar{T}})\). For functions \(f: \Gamma _1 \rightarrow {\mathbb {R}}\) we define
3.1 Lifting from One Face
The first result concerns the interpolation and continuity properties of \({\mathcal {E}}_k^{[1]}\).
Lemma 3.1
Let \(b \in C_c^{\infty }(T)\), \(k \in {\mathbb {N}}_0\), and \((s, p) \in {\mathcal {A}}_k\). Then, for all \(f \in W^{s-k-\frac{1}{p},p}(\Gamma _1)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), then \({\mathcal {E}}_k^{[1]}(f) \in {\mathcal {P}}_{N+k}(K)\).
The proof appears in Sect. 6.1. We now construct a lifting operator from \(\Gamma _1\).
Lemma 3.2
Let \(b \in C_c^{\infty }(T)\) with \(\int _{T} b(\varvec{x}) \,\textrm{d}{\varvec{x}} = 1\) and \(k \in {\mathbb {N}}_0\). We formally define the following operators for \(F = (f^0, f^1, \ldots , f^k)~\in ~L^p(\Gamma _1)^{k+1}\):
Then, for all \((s, p) \in {\mathcal {A}}_k\) and \(F \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _1)\), \({\mathcal {L}}_k^{[1]}(F)\) is well-defined and there holds
Moreover, if \(f^m \in {\mathcal {P}}_{N-m}(\Gamma _1)\), \(0 \le m \le k\), for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_k^{[1]}(F) \in {\mathcal {P}}_N(K)\).
Proof
We proceed by induction on k. The case \(k = 0\) follows immediately from Lemma 3.1. Now assume that the lemma is true for some \(k \in {\mathbb {N}}_0\) and let \((s, p) \in {\mathcal {A}}_{k+1}\) and \(F \in {{\,\textrm{Tr}\,}}_{k+1}^{s, p}(\Gamma _1)\) be as in the statement of the lemma. Then, we may apply the lemma to \({\tilde{F}} = (f^0, f^1, \ldots , f^k) \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _1)\) to conclude that for \(0 \le m \le k\) and
Thanks to the trace theorem (Theorem 2.4), there holds \(f^{k+1} - \partial _{\textbf{n}}^{k+1} {\mathcal {L}}_{k}^{[1]}(F)|_{\Gamma _1} \in W^{s-k-1-\frac{1}{p}, p}(\Gamma _1)\) with
and so (3.6) follows from Lemma 3.1. Additionally, if F satisfies \(f^m \in {\mathcal {P}}_{N-m}(\Gamma _1)\), \(0 \le m \le k+1\), for some \(N \in {\mathbb {N}}_0\), then \({\tilde{F}}\) satisfies the same condition, where the upper bound of m is restricted to k. Consequently, \({\mathcal {L}}_k^{[1]}({\tilde{F}}) \in {\mathcal {P}}_N(K)\) and so \(f^{k+1} - \partial _{\textbf{n}}^{k+1} {\mathcal {L}}_{k}^{[1]}({\tilde{F}})|_{\Gamma _1} \in {\mathcal {P}}_{N-k-1}(\Gamma _1)\). Thus, \({\mathcal {L}}_{k+1}^{[1]}(F) \in {\mathcal {P}}_N(K)\) thanks to Lemma 3.1. \(\square \)
3.2 Lifting from Two Faces
We now seek a lifting operator from \(\Gamma _2\) that has zero trace on \(\Gamma _1\). The operator will be a generalization of the form introduced in [46]. We first define an operator that lifts traces from \(\Gamma _1\) and has zero trace on \(\Gamma _2\), and then define the lifting operator from \(\Gamma _2\) in terms of this operator. To this end, denote by \(\omega _i\) the barycentric coordinates of \(T\) defined as follows.
Given nonnegative integers \(k, r \in {\mathbb {N}}_0\), a smooth compactly supported function \(b \in C_c^{\infty }(T)\), and a function \(f: T\rightarrow {\mathbb {R}}\), we define the operator \({\mathcal {M}}_{k, r}^{[1]}\) formally by the rule
Note that when \(r = 0\), we have \({\mathcal {M}}_{k, 0}^{[1]} = {\mathcal {E}}_k^{[1]}\). For functions \(f: \Gamma _1 \rightarrow {\mathbb {R}}\), we again abuse notation and set \({\mathcal {M}}_{k, r}^{[1]}(f):= {\mathcal {M}}_{k, r}^{[1]}(f \circ {\mathfrak {I}}_1)\).
The presence of the weight \(\omega _2^{-r}\) in the operator \({\mathcal {M}}_{k, r}^{[1]}\) means that derivatives of \(f: \Gamma _{1} \rightarrow {\mathbb {R}}\) up to order r have to vanish on edge \(\gamma _{12}\) in an appropriate sense. To this end, let \(s = m + \sigma \) with \(m \in {\mathbb {N}}_0\) and \(\sigma \in [0, 1)\) and \(1< p < \infty \). Given a face \(\Gamma _j\), \(1 \le j \le 4\), and \({\mathfrak {E}}\) a subset of the edges of \(\Gamma _j\), we define the following subspaces of \(W^{s, p}(\Gamma _j)\) with vanishing traces on the edges in \({\mathfrak {E}}\):
where the norm on \(W^{s, p}_{{\mathfrak {E}}}(\Gamma _j)\) is given by
and we recall that \(D_{\Gamma }\) is the surface gradient operator. When \({\mathfrak {E}}\) consists of only one edge \(\gamma \), we set \(W_{\gamma }^{s, p}(\Gamma _j):= W_{{\mathfrak {E}}}^{s, p}(\Gamma _j)\) and 
. One can readily verify that the spaces \(W_{{\mathfrak {E}}}^{s, p}(\Gamma _j)\) are Banach spaces and that the following relations hold:
Given a subset of edges \({\mathfrak {E}}\) of the reference triangle \(T\), we define the spaces \(W_{{\mathfrak {E}}}^{s, p}(T)\) analogously.
The first result states the continuity properties of \({\mathcal {M}}_{k, r}^{[1]}\).
Lemma 3.3
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), and \((s, p) \in {\mathcal {A}}_k\). Then, for all
\(f~\in ~W^{s-k-\frac{1}{p},p}(\Gamma _1) \cap W_{\gamma _{12}}^{\min \{s-k-\frac{1}{p}, r\}, p}(\Gamma _1)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\gamma _{12}} = 0\) for \(0 \le l \le r-1\), then \({\mathcal {M}}_{k, r}^{[1]}(f) \in {\mathcal {P}}_{N+k}(K)\).
The proof of Lemma 3.3 appears in Sect. 6.3. By mapping the other faces of \(K\) to \(\Gamma _1\) and mapping \(K\) onto itself in an appropriate fashion, we may define operators corresponding to these faces. In particular, we define the following operator corresponding to \(\Gamma _2\):
where \({\mathfrak {I}}_2(\varvec{x}):= (x_1, 0, x_2)\) and \({\mathfrak {R}}_{12}(\varvec{x}, z):= (x_1, z, x_2)\) for all \((\varvec{x}, z) \in K\).
Thanks to the chain rule and the smoothness of the mappings \({\mathfrak {I}}_2\) and \({\mathfrak {R}}_{12}\), the continuity and interpolation properties of \({\mathcal {M}}_{k, r}^{[2]}\) follow immediately from Lemma 3.3.
Corollary 3.4
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), and \((s, p) \in {\mathcal {A}}_k\). Then, for all \(f \in W^{s-k-\frac{1}{p},p}(\Gamma _2) \cap W_{\gamma _{12}}^{\min \{s-k-\frac{1}{p}, r\}, p}(\Gamma _2)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _2)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\gamma _{12}} = 0\) for \(0 \le l \le r-1\), then \({\mathcal {M}}_{k, r}^{[2]}(f) \in {\mathcal {P}}_{N+k}(K)\).
3.2.1 Regularity of Partially Vanishing Traces
The operator \({\mathcal {M}}_{k, r}^{[2]}\) lifts traces from \(\Gamma _2\) to \(K\) and has zero trace on \(\Gamma _1\), which are the properties we desired to correct the traces of \({\mathcal {L}}_k^{[1]}\) on \(\Gamma _2\). However, \({\mathcal {M}}_{k, r}^{[2]}\) acts on functions belonging to \(W^{s-k-\frac{1}{p},p}(\Gamma _2) \cap W_{\gamma _{12}}^{\min \{s-k - \frac{1}{p}, r\}, p}(\Gamma _2)\) rather than just functions in \(W^{s-k-\frac{1}{p},p}(\Gamma _2)\). The main result of this section characterizes one scenario in which traces belong to the space \(W^{s-k-\frac{1}{p},p}(\Gamma _2) \cap W_{\gamma _{12}}^{\min \{s-k - \frac{1}{p}, r\}, p}(\Gamma _2)\), and fortunately, we will encounter exactly this scenario in our construction.
We have the following result which characterizes the regularity of the restriction of a trace \(F \in {{\,\textrm{Tr}\,}}_k^{s,p}(\Gamma _i \cup \Gamma _j)\) to \(\Gamma _j\) when F vanishes on \(\Gamma _i\) and the first l components of F vanish on \(\Gamma _j\).
Lemma 3.5
Let \(k \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), \(1 \le l \le k\), and \(1 \le i, j \le 4\) with \(i \ne j\) be given. Suppose that \(F = (f^0, f^1, \ldots , f^k) \in {{\,\textrm{Tr}\,}}_k^{s,p}(\Gamma _i \cup \Gamma _j)\) satisfies
-
(i)
\(F = (0, 0, \ldots , 0)\) on \(\Gamma _i\);
-
(ii)
\(f_j^m = 0\) on \(\Gamma _j\) for \(0 \le m \le l-1\).
Then, there holds \(f^l_j \in W^{s - l - \frac{1}{p}, p}(\Gamma _j) \cap W^{\min \{s - l - \frac{1}{p}, k+1 \}, p}_{\gamma _{ij}}(\Gamma _j)\) and
Proof
Without loss of generality, assume that \(i < j\). We first show that for \(\alpha \in {\mathbb {N}}_0^2\) there holds
where \(\textbf{b}_{ij}\) and \(\textbf{b}_{ji}\) are defined in (2.12).
Step 1: \(0 \le \alpha _2 \le k-l\) and \(|\alpha | < \min \{s-l-2/p, k+1\}\). Manipulating definitions shows that
and so there holds
and using that \({\mathfrak {D}}_i^{l+\alpha _2}(F)|_{\Gamma _i} = 0\) by (i) gives
on \(T\). Equality (3.16) now follows from (2.18a).
Step 2: \(k-l+1 \le \alpha _2 \le k\) and \(|\alpha | < \min \{s-l-2/p, k+1\}\). The same arguments as in Step 1 show that
By construction, there exist constants \(a_1\) and \(a_2\) such that \(\textbf{n}_j = a_1 \textbf{b}_{ij} + a_2 \textbf{b}_{ji}\), and so
for some suitable constants \(\{c_r\}_{r=0}^{l}\). For \(0 \le r \le l-1\), \({\mathfrak {D}}^r_j(F) = 0\) by (ii), and so
Applying (i) then gives the following identity on \(T\):
Equality (3.16) then follows from (2.18a).
Step 3: \(f_j^l \in W^{s-l-\frac{1}{p}, p}(\Gamma _j) \cap W^{\min \{s - l - \frac{1}{p}, k+1 \} p}_{\gamma _{ij}}(\Gamma _j)\). For \(s - 2/p \notin {\mathbb {Z}}\), the inclusion \(f_j^l \in W^{s-l-\frac{1}{p}, p}(\Gamma _j) \cap W^{\min \{s - l - \frac{1}{p}, k+1 \}, p}_{\gamma _{ij}}(\Gamma _j)\) follows from (3.16), and (3.15a) and (3.15b) are an immediate consequence of the definition of the \(\Vert \cdot \Vert _{{{\,\textrm{Tr}\,}}_k^{s, p}, \Gamma _i \cup \Gamma _j}\) norm. For \(s - 2/p \in {\mathbb {Z}}\), and \(|\alpha | = \min \{s-l- 2/p, k+1\}\), there holds
and so the inclusion \(f_j^l \in W^{\min \{s - l - \frac{1}{p}, k + 1 \}, p}_{\gamma _{ij}}(\Gamma _j)\) follows from (3.18), (3.19) and (2.18b), while (3.15a) follows from the definition of the norm. \(\square \)
3.2.2 Construction of Lifting
In the following lemma, we construct the lifting operator \({\mathcal {L}}_{k}^{[2]}\) in the same fashion as \({\mathcal {L}}_k^{[1]}\) (3.5), replacing the use of \({\mathcal {E}}_{m}^{[1]}\) with \({\mathcal {M}}_{m, k+1}^{[2]}\).
Lemma 3.6
Let \(b \in C_c^{\infty }(T)\) with \(\int _{T} b(\varvec{x}) \,\textrm{d}{\varvec{x}} = 1\), \(k \in {\mathbb {N}}_0\), and \({\mathcal {S}} = \{1,2\}\). For \(F = (f^0, f^1, \ldots , f^k) \in L^p(\Gamma _1 \cup \Gamma _2)^{k+1}\), we formally define the following operators:
Then, for all \((s, p) \in {\mathcal {A}}_k\) and \(F \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\), \({\mathcal {L}}_{k}^{[2]}(F)\) is well-defined and there holds
Moreover, if for some \(N \in {\mathbb {N}}_0\), F satisfies (2.20), then \({\mathcal {L}}_{k}^{[2]}(F) \in {\mathcal {P}}_N(K)\).
Proof
Let \(k \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \(f \in {{\,\textrm{Tr}\,}}_k^{s, p}({\mathcal {S}})\) be given.
Step 1: \(m = 0\). Thanks to Lemma 3.2, the traces \(G = (g^0, g^1, \ldots , g^k)\) given by
satisfy the hypotheses of Lemma 3.5 with \((i, j) = (1, 2)\) and \(l=1\). Thanks to Lemma 3.5 and Corollary 3.4, \({\mathcal {M}}_{0, k+1}^{[2]}(g_2^0)\), and hence \({\mathcal {L}}_{k, 0}^{[2]}(F)\), is well-defined with
Applying (3.6) and (3.13a) gives
and applying (3.6) and (3.15) gives
Moreover, if F satisfies (2.20) and for some \(N \in {\mathbb {N}}_0\), then \(F \in {{\,\textrm{Tr}\,}}^{s, p}_k(\Gamma _{{\mathcal {S}}})\) by Lemma 2.5 and \({\mathcal {L}}_k^{[1]}(F) \in {\mathcal {P}}_N(K)\) by Lemma 3.2. Thus, the trace G satisfies (2.20) for \(\{i, j\} \subseteq \{1,2\}\) and \(G \in {{\,\textrm{Tr}\,}}^{s, p}_k(\Gamma _{{\mathcal {S}}})\) for all \((s, p) \in {\mathcal {A}}_k\). By Lemma 3.5, \(g_2^0 \in W_{\gamma _{12}}^{k + 1, p}(\Gamma _2)\) for all \(p \in (1,\infty )\), and so \(D_{\Gamma }^l g^0_2|_{\gamma _{12}} = 0\) for \(0 \le l \le k\). Thanks to Corollary 3.4, \({\mathcal {L}}_{k, 0}^{[2]}(F) \in {\mathcal {P}}_{N}(K)\).
Step 2: Induction on m. Assume that for some m such that \(0 \le m \le k-1\), \({\mathcal {L}}^{[2]}_{k, m}(F)\) is well-defined and satisfies
and
Additionally assume that if F satisfies (2.20) for \(\{i, j\} \subseteq \{1,2\}\) and for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k, m}^{[2]}(F) \in {\mathcal {P}}_N(K)\).
Thanks to (3.23), the traces \(G = (g^0, g^1, \ldots , g^k)\) given by
satisfy the hypotheses of Lemma 3.5 with \((i, j) = (1, 2)\) and \(l=m+1\). Thanks to Lemma 3.5 and Corollary 3.4, \({\mathcal {M}}_{m+1, k+1}^{[2]}(g_2^{m+1})\), and hence \({\mathcal {L}}_{k, m+1}^{[2]}(F)\) is well-defined with
Applying (3.23) and (3.13a) gives (3.23) for \(m+1\), while applying (3.24) and (3.15) gives (3.24) for \(m+1\).
Moreover, if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k, m}^{[2]}(F) \in {\mathcal {P}}_N(K)\) by assumption and so the trace G satisfies (2.20) and \(G \in {{\,\textrm{Tr}\,}}^{s, p}_k(\Gamma _{{\mathcal {S}}})\) for all \((s, p) \in {\mathcal {A}}_k\). By Lemma 3.5, \(g_2^{m+1} \in W_{\gamma _{12}}^{k + 1, p}(\Gamma _2)\) for all \(p \in (1,\infty )\), and so \(D_{\Gamma }^l g^{m+1}_2|_{\gamma _{12}} = 0\) for \(0 \le l \le k\). Thanks to Corollary 3.4, \({\mathcal {L}}_{k, m+1}^{[2]}(F) \in {\mathcal {P}}_{N}(K)\). \(\square \)
3.3 Lifting from Three Faces
We continue in the spirit of the previous two sections and define another lifting operator from \(\Gamma _1\) with vanishing traces on \(\Gamma _2\) and \(\Gamma _3\). Given nonnegative integers \(k, r \in {\mathbb {N}}_0\), a smooth compactly supported function \(b \in C_c^{\infty }(T)\), and a function \(f: T\rightarrow {\mathbb {R}}\), we define the operator \({\mathcal {S}}_{k, r}^{[1]}\) formally by the rule
Note that when \(r = 0\), we have \({\mathcal {S}}_{k, 0}^{[1]} = {\mathcal {E}}_k^{[1]}\). For functions \(f: \Gamma _1 \rightarrow {\mathbb {R}}\), we again abuse the notation and set \({\mathcal {S}}_{k, r}^{[1]}(f):= {\mathcal {S}}_{k, r}^{[1]}(f \circ {\mathfrak {I}}_1)\).
We require one additional family of spaces with vanishing traces. Let \(s = m + \sigma \) with \(m \in {\mathbb {N}}_0\) and \(\sigma \in [0, 1)\) and \(1< p < \infty \). Given \(r \in {\mathbb {N}}\), a face \(\Gamma _j\), \(1 \le j \le 4\), and \({\mathfrak {E}}\) a subset of the edges of \(\Gamma _j\), we define the following subspaces of \(W^{s, p}_{{\mathfrak {E}}}(\Gamma _j)\):
where the norm on \(W^{s, p}_{{\mathfrak {E}}, r}(\Gamma _j)\) is given by
where \(\textbf{t}_{\gamma }\) is a unit-tangent vector on the edge \(\gamma \in {\mathfrak {E}}\). For \(r = 0\), we set \(W_{{\mathfrak {E}}, 0}^{s, p}(\Gamma _j):= W^{s, p}(\Gamma _j)\). When \({\mathfrak {E}}\) consists of only one element \(\gamma \), we set \(W_{\gamma , r}^{s, p}(\Gamma _j):= W_{{\mathfrak {E}}, r}^{s, p}(\Gamma _j)\) and 
. One can again verify that \(W_{{\mathfrak {E}}, r}^{s, p}(\Gamma _j)\) are Banach spaces and that the following analogue of (3.10) holds:
The following result shows that the continuity of \({\mathcal {S}}_{k, r}^{[1]}\) can be characterized with these spaces.
Lemma 3.7
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \({\mathfrak {E}} = \{ \gamma _{12}, \gamma _{13} \}\). Then, for all \(f \in W^{s-k-\frac{1}{p}, p}_{{\mathfrak {E}}, r}(\Gamma _1)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\gamma _{12}} = D_{\Gamma }^{l} f|_{\gamma _{13}} = 0\) for \(0 \le l \le r-1\), then \({\mathcal {S}}_{k, r}^{[1]}(f) \in {\mathcal {P}}_{N+k}(K)\).
The proof of Lemma 3.7 appears in Sect. 6.4. We define the analogous operator associated to \(\Gamma _3\) as follows.
where \({\mathfrak {I}}_3(\varvec{x}):= (0, x_2, x_1)\) and \({\mathfrak {R}}_{13}(\varvec{x}, z):= (z, x_2, x_1)\) for all \((\varvec{x}, z) \in K\). Thanks to the chain rule and the smoothness of the mappings \({\mathfrak {I}}_3\) and \({\mathfrak {R}}_{13}\), the continuity and interpolation properties of \({\mathcal {S}}_{k, r}^{[3]}\) follow immediately from Lemma 3.7.
Corollary 3.8
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \({\mathfrak {E}} = \{ \gamma _{13}, \gamma _{23} \}\). Then, for all \(f \in W^{s-k-\frac{1}{p}, p}_{{\mathfrak {E}}, r}(\Gamma _3)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _3)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\gamma _{13}} = D_{\Gamma }^{l} f|_{\gamma _{23}} = 0\) for \(0 \le l \le r-1\), then \({\mathcal {S}}_{k, r}^{[3]}(f) \in {\mathcal {P}}_{N+k}(K)\).
We also have the following analogue of Lemma 3.5.
Lemma 3.9
Let \(k \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), \(1 \le l \le k\), and \(1 \le i, j \le 4\) with \(i \ne j\) be given. Suppose that \(F = (f^0, f^1, \ldots , f^k) \in {{\,\textrm{Tr}\,}}_k^{s,p}(\Gamma _i \cup \Gamma _j)\) satisfies
-
(i)
\(F = (0, 0, \ldots , 0)\) on \(\Gamma _i\);
-
(ii)
\(f_j^m = 0\) on \(\Gamma _j\) for \(0 \le m \le l-1\).
Then, there holds \(f^l_j \in W_{\gamma _{ij}, k+1}^{s - l - \frac{1}{p}, p}(\Gamma _j)\) and
Proof
The result follows from applying inequality (3.20) and identity (3.19). \(\square \)
We now construct the lifting operator \({\mathcal {L}}_{k}^{[3]}\) in the same fashion as \({\mathcal {L}}_k^{[2]}\) (3.21), replacing the use of \({\mathcal {M}}_{m, k}^{[2]}\) with \({\mathcal {S}}_{m, k}^{[3]}\).
Lemma 3.10
Let \(b \in C_c^{\infty }(T)\) with \(\int _{T} b(\varvec{x}) \,\textrm{d}{\varvec{x}} = 1\), \(k \in {\mathbb {N}}_0\), and \({\mathcal {S}} = \{1,2,3\}\). For \(F = (f^0, f^1, \ldots , f^k) \in L^p(\Gamma _{{\mathcal {S}}})^{k+1}\), we formally define the following operators:
Then, for all \((s, p) \in {\mathcal {A}}_k\) and \(F \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\), there holds
Moreover, if F satisfies (2.20) and for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k}^{[3]}(F) \in {\mathcal {P}}_N(K)\).
Proof
Let \(k \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \(f \in {{\,\textrm{Tr}\,}}_k^{s, p}(\Gamma _{{\mathcal {S}}})\) be given. Let \({\mathfrak {E}} = \{ \gamma _{13}, \gamma _{23} \}\).
Step 1: \(m = 0\). Thanks to Lemma 3.6, the traces \(G = (g^0, g^1, \ldots , g^k)\) given by
satisfy the hypotheses of Lemma 3.9 with \((i, j) \in \{ (1, 3), (2,3)\}\) and \(l=1\). Thanks to (3.27) and Lemma 3.9, \(g^0_3 \in W_{{\mathfrak {E}}, k+1}^{s - \frac{1}{p}, p}(\Gamma _3)\). Consequently, \({\mathcal {S}}_{0, k+1}^{[3]}(g_3^0)\) is well-defined by Corollary 3.8, and hence \({\mathcal {L}}_{k, 0}^{[3]}(F)\) is well-defined with
Applying (3.22) and (3.30a) gives
and applying (3.10), (3.22), and (3.15) gives
Moreover, if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_k^{[2]}(F) \in {\mathcal {P}}_N(K)\) by Lemma 3.6, and so the trace G satisfies (2.20) and \(G \in {{\,\textrm{Tr}\,}}^{s, p}_k(\Gamma _{{\mathcal {S}}})\) for all \((s, p) \in {\mathcal {A}}_k\) thanks to Lemma 2.5. By (3.27) and Lemma 3.9, \(g_3^0 \in W_{{\mathfrak {E}}}^{k + 1, p}(\Gamma _3)\) for all \(p \in (1, \infty )\), and so \(D_{\Gamma }^l g^0_3|_{\gamma _{13}} = D_{\Gamma }^l g^0_3|_{\gamma _{23}} = 0\) for \(0 \le l \le k\). Thanks to Corollary 3.8, \({\mathcal {L}}_{k, 0}^{[3]}(F) \in {\mathcal {P}}_{N}(K)\).
Step 2: Induction on m . Assume that for some m such that \(0 \le m \le k-1\), \({\mathcal {L}}^{[3]}_{k, m}(F)\) is well-defined and satisfies
and
Additionally assume that if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k, m}^{[3]}(F) \in {\mathcal {P}}_N(K)\).
Thanks to (3.35), the traces \(G = (g^0, g^1, \ldots , g^k)\) given by
satisfy the hypotheses of Lemma 3.9 with \((i, j) \in \{ (1, 3), (2,3)\}\) and \(l=m+1\). Thanks to (3.27) and Lemma 3.9, there holds \(g^{m+1}_3 \in W_{{\mathfrak {E}},k+1}^{s-m-1-\frac{1}{p}, p}(\Gamma _3)\). Consequently, \({\mathcal {S}}_{m+1, k+1}^{[3]}(g_3^{m+1})\) is well-defined by Corollary 3.8, and hence \({\mathcal {L}}_{k, m+1}^{[3]}(F)\) is well-defined with
Applying (3.35) and (3.30a) gives (3.35) for \(m+1\), while applying (3.27), (3.36), and (3.32) gives (3.36) for \(m+1\).
Moreover, if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k, m}^{[3]}(F) \in {\mathcal {P}}_N(K)\) by assumption and so the trace G satisfies (2.20) and \(G \in {{\,\textrm{Tr}\,}}^{s, p}_k(\Gamma _{{\mathcal {S}}})\) for all \((s, p) \in {\mathcal {A}}_k\) thanks to Lemma 2.5. By (3.10) and Lemma 3.5, \(g_3^{m+1} \in W_{{\mathfrak {E}}}^{k + 1, p}(\Gamma _3)\) for all \(p \in (1,\infty )\), and so \(D_{\Gamma }^l g^{m+1}_3|_{\gamma _{13}} = D_{\Gamma }^l g_3^{m+1}|_{\gamma _{23}} = 0\) for \(0 \le l \le k\). Thanks to Corollary 3.8, \({\mathcal {L}}_{k, m+1}^{[3]}(F) \in {\mathcal {P}}_{N}(K)\). \(\square \)
3.4 Lifting from Four Faces
To complete the construction of the lifting operator from the entire boundary, we define one final single face lifting operator from \(\Gamma _1\) that vanishes on the remaining faces. Given nonnegative integers \(k, r \in {\mathbb {N}}_0\), a smooth compactly supported function \(b \in C_c^{\infty }(T)\), and a function \(f: T\rightarrow {\mathbb {R}}\), we define the operator \({\mathcal {R}}_{k, r}^{[1]}\) formally by the rule
Note that when \(r = 0\), we have \({\mathcal {R}}_{k, r}^{[1]} = {\mathcal {E}}_k^{[1]}\). For functions \(f: \Gamma _1 \rightarrow {\mathbb {R}}\), we again abuse notation and set \({\mathcal {R}}_{k, r}^{[1]}(f):= {\mathcal {R}}_{k, r}^{[1]}(f \circ {\mathfrak {I}}_1)\). The weighted spaces \(W_{{\mathfrak {E}}, r}^{s, p}(\Gamma _1)\) again play a role in the continuity of \({\mathcal {R}}_{k, r}^{[1]}\) as the following result shows.
Lemma 3.11
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \({\mathfrak {E}} = \{ \gamma _{12}, \gamma _{13}, \gamma _{14} \}\). Then, for all \(f \in W_{{\mathfrak {E}}, r}^{s-k - \frac{1}{p}, p}(\Gamma _1)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\partial \Gamma _1} = 0\) for \(0 \le l \le r-1\), then \({\mathcal {R}}_{k, r}^{[1]}(f) \in {\mathcal {P}}_{N+k}(K)\).
The proof of Lemma 3.11 appears in Sect. 6.5. The analogous operator associated to \(\Gamma _4\) is given by
where \({\mathfrak {I}}_4(\varvec{x}):= (x_1, x_2, 1 - x_1 - x_2)\) and \({\mathfrak {R}}_{14}(\varvec{x}, z):= (x_1, x_2, 1-x_1-x_2-z)\) for all \((\varvec{x}, z) \in K\). Thanks to the chain rule and the smoothness of the mappings \({\mathfrak {I}}_4\) and \({\mathfrak {R}}_{14}\), the continuity and interpolation properties of \({\mathcal {R}}_{k, r}^{[4]}\) follow immediately from Lemma 3.11.
Corollary 3.12
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \({\mathfrak {E}} = \{ \gamma _{14}, \gamma _{24}, \gamma _{34} \}\). Then, for all \(f \in W_{{\mathfrak {E}}, r}^{s-k- \frac{1}{p}, p}(\Gamma _4)\), there holds
and
Moreover, if \(f \in {\mathcal {P}}_N(\Gamma _4)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\partial \Gamma _4} = 0\) for \(0 \le l \le r-1\), then \({\mathcal {R}}_{k, r}^{[4]}(f) \in {\mathcal {P}}_{N+k}(K)\).
Finally, we construct the lifting operator \({\mathcal {L}}_{k}^{[4]}\) in the same fashion as \({\mathcal {L}}_k^{[3]}\) (3.33), replacing the use of \({\mathcal {S}}_{m, k+1}^{[3]}\) with \({\mathcal {R}}_{m, k+1}^{[4]}\).
Lemma 3.13
Let \(b \in C_c^{\infty }(T)\) with \(\int _{T} b(\varvec{x}) \,\textrm{d}{\varvec{x}} = 1\) and \(k \in {\mathbb {N}}_0\). For \(F = (f^0, f^1, \ldots , f^k) \in L^p(\partial K)^{k+1}\), we formally define the following operators:
Then, for all \((s, p) \in {\mathcal {A}}_k\) and \(F \in {{\,\textrm{Tr}\,}}_k^{s, p}(\partial K)\), there holds
Moreover, if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_k^{[4]}(F) \in {\mathcal {P}}_N(K)\).
Proof
Let \(k \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \(f \in {{\,\textrm{Tr}\,}}_k^{s, p}(\partial K)\) be given and set \({\mathfrak {E}}:= \{ \gamma _{14}, \gamma _{24}, \gamma _{34} \}\).
Step 1: \(m = 0\). Thanks to Lemma 3.10, the traces \(G = (g^0, g^1, \ldots , g^k)\) given by
satisfies the hypotheses of Lemma 3.9 with \((i, j) \in \{ (1, 4), (2,4), (3,4)\}\) and \(l=1\). Thanks to (3.27) and Lemma 3.9, \(g^0_4 \in W_{{\mathfrak {E}},k+1}^{ s-\frac{1}{p}, p}(\Gamma _4)\). Consequently, \({\mathcal {R}}_{0, k+1}^{[4]}(g_4^0)\) is well-defined by Corollary 3.12, and hence \({\mathcal {L}}_{k, 0}^{[4]}(F)\) is well-defined with
Applying (3.34) and (3.40a) gives
and applying (3.27), (3.34), and (3.32) gives
Moreover, if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_k^{[3]}(F) \in {\mathcal {P}}_N(K)\) by Lemma 3.6, and so the trace G satisfies (2.20) and \(G \in {{\,\textrm{Tr}\,}}^{s, p}_k(\partial K)\) for all \((s, p) \in {\mathcal {A}}_k\) thanks to Lemma 2.5. By (3.27) and Lemma 3.9, \(g_4^0 \in W_{{\mathfrak {E}}}^{k + 1, p}(\Gamma _4)\) for all \(p \in (1,\infty )\), and so \(D_{\Gamma }^l g^0_4|_{\partial \Gamma _4} = 0\) for \(0 \le l \le k\). Thanks to Corollary 3.12, \({\mathcal {L}}_{k, 0}^{[4]}(F) \in {\mathcal {P}}_{N}(K)\).
Step 2: Induction on m . Assume that for some \(0 \le m \le k-1\), \({\mathcal {L}}^{[4]}_{k, m}(F)\) is well-defined and satisfies
and
Additionally assume that if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k, m}^{[4]}(F) \in {\mathcal {P}}_N(K)\).
Thanks to (3.44), the traces \(G = (g^0, g^1, \ldots , g^k)\) given by
satisfies the hypotheses of Lemma 3.9 with \((i, j) \in \{ (1, 4), (2,4), (3,4) \}\) and \(l=m+1\). Thanks to (3.27) and Lemma 3.9, \(g^{m+1}_4 \in W_{{\mathfrak {E}},k+1}^{s-m-1-\frac{1}{p}, p}(\Gamma _4)\). Consequently, \({\mathcal {R}}_{m+1, k+1}^{[4]}(g_4^{m+1})\) is well-defined by Corollary 3.12, and hence \({\mathcal {L}}_{k, m+1}^{[4]}(F)\) is well-defined with
Applying (3.44) and (3.40a) gives (3.44) for \(m+1\), while applying (3.27), (3.45), and (3.32) gives (3.45) for \(m+1\).
Moreover, if F satisfies (2.20) for some \(N \in {\mathbb {N}}_0\), then \({\mathcal {L}}_{k, m}^{[4]}(F) \in {\mathcal {P}}_N(K)\) by assumption and so the trace G satisfies (2.20) and \(G \in {{\,\textrm{Tr}\,}}^{s, p}_k(\partial K)\) for all \((s, p) \in {\mathcal {A}}_k\) thanks to Lemma 2.5. By (3.27) and Lemma 3.9, \(g_4^{m+1} \in W_{{\mathfrak {E}}}^{k + 1, p}(\Gamma _4)\) for all \(p \in (1,\infty )\), and so \(D_{\Gamma }^l g^{m+1}_4|_{ \partial \Gamma _4 } = 0\) for \(0 \le l \le k\). Thanks to Corollary 3.12, \({\mathcal {L}}_{k, m+1}^{[4]}(F) \in {\mathcal {P}}_{N}(K)\). \(\square \)
3.5 Proof of Theorem 2.6
Let \(b \in C_c^{\infty }(T)\) be any smooth function satisfying \(\int _{T} b(\varvec{x}) \,\textrm{d}{\varvec{x}} = 1\). Then, \({\mathcal {L}}_k:= {\mathcal {L}}_k^{[4]}\), where \({\mathcal {L}}_{k}^{[4]}\) is defined in (3.42) satisfies the desired properties thanks to Lemma 3.13. \(\square \)
4 Whole Space Operators
In this section, we examine the continuity properties of the following operators, which are the whole space extensions of the lifting operators \({\mathcal {E}}_k^{[1]}\) (3.1): Given \(k \in {\mathbb {N}}_0\), \(\chi \in C^{\infty }_c({\mathbb {R}})\), and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) and a function \(f: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}\), we define the lifting operator \(\tilde{{\mathcal {E}}}_k\) by the rule
We use the notation \(\tilde{{\mathcal {E}}}_k[\chi , b]\) when we want to make the dependence on \(\chi \) and b explicit. The advantage of working with the operator \(\tilde{{\mathcal {E}}}_k\) is that we shall capitalize on the abundance of equivalent \(W^{s, p}(U)\)-norms when U is all of \({\mathbb {R}}^d\) or the half-space \({\mathbb {R}}^{d}_+ = {\mathbb {R}}^{d-1} \times (0, \infty )\), \(d > 1\). In particular, we recall the following norm-equivalence on \(W^{s, p}(U)\), \(0< s < 1\), \(1< p < \infty \), with \(U = {\mathbb {R}}^d\) or \({\mathbb {R}}^d_+\) (see e.g. [44, Theorems 6.38 & 6.61]):
The main result of this section is the following analogue of Lemma 3.1.
Theorem 4.1
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) with \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\), \(b \in C^{\infty }_c({\mathbb {R}}^2)\) with \({{\,\textrm{supp}\,}}b \subset T\), and \(k \in {\mathbb {N}}_0\) be given. Then, for \((s, p) \in {\mathcal {A}}_k \cup (k+\frac{1}{2}, 2)\), there holds
The proof of Theorem 4.1 appears in Sect. 4.3.
4.1 Continuity of \(\tilde{{\mathcal {E}}}_0\)
We begin by recording the particular case of Theorem 4.1 with \(k=0\), which follows from the same arguments as in the proof of [44, Theorem 9.21].
Lemma 4.2
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) with \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) with \({{\,\textrm{supp}\,}}b \subset T\). Then, for \(1< p < \infty \) and \(1/p< s < 1\), there holds
When \(p=2\), the above result is also true for \(s = 1/2\) as the following lemma shows.
Lemma 4.3
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) with \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) with \({{\,\textrm{supp}\,}}b \subset T\). Then, there holds
Proof
By density, it suffices to consider \(f \in C^{\infty }_{c}({\mathbb {R}}^2)\). For \(k \in {\mathbb {N}}_0\) define
Step 1: \(H^{1/2}({\mathbb {R}}^3_+)\) bound for \(g_0\). Thanks to (4.2), there holds
We now follow the steps in the proof of [18, Theorem 2.2]. Let \({\hat{\cdot }}\) denote the Fourier transform with respect to the \(\varvec{x}\)-variable. Then,
where we used the following convolution identity for \(z > 0\):
Similarly, there holds
Thanks to a change of variables, we obtain
which is finite since \({\hat{b}}\) is a Schwartz function, and so \(| g_0 |_{\frac{1}{2}, 2, {\mathbb {R}}_+^3} \lesssim _{b} \Vert f\Vert _{2, {\mathbb {R}}^2}.\)
Step 2: \(H^{1/2}({\mathbb {R}}^3_+)\) bound on \(\tilde{{\mathcal {E}}}_0(f)\). For \(i=1,2\), there holds
where we used (4.2) and step 1. Thanks to the relation
we obtain
Now, applying Hardy’s inequality [40, Theorem 327] gives
and so
Applying Young’s inequality to the convolution form of \(g_0\), (4.7) then gives
Inequality (4.5) now follows on collecting results and applying (4.2). \(\square \)
We shall also need the stability of the lifting of the derivative of a smooth function.
Lemma 4.4
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\). For \(1< p < \infty \), there holds
Proof
Let \(1< p < \infty \), \(f \in C_c^{\infty }({\mathbb {R}}^2)\), and \(i \in \{1,2\}\). Integrating by parts gives
Since \(b \in C^{\infty }_c({\mathbb {R}}^2)\), there holds \(\int _{{\mathbb {R}}} (\partial _i b)(\varvec{y}) \,\textrm{d}{y_i} = 0\), and so
Applying Hölder’s inequality, we obtain
Integrating over \({\mathbb {R}}^3_+\) then gives
Inequality (4.8) now follows from summing over i and applying (4.2). \(\square \)
4.2 Continuity of \(\tilde{{\mathcal {E}}}_k\)
We now show how the continuity of the operator \(\tilde{{\mathcal {E}}}_{0}\) can be used to deduce the continuity of \(\tilde{{\mathcal {E}}}_k\) for \(k \in {\mathbb {N}}_0\). We begin with a partial result.
Lemma 4.5
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) be as in Theorem 4.1 and \(k \in {\mathbb {N}}_0\) be given. Then, for \(1< p < \infty \), there holds
and for \(1/p< s < 1\) or \((s, p) = (\frac{1}{2}, 2)\), there holds
Proof
Let \(k \in {\mathbb {N}}_0\), \(1< p < \infty \), and \(f \in C^{\infty }_c({\mathbb {R}}^2)\). Since the function \({\tilde{\chi }}:= z^k \chi \in C^{\infty }_c({\mathbb {R}})\) with \({{\,\textrm{supp}\,}}{\tilde{\chi }} = {{\,\textrm{supp}\,}}\chi \), we have \(\tilde{{\mathcal {E}}}_{k}[\chi , b](f) = \tilde{{\mathcal {E}}}_0[{\tilde{\chi }}, b](f)\). Consequently, it suffices to prove (4.9) and (4.10) in the case \(k=0\). To this end, we apply Jensen’s inequality to the identity (4.7) to obtain
and (4.9) follows. Inequality (4.10) for \(1/p< s < 1\) is an immediate consequence of (4.4), while the case \((s, p) = (\frac{1}{2}, 2)\) follows from Lemma 4.3. \(\square \)
For more precise results, we shall show the effect of taking partial derivatives of \(\tilde{{\mathcal {E}}}_k(f)\) on the index k and on the function f. To this end, we recall an integration-by-parts formula for tensors. Given two d-dimensional tensors S and T, let S : T denote the usual tensor contraction
where we are using Einstein summation notation. Given a d-dimensional tensor S with \(d \ge 0\) and \(k \ge 0\), let \(D^k S\) denote the k-th derivative tensor of S:
and let \({{\,\textrm{div}\,}}S\) denote the \((d-1)\)-dimensional tensor given by
while \({{\,\textrm{div}\,}}^k S\), \(0 \le k \le d\), denotes k applications of \({{\,\textrm{div}\,}}\) to S. With this notation, we have the following integration by parts formula for symmetric, smooth, compactly supported tensors S and T of dimension d and \(0 \le k \le d\), respectively:
With this notation in hand, we have the following identity that shows that the derivatives of \(\tilde{{\mathcal {E}}}_k(f)\) are linear combinations of liftings of derivatives of f.
Lemma 4.6
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\), \(b \in C^{\infty }_c({\mathbb {R}}^2)\), and \(k \in {\mathbb {N}}_0\) be given. For all \(\alpha \in {\mathbb {N}}_0^{3}\) and \(f \in C^{\infty }_c({\mathbb {R}}^2)\), there holds
for suitable \(\chi _i \in C_c^{\infty }({\mathbb {R}})\) and \(\max \{ |\alpha |-i-k, 0\}\)-dimensional tensors \(B_{ki\alpha }\) with entries in \(C^{\infty }_c({\mathbb {R}}^2)\).
Proof
Let \(f \in C^{\infty }_c({\mathbb {R}}^2)\) and let \(g_k\) be defined as in (4.6). For integers \(m \ge k\), there holds
where \(c_{kmj}\) are suitable constants, \(\varvec{y}^{\otimes n}\) is the tensor product of n copies of \(\varvec{y}\), and \(D_{\varvec{y}}\) denotes the derivative operator with respect to \(\varvec{y}\). For \(0 \le m < k\), there holds
Consequently, there holds
Now let \(\beta \in {\mathbb {N}}_0^2\) with \(|\beta | \ge k\). Then, there holds
where \(D_{\varvec{x}}^{\beta }:= \partial _{x_1}^{\beta _1} \partial _{x_2}^{\beta _2}\) and \({\tilde{\beta }} \in {\mathbb {N}}_0^2\) is any fixed multi-index such that \(|{\tilde{\beta }}| = k\) and \(\beta - {\tilde{\beta }} \in {\mathbb {N}}_0^2\). Similar arguments show that for \(\beta \in {\mathbb {N}}_0^2\) with \(|\beta | < k\), there holds
Collecting results, for any \(\alpha \in {\mathbb {N}}_0^3\), there holds
for suitable \(\max \{ |\alpha |-k, 0\}\)-dimensional tensors \(B_{k\alpha }\) with entries in \(C^{\infty }_c({\mathbb {R}}^2)\). Equality (4.11) now follows from the product rule. \(\square \)
4.3 Proof of Theorem 4.1
Let \(k \in {\mathbb {N}}_0\), \(1< p < \infty \), and \(f \in C^{\infty }_c({\mathbb {R}}^2)\). For \(\alpha \in {\mathbb {N}}_0^3\), (4.11) gives
where \(\chi _i \in C^{\infty }_c({\mathbb {R}})\) and \(b_{ki\beta } \in C^{\infty }_c({\mathbb {R}}^2)\) are suitable functions depending on \(\chi \) and b respectively and \(0 \le \sigma < 1\).
Step 1: \(L^p\) bounds on derivatives. For \(k + i - |\alpha | \ge 0\) (so that \(|\beta | = 0\)), (4.9) gives
For \(k + i - |\alpha | < 0\) (so that \(|\alpha | \ge k\) and \(|\beta | \ge 1\)), there exists \(j \in \{1, 2\}\) such that \(\beta _j \ge 1\), and so we apply (4.8) to obtain
Consequently, for all \(f \in C^{\infty }_c({\mathbb {R}}^2)\), there holds
By density, (4.13) holds for all \(f \in W^{m-k-1/p, p}({\mathbb {R}}^2)\).
Step 2: The case \(s \ge k+1\). Inequality (4.3) for real \(s \ge k+1\) with \((s, p) \in {\mathcal {A}}_k\) follows from (4.13) using a standard interpolation argument.
Step 3: The case \(k+1/p \le s < k+1\). For \(s = k + \sigma \), where \(1/p< \sigma < 1\) or \((\sigma , p) = (1/2, 2)\), we take \(|\alpha | = k\) in (4.12) and apply (4.10) to obtain
which completes the proof. \(\square \)
5 Weighted \(L^p\) Continuity of Whole-Space Operators
In the previous section in Theorem 4.1, we established that the lifting operators \(\tilde{{\mathcal {E}}}_k\) are continuous from \(W^{s-k-1/p, p}({\mathbb {R}}^2)\) to \(W^{s, p}({\mathbb {R}}^3_+)\) provided that \(s > k + 1/p\). We now turn to the stability of the operator \(\tilde{{\mathcal {E}}}_k\) with respect to lower-order Sobolev spaces. In particular, we seek to obtain bounds on \(\Vert \tilde{{\mathcal {E}}}_k(f)\Vert _{s, p, {\mathcal {O}}_1}\) for \(0 \le s < k+1/p\), where \({\mathcal {O}}_1:= (0, \infty )^3 \supset K\) is the first octant. It turns out that one suitable space for the lifted function f is a weighted \(L^p\) space. Let \({\mathcal {Q}}_1 = (0, \infty )^2 \supset T\) denote the first quadrant and let \(\rho \in L^{\infty }({\mathcal {Q}}_1)\) be a weight function that satisfying \(\rho > 0\) almost everywhere. Then, for \(1< p < \infty \), define
The weight that will appear in our estimates are powers of \(\omega _1\) (3.7) extended to all of \({\mathbb {R}}^2\) by
In particular, the main result of this section is as follows.
Theorem 5.1
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) be as in Theorem 4.1 and \(k \in {\mathbb {N}}_0\) be given. For \(1< p < \infty \) and \(0 \le s < k + 1/p\), there holds
The proof proceeds in several steps and appears in Sect. 5.3.
5.1 Auxiliary Results
We begin by recording a number of technical lemmas. Throughout the rest of the section we use the notation 
.
Lemma 5.2
For \(1 \le p < \infty \) and \(0< h < \infty \), there holds
Proof
The result follows on applying Hölder’s inequality and changing the order of integration:
\(\square \)
Lemma 5.3
Let \(1< p < \infty \), \(0< s < 1\), and \(0 \le a \le \infty \). Then, there holds
Proof
The proof follows the same arguments as those used in the proof of [44, Theorem 1.28], which considers the case \(a = \infty \). The full details are given below.
By symmetry, there holds
Performing a change of variable and applying Hardy’s inequality [44, Theorem 1.3], we obtain
Thus,
which completes the proof. \(\square \)
Lemma 5.4
Let \(1< p < \infty \) and \(0< s < 1/p\). For all \(f \in L^p({\mathcal {Q}}_1; \omega _1^{1-sp} \,\textrm{d}{\varvec{x}})\), there holds
Proof
Applying (5.4) and using that \(0< z < 2\) gives
Moreover, there holds
The result now follows on integrating over \(0< x_2 < \infty \). \(\square \)
Lemma 5.5
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) be as in Theorem 4.1 and \(1< p < \infty \). Let \(k \in {\mathbb {N}}_0\) and \(f \in L^p({\mathcal {Q}}_1; \omega _1^{1+kp} \,\textrm{d}{\varvec{x}})\). For \(0< t < 2\), there holds
where \(g_k\) is defined in (4.6).
Proof
Let \(k \in {\mathbb {N}}_0\), \(1< p < \infty \), and \(f \in L^p({\mathcal {Q}}_1; \omega _1^{1+kp} \,\textrm{d}{\varvec{x}})\) be given. Let \(z \in (0, t)\). Then, for \(\varvec{x} \in {\mathcal {Q}}_1\) and \(\varvec{y} \in (0, 1)^2\), there holds \(z \le \min \{ x_1 + z y_1, t \}/y_1\), and so
Integrating over \(x_2 \in (0, \infty )\) and applying (5.4) to the function
and using that \(0< z< t < 2\) gives
Hardy’s inequality [40, Theorem 327] then shows that, for \(0< x_2 < \infty \), there holds
and so
Integrating over \(x_1\) and changing the order of integration gives
which completes the proof. \(\square \)
5.2 Continuity of \(\tilde{{\mathcal {E}}}_0\)
In this section, we prove Theorem 5.1 in the case \(k=0\). We will utilize the following equivalent norm on \(W^{s, p}({\mathcal {O}}_1)\).
Lemma 5.6
For all \(p \in (1,\infty )\), \(s \in (0, 1)\), and \(f \in W^{s, p}({\mathcal {O}}_1)\), there holds
Proof
Let \(f \in W^{s, p}({\mathcal {O}}_1)\). Thanks to [44, Theorem 6.38], there holds
and (5.8) now follows on noting that
\(\square \)
We now estimate each term in (5.8). The first result deals with terms involving translations in the first two coordinate directions.
Lemma 5.7
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) be as in Theorem 4.1. For \(1< p < \infty \), \(0< s < 1/p\), and \(1 \le i \le 2\), there holds
for all \(f \in C^{\infty }_c({\mathcal {Q}}_1)\).
Proof
Let \(1< p < \infty \), \(0< s < 1/p\), and \(f \in C^{\infty }_c({\mathcal {Q}}_1)\) be given. Let \(g_0(\cdot ,\cdot )\) be as in (4.6) with \(k=0\) and let \({\tilde{g}}(\varvec{x}, z, t):= g_0(\varvec{x} + t\textbf{e}_i, z) - g_0(\varvec{x}, z)\).
Step 1. Let \(1 \le i \le 2\). We will show that
We begin by decomposing the above integral into two terms:
Part (a): \(A_i\). Let \(0< t < 1\). Then, \(f(\cdot + t\textbf{e}_i) - f(\cdot ) \in L^p({\mathcal {Q}}_1, \omega _1\,\textrm{d}{\varvec{x}})\) and
Integrating (5.7) over \(0< t < 1\) then gives
For \(i = 1\), there holds
On the other hand, note that for any \(0 \le u < \infty \) and \({\tilde{f}}(\varvec{x}) = f(\varvec{x} + u\textbf{e}_2)\), there holds
The bound \(A_i \lesssim _{b, s, p} \Vert \omega _1^{\frac{1}{p} - s} f \Vert _{p, {\mathcal {Q}}_1}^p\) now follows on performing a change of variables and collecting results.
Part (b): \(B_i\). Using identity (4.7), we obtain
Writing \(z^{-2} |\partial _i b| = (z^{-2} |\partial _i b|)^{1-1/p} (z^{-2} |\partial _i b|)^{1/p}\) and applying Hölder’s inequality gives
Integrating over \(\varvec{x}\) gives
Integrating over z and t, we obtain
Applying (5.6), we obtain \(B_i \lesssim _{b, s, p} \Vert \omega _1^{\frac{1}{p}-s} f \Vert _{p, {\mathcal {Q}}_1}^p\), which completes the proof of (5.10).
Step 2. Since \({{\,\textrm{supp}\,}}\chi \subset B(0, 2)\), there holds
which completes the proof. \(\square \)
The next result deals with the term involving a translation in the z-direction.
Lemma 5.8
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) be as in Theorem 4.1. For \(1< p < \infty \) and \(0< s < 1/p\), there holds
for all \(f \in C^{\infty }_c({\mathcal {Q}}_1)\).
Proof
Let \(1< p < \infty \), \(0< s < 1/p\), and \(f \in C^{\infty }_c({\mathcal {Q}}_1)\) be given. Let \(g_0(\cdot , \cdot )\) be defined as in (4.6).
Step 1. We will first show that
Applying (5.5) gives
Applying identity (4.7), we obtain
For \(\varvec{y} \in (x_1, x_1+z) \times (x_2, x_2 + z)\), there holds
where we used that b and Db are uniformly bounded. Since \({{\,\textrm{supp}\,}}b \subset (0, 1)^2\), we obtain
Inequality (5.12) now follows on integrating (5.13) over \(\varvec{x} \in {\mathcal {Q}}_1\) and applying (5.6).
Step 2. For \(0< t < 2\) and \(\varvec{x} \in {\mathcal {Q}}_1\), we add and subtract \(\chi (z+t)g_0(\varvec{x}, t)\) to obtain
For the first term, we use that \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\) and apply (5.12) to obtain
For the second term, we again use that \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\) as well as the assumption \(0< s < 1/p\):
Inequality (5.11) now follows from (5.7). \(\square \)
We now obtain Theorem 5.1 in the case \(k=0\).
Lemma 5.9
Let \(\chi \in C^{\infty }_c({\mathbb {R}})\) and \(b \in C^{\infty }_c({\mathbb {R}}^2)\) be as in Theorem 4.1. For \(1< p < \infty \) and \(0 \le s < 1/p\), there holds
Proof
The case \(s = 0\) follows on taking \(t=2\) in (5.7) and using the fact that \(\Vert \tilde{{\mathcal {E}}}_0(f) \Vert _{p, {\mathcal {O}}_1} \lesssim _{\chi , p} \Vert g \Vert _{p, {\mathcal {Q}}_1 \times (0, 2)}\), where g is defined in (4.6). The case \(0< s < 1/p\) follows from the norm equivalence (5.8), the bounds (5.9) and (5.11), and the density of \(C^{\infty }_c({\mathcal {O}}_1)\) in \(L^p({\mathcal {O}}_1, \omega _1^{1-sp} \,\textrm{d}{\varvec{x}})\). \(\square \)
5.3 Proof of Theorem 5.1
Let \(k \in {\mathbb {N}}_0\), \(1< p < \infty \).
Step 1: \(s=0\). Taking \(t=2\) in (5.7) and using the fact that \(\Vert \tilde{{\mathcal {E}}}_k(f) \Vert _{p, {\mathcal {O}}_1} \lesssim _{\chi , p} \Vert g_k \Vert _{p, {\mathcal {Q}}_1 \times (0, 2)}\), where \(g_k\) is defined in (4.6), we obtain (5.3) in the case \(s = 0\).
Step 2: \(s \in \{1, 2, \ldots , k\}\). Let \(f \in C^{\infty }_c({\mathcal {Q}}_1)\). Applying (4.11) with \(|\alpha | \le k\), we obtain
where \(\chi _i \in C^{\infty }_c({\mathbb {R}})\) and \(b_{ki} \in C^{\infty }_c({\mathbb {R}}^2)\) are suitable functions depending on \(\chi \) and b respectively and \(0 \le \sigma < 1\). Applying (5.3) with \(s=0\) then gives
where we used that \(k + i - |\alpha | \ge k-m\) for \(0 \le i \le \alpha _3\). By density, (5.3) holds for \(s \in \{0, 1, \ldots , k\}\).
Step 3: \(0 \le s \le k\). This case follows from interpolating Step 2 (see e.g. [23, Theorem 14.2.3] and [17, Theorem 5.4.1]).
Step 4: \(k< s < k+1/p\). Let \(\sigma = s - k\) so that \(0< \sigma < 1/p\). Setting \({\tilde{\chi }}_{ik\alpha }(z):= z^{k+i-|\alpha |} \chi _i \in C^{\infty _c}({\mathcal {Q}}_1)\) so that \({{\,\textrm{supp}\,}}{\tilde{\chi }}_{ik\alpha } \in (-2, 2)\) and applying (5.15) and (5.14) then gives
Inequality (5.3) now follows. \(\square \)
6 Continuity of Fundamental Operators
In this section, we prove the continuity and interpolation properties of the four fundamental operators \({\mathcal {E}}_k^{[1]}\) defined in (3.1), \({\mathcal {M}}_{k, r}^{[1]}\) defined in (3.8), \({\mathcal {S}}_{k, r}^{[1]}\) defined in (3.25), and \({\mathcal {R}}_{k, r}^{[1]}\) defined in (3.37). We begin with the properties of \({\mathcal {E}}_k^{[1]}\), which rely on the results of Sect. 4. Then, in Sect. 6.2, we show that the four fundamental operators are continuous from weighted \(L^p\) spaces (5.1) to \(W^{s, p}(K)\) for small s, which will be useful for the analysis of \({\mathcal {M}}_{k, r}^{[1]}\), \({\mathcal {S}}_{k, r}^{[1]}\), and \({\mathcal {R}}_{k, r}^{[1]}\). This section concludes with the proofs of Lemmas 3.3, 3.7 and 3.11.
6.1 Proof of Lemma 3.1
Step 1: Continuity (3.4). Let \({\tilde{b}}\) denote the extension by zero of b to \({\mathbb {R}}^2\) and let \(\chi \in C^{\infty }_c({\mathbb {R}})\) with \(\chi \equiv 1\) on \((-1, 1)\) and \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\). Let \(f \in W^{s-k-\frac{1}{p}, p}(T)\) be given and let \({\tilde{f}}\) denote a bounded extension f to \({\mathbb {R}}^2\) satisfying \(\Vert {\tilde{f}}\Vert _{s, p, {\mathbb {R}}^2} \lesssim _{s, p} \Vert f\Vert _{s, p, T}\); see e.g. [34]. Thanks to the identity
where \({\tilde{E}}_k\) is defined in (4.1), inequality (3.4) immediately follows from (4.3) and the smoothness of the mapping \({\mathfrak {I}}_1\) defined in (3.2).
Step 2: Trace property (3.3). Direct computation shows that (3.3) holds.
Step 3: Polynomial preservation. If \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), then direct inspection reveals that \({\mathcal {E}}_k^{[1]}(f) \in {\mathcal {P}}_{N+k}(K)\). \(\square \)
6.2 Weighted Continuity
We begin with the continuity of \({\mathcal {E}}_k^{[1]}\).
Lemma 6.1
Let \(b \in C_c^{\infty }(T)\), \(k \in {\mathbb {N}}_0\), \(1< p < \infty \), and \(0 \le s < k+1/p\) or \((s, p) = (k+\frac{1}{2}, 2)\). Then, for all \(t_1, t_2, t_3 \in [0, \infty )\) such that \(t_1 + t_2 + t_3 = k - s + 1/p\), there holds
where \(\omega _i\) are defined in (3.7).
Proof
Let \(t = k-s+1/p\).
Step 1: \(t_2 = t_3 = 0\). Let \({\tilde{b}}\) denote the extension by zero of b to \({\mathbb {R}}^2\) and let \(\chi \in C^{\infty }_c({\mathbb {R}})\) with \(\chi \equiv 1\) on \((-1, 1)\) and \({{\,\textrm{supp}\,}}\chi \in (-2, 2)\). Let \(f \in C^{\infty }_c(T)\) be given and let \({\tilde{f}}\) denote the extension by zero of f to \({\mathbb {R}}^2\). Thanks to the identity (6.1), (6.2) with \(t_2 = t_3 = 0\) follows from (5.3), where we recall that \(\omega _1\) is extended to \({\mathbb {R}}^2\) by (5.2), and a standard density argument. The case \((s, p) = (k+\frac{1}{2}, 2)\) follows from a similar argument using (4.3).
Step 2: \(t_1 = t_3 = 0\). We define transformations \({\mathfrak {F}}_1: T\rightarrow T\) and \({\mathfrak {G}}_1: K\rightarrow K\) as follows:
Then, a change of variable shows that \({\mathcal {E}}_k^{[1]}(f) \circ {\mathfrak {G}}_1 = {\mathcal {E}}_k^{[1]}[b \circ {\mathfrak {F}}_1](f \circ {\mathfrak {F}}_1)\), and so
where we applied Step 1 in the middle inequality.
Step 3: \(t_3 = 0\). Applying Steps 1 and 2 and interpolating between \(L^p(T; \omega _1^{tp} d\varvec{x})\) and \(L^p(T; \omega _2^{tp} d\varvec{x})\) (see e.g. [17, Theorem 5.4.1]) then gives (6.2).
Step 4: \(t_1 = t_2 = 0\). We define transformations \({\mathfrak {F}}_2: T\rightarrow T\) and \({\mathfrak {G}}_2: K\rightarrow K\) as follows:
A change of variables then gives \({\mathcal {E}}_k^{[1]}(f) \circ {\mathfrak {G}}_2 = {\mathcal {E}}_k^{[1]}[b \circ {\mathfrak {F}}_2](f \circ {\mathfrak {F}}_2)\), and so
where we applied Step 1 in the middle inequality.
Step 5: General case. Applying Steps 3 and 4 and interpolating between \(L^p(T; (\omega _1^{r_1} \omega _2^{r_2})^p d\varvec{x})\) with \(r_1, r_2 \in {\mathbb {R}}_+\) with \(r_1 + r_2 = t\) and \(L^p(T; \omega _3^{tp}d\varvec{x})\) (see e.g. [17, Theorem 5.4.1]) gives (6.2). \(\square \)
We now turn to the continuity of \({\mathcal {M}}_{k, r}^{[1]}\).
Lemma 6.2
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \(1< p < \infty \), and \(0 \le s < k+1/p\) or \((s, p) = (k+\frac{1}{2}, 2)\). Then, for all \(t_1, t_2, t_3 \in [0, \infty )\) such that \(t_1 + t_2 + t_3 = k - s + 1/p\), there holds
where \(\omega _i\) are defined in (3.7).
Proof
Let \(0 \le s < k + 1/p\) or \((s, p) = (k+\frac{1}{2}, 2)\). We proceed by induction on r. The case \(r=0\) follows from (6.2), so assume that (6.5) holds for some \(r \in {\mathbb {N}}_0\). Direction computation gives
which leads to the following identity
Consequently, there holds
Applying (6.5) with \(\tau _1 = t_1\), \(\tau _2 = t_2 + 1\) and \(\tau _3 = t_3\) gives
and so \(\Vert {\mathcal {M}}_{k, r+1}^{[1]}(f) \Vert _{s, p, K} \lesssim _{b, k, r, s, p} \Vert \omega _1^{t_1} \omega _2^{t_2} \omega _3^{t_3} f \Vert _{p, T}\), which completes the proof. \(\square \)
It will be convenient to define a three-parameter version of \({\mathcal {S}}_{k, r}^{[1]}\) as follows:
for \(k, r, q \in {\mathbb {N}}_0\). This three-parameter version satisfies the same continuity properties as \({\mathcal {E}}_k^{[1]}\) and \({\mathcal {M}}_{k, r}^{[1]}\).
Lemma 6.3
Let \(b \in C_c^{\infty }(T)\), \(k, r, q \in {\mathbb {N}}_0\), \(1< p < \infty \), and \(0 \le s < k+1/p\) or \((s, p) = (k+\frac{1}{2}, 2)\). Then, for all \(t_1, t_2, t_3 \in [0, \infty )\) such that \(t_1 + t_2 + t_3 = k - s + 1/p\), there holds
where \(\omega _i\) are defined in (3.7).
Proof
Let \(0 \le s < k + 1/p\) or \((s, p) = (k+\frac{1}{2}, 2)\). We proceed by induction on q. The case \(q=0\) follows from (6.5), so assume that (6.8) holds for some \(q \in {\mathbb {N}}_0\). Direct computation gives
which leads to the following identity
Consequently, there holds
Applying (6.8) with \(\tau _1 = t_1 + 1\), \(\tau _2 = t_2\) and \(\tau _3 = t_3\) gives
and so \(\Vert {\mathcal {S}}_{k, r, q+1}^{[1]}(f) \Vert _{s, p, K} \lesssim _{b, k, r, q, s, p} \Vert \omega _1^{t_1} \omega _2^{t_2} \omega _3^{t_3} f \Vert _{p, T}\). \(\square \)
6.3 Proof of Lemma 3.3
Step 1: Continuity (3.12). We first show that (3.12) holds with \(\Gamma _1\) replaced by \(T\) and \(\gamma _{12}\) replaced by \(\gamma _{2}\), where we recall that the edges of \(T\) are labeled as in Fig. 1b: For all \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k \cup \{ (k+\frac{1}{2}, 2) \}\), and \(f~\in ~W^{s-k-\frac{1}{p},p}(T) \cap W_{\gamma _{2}}^{\min \{s-k- \frac{1}{p}, r\}, p}(T)\), there holds
We proceed by induction on r. The case \(r=0\) follows from (3.4) and (6.2), so assume that (6.10) holds for some \(r \in {\mathbb {N}}_0\) and all \(k \in {\mathbb {N}}_0\) and \((s, p) \in {\mathcal {A}}_k\). Let \(k \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k \cup \{(k+\frac{1}{2}, 2)\}\), and \(f \in W^{s-k-\frac{1}{p},p}(T) \cap W_{\gamma _{2}}^{\min \{s-k- \frac{1}{p}, r+1\}, p}(T)\) be given. Thanks to (6.6), there holds
Part (a): \(k+1/p \le s \le k+1+1/p\). Thanks to Theorem A.3, there holds \(\omega _2^{-1} f \in L^p(T; \omega _2^{(k-s+1)p+1} \,\textrm{d}{\varvec{x}})\) and (A.9) and (A.7) give
Consequently, we apply (6.5) to obtain
Part (b): \(k+1+1/p < s \le k + r + 1 + 1/p\). Theorem A.3 shows that \(\omega _2^{-1} f \in W_{\gamma _2}^{s-k-1-\frac{1}{p}, p}(T)\) and (6.10) and (A.8) then give
Part (c): \(s > k + r + 1 + 1/p\). Thanks to Theorem A.3, there holds \(\omega _2^{-1} f \in W^{s-k-1-\frac{1}{p}, p}(T) \cap W^{r}_{\gamma _2}(T)\), and so we apply (6.10) and (A.7) to obtain
Inequality (6.10) for \(r + 1\) now follows from the triangle inequality. The smoothness of the mapping \({\mathfrak {I}}_1\) defined in (3.2) then gives (3.12).
Step 2: Trace properties (3.11a) and (3.11b). Direct computation shows that (3.11a) and (3.11b) hold.
Step 3: Polynomial preservation. Suppose that \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\gamma _{12}} = 0\) for \(0 \le l \le r-1\). Then, \(f \circ {\mathfrak {I}}_1 = \omega _2^r g\) for some \(g \in {\mathcal {P}}_{N-r}(T)\), and so \({\mathcal {M}}_{k, r}^{[1]}(f) = x_2^r {\mathcal {E}}_{k}^{[1]}(g) \in {\mathcal {P}}_{N+k}(K)\) thanks to Lemma 3.1. \(\square \)
6.4 Proof of Lemma 3.7
Step 1: Continuity (3.29). We first show that the following analogue of (3.29) holds: Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k \cup \{(k+1/2, 2)\}\), and \({\mathfrak {E}} = \{\gamma _1, \gamma _2\}\). For all \(f \in W_{{\mathfrak {E}}, r}^{s-k- \frac{1}{p}, p}(T)\), there holds
We proceed by induction on r. The case \(r=0\) follows from (3.4) and (6.2). Now let \(r \in {\mathbb {N}}_0\) be given, and assume that (6.11) holds for all \(k \in {\mathbb {N}}_0\) and \((s, p) \in {\mathcal {A}}_k \cup \{(k+1/2, 2)\}\).
Let \(k\in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k \cup \{(k+1/2, 2)\}\), and \(f \in W_{{\mathfrak {E}}, r+1}^{s-k - \frac{1}{p}, p}(T)\) be given. Then, applying (6.6) and (6.9) gives
where \(S_{k, r, q}^{[1]}\) is defined in (6.7), and so
Consequently, we obtain
Part (a). We first consider the terms \(\Vert {\mathcal {S}}^{[1]}_{k+1, r}[\omega _i b](\omega _i^{-1} f) \Vert _{s, p, K}\), \(1 \le i \le 2\). For \(k+1/p \le s \le k + 1 + 1/p\), Theorem A.3 shows that \(\omega _i^{-1} f \in L^p(T; \omega _i^{(k-s+1)p + 1} \,\textrm{d}{\varvec{x}})\) and (A.9) and (A.8) gives
for \(1 \le i \le 2\). Applying (6.8) then gives
Now let \(s > k + 1 + 1/p\). Corollary A.4 shows that \(\omega _i^{-1} f \in W_{{\mathfrak {E}}, r}^{s-k-1-\frac{1}{p}, p}(T)\) and (A.11) then gives
Inequality (6.13) then follows from (6.11).
Part (b). We now turn to the term \(\Vert {\mathcal {S}}^{[1]}_{k+2, r}[\omega _1 \omega _2 b]((\omega _1 \omega _2)^{-1}f) \Vert _{s, p, K}\). Assume first that \(k + 1/p \le s \le k + 1 + 1/p\). Theorem A.3 shows that \((\omega _1 \omega _2)^{-1} f \in L^p(T; \omega _1^p \omega _2^{(k-s+1)p + 1} \,\textrm{d}{\varvec{x}})\), and (A.9) and (A.8) give
Applying (6.8) then gives
Now assume that \(k + 1 + 1/p < s \le k + 2 + 1/p\). Thanks to Corollary A.4, \(\omega _2^{-1} f \in W^{s-k-1-\frac{1}{p}, p}_{\gamma _1, r+1}(T)\), and so Theorem A.3 gives \((\omega _1 \omega _2)^{-1} f \in L^p(T; \omega _1^{(k-s+2)p + 1} \,\textrm{d}{\varvec{x}})\). Inequalities (A.11b) and (A.8) then give
Applying (6.8) then gives (6.14).
Now assume that \(s > k + 2 + 1/p\). Two applications of Corollary A.4 show that \((\omega _1 \omega _2)^{-1} f \in W^{s-k-2-\frac{1}{p}, p}_{{\mathfrak {E}}, r}(T)\) and (A.11b) and (3.27) give
Applying (6.11) then gives (6.14). Inequality (6.11) for \(r+1\) now follows from the triangle inequality, (6.13), and (6.14). The smoothness of the mapping \({\mathfrak {I}}_1\) defined in (3.2) then gives (3.29).
Step 2: Trace properties (3.28a) and (3.28b). Direct computation shows that (3.28a) and (3.28b) hold.
Step 3: Polynomial preservation. Suppose that \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\gamma _{12}} = D_{\Gamma }^{l} f|_{\gamma _{13}} = 0\) for \(0 \le l \le r-1\). Then, \(f \circ {\mathfrak {I}}_1 = (\omega _1 \omega _2)^r g\) for some \(g \in {\mathcal {P}}_{N-2r}(T)\), and so \({\mathcal {S}}_{k, r}^{[1]}(f) = (x_1 x_2)^r {\mathcal {E}}_{k}^{[1]}(g) \in {\mathcal {P}}_{N+k}(K)\) thanks to Lemma 3.1. \(\square \)
6.5 Proof of Lemma 3.11
Step 1: Continuity (3.39). We first show that the following analogue of (3.39) holds: For \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \({\mathfrak {E}} = \{\gamma _1, \gamma _2, \gamma _3\}\), there holds
Part (a): Variants of \({\mathcal {S}}_{k, r}^{[1]}\). We begin with a brief aside. Let \({\mathfrak {E}}_{ij} = \{\gamma _i, \gamma _j\}\) for \(1 \le i < j \le 3\). Formally define the following analogue of \({\mathcal {S}}_{k, r}^{[1]}\) (3.25):
where \({\mathfrak {F}}_2\) and \({\mathfrak {G}}_2\) are defined in (6.4). Note that for any \(s \ge 0\) and \(r \in {\mathbb {N}}_0\), there holds \(f \in W^{s, p}_{{\mathfrak {E}}_{13}, r}(T)\) if and only if \(f \circ {\mathfrak {F}}_2 \in W^{s, p}_{{\mathfrak {E}}_{12}, r}(T)\). Thanks to Lemma 3.7, for \(b \in C^{\infty }_c(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), there holds
where we used that \(\Vert f\Vert _{t, p, T} \approx _{t, p} \Vert f \circ {\mathfrak {F}}_2\Vert _{t, p, T}\) and 
. Analogous arguments show that the operator
satisfies the following for \(b \in C^{\infty }_c(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\):
Part (b): Key identity for \({\mathcal {R}}_{k, r}^{[1]}\). Thanks to Lemma C.1, there holds
where \(\lambda _1:= x_1\), \(\lambda _2:= x_2\), \(\lambda _3:= 1 - x_1 - x_2 - z\), m(i, j) is the lone element of \(\{1,2,3\} {\setminus } \{i, j\}\), \(d_{ln}^{(ij)}\) and \(d_{ln}^{(ji)}\) are suitable constants, and \(S_{k, r}^{[1], (12)}:= {\mathcal {S}}_{k, r}^{[1]}\).
Let \(b \in C_c^{\infty }(T)\), \(k, r \in {\mathbb {N}}_0\), \((s, p) \in {\mathcal {A}}_k\), and \(f \in W^{s-k-\frac{1}{p}, p}_{{\mathfrak {E}}, r}(T) \) be given. For any \(n \in {\mathbb {N}}_0\) and real \(t \ge 0\), the mapping \(g \mapsto \omega _i^{n} g\) is continuous from \(W^{t, p}_{{\mathfrak {E}}, r}(T)\) to \(W^{t, p}_{{\mathfrak {E}}, r}(T)\). Similarly, for any \(\alpha \in {\mathbb {N}}_0^3\), the mapping \(g \mapsto \lambda _1^{\alpha _1} \lambda _2^{\alpha _2} \lambda _3^{\alpha _3} g\) is continuous from \(W^{s, p}(K)\) to \(W^{s, p}(K)\). Consequently, (6.15) follows from the triangle inequality, (3.10), (3.29), (6.16), and (6.17). The smoothness of the mapping \({\mathfrak {I}}_1\) (3.2) then gives (3.39).
Step 2: Trace properties (3.38a) and (3.38b). Direct computation shows that (3.38a) and (3.38b) hold.
Step 3: Polynomial preservation. Suppose that \(f \in {\mathcal {P}}_N(\Gamma _1)\), \(N \in {\mathbb {N}}_0\), satisfies \(D_{\Gamma }^{l} f|_{\partial T} = 0\) for \(0 \le l \le r-1\). Then, \(f \circ {\mathfrak {I}}_1 = (\omega _1 \omega _2 \omega _3)^r g\) for some \(g \in {\mathcal {P}}_{N-3r}(T)\), and so \({\mathcal {R}}_{k, r}^{[1]}(f) = (x_1 x_2 (1-x_1-x_2-z))^r {\mathcal {E}}_{k}^{[1]}(g) \in {\mathcal {P}}_{N+k}(K)\) thanks to Lemma 3.1. \(\square \)
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Communicated by Rob Stevenson.
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Charles Parker acknowledges that this material is based upon work supported by the National Science Foundation under Award No. DMS-2201487.
Appendices
Properties of Spaces with Vanishing Traces
In this section, we show that smooth functions with vanishing traces are dense in the space \(W^{s, p}_{{\mathfrak {E}}}(T)\) (3.9) and that functions in \(W^{s, p}_{{\mathfrak {E}}, r}(T)\) (3.26) satisfy a Hardy inequality.
1.1 A Density Result
We begin with a density result for the spaces \(W^{s,p}_{{\mathfrak {E}}}(T)\) defined in Sect. 3.2.
Lemma A.1
Let \({\mathfrak {E}} \subseteq \{ \gamma _1, \gamma _2, \gamma _3 \}\) and define
For \(1< p < \infty \) and \(0 \le s < \infty \), the space \(C^{\infty }_{{\mathfrak {E}}}(T)\) is dense in \(W^{s,p}_{{\mathfrak {E}}}(T)\).
Proof
Let \({\mathfrak {E}} \subseteq \{ \gamma _1, \gamma _2, \gamma _3 \}\) and \(1< p < \infty \) be given.
Step 1: \(0 \le s < 1/p\). The space \(C^{\infty }_c(T) \subseteq C^{\infty }_{{\mathfrak {E}}}(T)\) is dense in \(W^{s, p}(T) = W_{{\mathfrak {E}}}^{s, p}(T)\) (see e.g. [38, Theorem 1.4.5.2]).
Step 2: \(s \ge 1/p\) and \({\mathfrak {E}} = \{\gamma _1\}\). Let \(s = m + \sigma \) with \(m \in {\mathbb {N}}_0\) and \(\sigma \in [0, 1)\), and let \(f \in W^{s, p}_{{\mathfrak {E}}}(T)\). For \(n \in {\mathbb {N}}\), we construct a partition of unity on \(T\) as follows. Let \(\{ \varvec{a}_i \}_{i=1}^{3}\) denote the vertices of T labeled counterclockwise as in Fig. 1b and define the following sets:
where we use the notation \(B(\varvec{x}, r)\) to denote the ball of radius r centered at \(\varvec{x}\). By construction, \(T\subset \bigcup _{i=0}^{3n} {\mathcal {U}}_{i}\), and so there exists a partition of unity \(\{ \phi _i \in C^{\infty }_c({\mathcal {U}}_i): 0 \le i \le 3n\}\) satisfying
We denote \(f_i:= \phi _i f\) for \(0 \le i \le 3n-3\) and set \({\mathcal {V}}_i:= {\mathcal {U}}_i \cap T\) for \(0 \le i \le 3n\).
Let \(\{ \delta _i \}_{i=0}^{3n-3}\) be arbitrary positive constants. The construction proceeds in several parts.
Part (a). The function \(f_0:= \phi _0 f\) satisfies
By [38, Theorem 1.4.5.2], there exists a \(\psi _0 \in C^{\infty }_c({\mathcal {U}}_0)\) satisfying
Since \(f \in W^{s, p}(T)\), we apply the same argument to \(f_i:= \phi _i f\) on \({\mathcal {V}}_i\), \(1 \le i \le n-1\), to show that there exists a \(\psi _i \in C^{\infty }_c({\mathcal {V}}_i)\) satisfying
Part (b). For \(n \le i \le 3n-2\), \(f \in W^{s, p}({\mathcal {V}}_i)\), and so there exists a \(\rho _i \in C^{\infty }(\bar{{\mathcal {V}}}_i)\) satisfying \(\Vert f - \rho _i\Vert _{s, p, {\mathcal {V}}_i} \le \delta _i n^{m+2}\) thanks to [38, Theorem 1.4.5.2]. Then, the function \(\psi _i:= \phi _i \rho _i\) satisfies
where we used [44, Theorem 6.3] to conclude that
Moreover, when \(\sigma p=1\), we have
Part (c). For \(3n-1 \le i \le 3n\), we will show that
Thanks to [44, Theorem 6.3], there holds
Similar computations show that for \(\sigma p =1 \), there holds
Since \({\mathcal {V}}_i \cap \gamma _1 \ne \emptyset \), Poincaré’s inequality gives
and so \(\Vert f_i \Vert _{s, p, {\mathcal {V}}_i} \lesssim _{s, p} |D^m f|_{\sigma , p, {\mathcal {V}}_i}\). Moreover, if \(\sigma p=1\), then \(D^{m-j} f \in W_{{\mathfrak {E}}}^{1,p}({\mathcal {V}}_i)\) for \(1 \le j \le m\), and so [24, Theorem 5.2], [35, Theorem 3.2], and a standard scaling argument give
and so \(\Vert \omega _1^{-\sigma } D^m f_i\Vert _{p, {\mathcal {V}}_i} \lesssim _{s, p} |D^m f|_{\sigma , p, {\mathcal {V}}_i}\). Equality (A.1) now follows from that fact that \( |D^m f|_{\sigma , p, {\mathcal {V}}_i} \rightarrow 0\) as \(n \rightarrow \infty \) since \(|{\mathcal {V}}_i| \rightarrow 0\) as \(n \rightarrow \infty \).
Part (d). Let \(\epsilon > 0\) be given. First, choose n large enough so that
Then, for \(\{\delta _i\}_{i=0}^{3n-2}\) chosen sufficiently small, we construct \(\psi _i\) as above so that
Let \({\tilde{\psi }}_i\) denote the extension of \(\psi _i\) by zero to \(T{\setminus } {\mathcal {U}}_i\), \(0 \le i \le 3n-2\) and set \({\tilde{\psi }}_{j} \equiv 0\) for \(3n-1 \le j \le 3n\). Then, \({\tilde{\psi }}_i \in C^{\infty }_{{\mathfrak {E}}}(T)\), and the function \(\psi = \sum _{i=0}^{3n} {\tilde{\psi }}_i\) then satisfies \(\psi \in C^{\infty }_{{\mathfrak {E}}}(T)\) and
which shows that \(C^{\infty }_{{\mathfrak {E}}}(T)\) is dense in \(W^{s,p}_{{\mathfrak {E}}}(T)\).
Step 3: \({\mathfrak {E}} = \{\gamma _2\}\) or \({\mathfrak {E}} = \{\gamma _3\}\). If \({\mathfrak {E}} = \{\gamma _2\}\), the density of \(C^{\infty }_{\gamma _2}(T)\) in \(W_{\gamma _2}^{s, p}(T)\) follows from the fact that \(f \in W_{\gamma _2}^{s, p}(T)\) if and only if \(f \circ {\mathfrak {F}}_2^{-1} \in W_{\gamma _1}^{s, p}(T)\), where \({\mathfrak {F}}_2^{-1}(x_1, x_2) = (1-x_1-x_2, x_1)\) is the inverse of \({\mathfrak {F}}_2\) defined in (6.4). The case \({\mathfrak {E}} = \{\gamma _3\}\) follows from similar arguments using the mapping \({\mathfrak {F}}_2\).
Step 4: \(|{\mathfrak {E}}| = 2\). Now let \({\mathfrak {E}} = \{\gamma _1,\gamma _2\}\). The density of \(C^{\infty }_{{\mathfrak {E}}}(T)\) in \(W^{s,p}_{{\mathfrak {E}}}(T)\) may be shown using a similar construction to the case \({\mathfrak {E}} = \{\gamma _1\}\). In particular, we apply the construction of Step 1 Part (a) for \(0 \le i \le 2n-2\) and \(i = 3n\), Part (b) for \(2n-1 \le i \le 3n-3\), Part (c) for \(3n-2 \le i \le 3n-1\), and proceed analogously as in Part (d). The remaining cases for \(|{\mathfrak {E}}| = 2\) are proved along similar lines.
Step 5: \({\mathfrak {E}} = \{\gamma _1,\gamma _2,\gamma _3\}\). This case is a restatement of [38, Lemma 1.4.5.2]. \(\square \)
1.2 Hardy Inequalities
First, we construct a bounded averaging operator.
Lemma A.2
There exists a linear operator \({\mathcal {H}}_1\) satisfying the following properties:
-
(i)
\({\mathcal {H}}_1\) maps \(C({\bar{T}})\) boundedly into \(C({\bar{T}})\), and there holds
$$\begin{aligned} {\mathcal {H}}_1(f)(\varvec{x}) = \frac{1}{x_1} \int _{0}^{x_1} f(u, x_2) \,\textrm{d}{u} = \int _{0}^{1} f(u x_1 , x_2) \,\textrm{d}{u} \qquad \forall \varvec{x} \in T. \end{aligned}$$(A.2) -
(ii)
\({\mathcal {H}}_1\) maps \(W^{s, p}_{{\mathfrak {E}}, r}(T)\) boundedly into \(W^{s, p}_{{\mathfrak {E}}, r}(T)\) and for all \(p \in (1,\infty )\), \(s \in [0, \infty )\), \(r \in {\mathbb {N}}_0\), and \({\mathfrak {E}} \in \{ \emptyset , \{\gamma _1\}, \{\gamma _1, \gamma _2\} \}\). In particular,
(A.3)
Proof
Step 1: Continuity on \(C({\bar{T}})\). Let \(f \in C({\bar{T}})\) and define \({\mathcal {H}}_1(f)\) by (A.2). Elementary arguments show that \({\mathcal {H}}_1(f) \in C({\bar{T}})\) with \(\Vert {\mathcal {H}}_1(\phi _i)\Vert _{\infty , T} \le \Vert \phi _i\Vert _{\infty , T}\).
Step 2: Extension to \(W^{s, p}_{{\mathfrak {E}}, r}(T)\) when \({\mathfrak {E}} = \emptyset \). Let \(f \in C^{\infty }({\bar{T}})\) and \(1< p < \infty \). For \(\alpha \in {\mathbb {N}}_0^2\), there holds
and so \({\mathcal {H}}_1(f) \in C^{\infty }({\bar{T}})\). Moreover, Hardy’s inequality [40, Theorem 327] gives
Consequently, we obtain
Since \(C^{\infty }({\bar{T}})\) is dense in \(W^{s, p}(T)\) [38, Theorem 1.4.5.2], \({\mathcal {H}}_1\) can be continuously extended to a linear operator from \(W^{s, p}(T)\) into \(W^{s, p}(T)\) for \(s \in {\mathbb {N}}_0\). The case for non-integer \(s \in (0,\infty )\) follows from interpolation.
Step 3: Inequality (A.3) when \({\mathfrak {E}} = \{\gamma _1\}\). The case \(r = 0\) follows from Step 2, so let \(r \in {\mathbb {N}}\). Assume first the \(s \le r\) and let \(f \in C^{\infty }_{\gamma _1}(T)\). Equation A.4 shows that \({\mathcal {H}}_1(f) \in C^{\infty }_{\gamma _1}(T)\). Moreover, for \(s = m + 1/p\), \(m \in {\mathbb {N}}_0\), we apply Hardy’s inequality [40, Theorem 327] to obtain
for all \(\alpha \in {\mathbb {N}}_0\) with \(|\alpha | = m\). Thus, 
for all \(f \in C^{\infty }_{\gamma _1}(T)\). By density (Lemma A.1), \({\mathcal {H}}_1\) maps \(W_{\gamma _1, r}^{s, p}(T)\) boundedly into \(W_{\gamma _1, r}^{s, p}(T)\) for all \(p \in (1,\infty )\) and \(s \in [0, r]\).
Now let \(s > r\) and \(f \in W_{\gamma _1, r}^{s, p}(T)\). Step 2 and the arguments above show that \({\mathcal {H}}_1(f) \in W^{s, p}(T) \cap W^{r, p}_{\gamma _1}(T)\), and so \({\mathcal {H}}_1(f) \in W^{s, p}_{\gamma _1, r}(T)\) if \(s-1/p \notin {\mathbb {Z}}\) with 
. Now let \(s = m + 1/p\) for some \(m \in {\mathbb {N}}\). Then, \(f \in C({\bar{T}})\) and thanks to Step 1 and (A.5), we have
Consequently, \({\mathcal {H}}_1(f) \in W^{s, p}_{\gamma _1, r}(T)\) and (A.3) holds when \({\mathfrak {E}} = \{\gamma _1\}\).
Step 4: Inequality (A.3) when \({\mathfrak {E}} = \{\gamma _1, \gamma _2\}\). Again let \(r \in {\mathbb {N}}\). Assume first that \(s \le r\). As above, \({\mathcal {H}}_1(f) \in C^{\infty }_{{\mathfrak {E}}}(T)\) for any \(f \in C^{\infty }_{{\mathfrak {E}}}(T)\) by (A.4), and for \(s = m + 1/p\), \(m \in {\mathbb {N}}_0\), Hardy’s inequality [40, Theorem 327] gives
for all \(\alpha \in {\mathbb {N}}_0\) with \(|\alpha | = m\). Inequality (A.3) now follows from Step 2 and (3.10). By density, \({\mathcal {H}}_1\) maps \(W_{{\mathfrak {E}}, r}^{s, p}(T)\) boundedly into \(W_{{\mathfrak {E}}, r}^{s, p}(K)\) for all \(p \in (1,\infty )\) and \(s \in [0, r]\).
Now let \(s > r\) and \(f \in W^{s, p}_{{\mathfrak {E}}, r}(T)\). Arguing analogously as in Step 3, we have \({\mathcal {H}}_1(f) \in W^{s, p}_{{\mathfrak {E}}, r}(T)\) if \(s - \frac{1}{p} \notin {\mathbb {Z}}\) with 
. Moreover, (A.6) gives
Consequently, \({\mathcal {H}}_1(f) \in W^{s, p}_{{\mathfrak {E}}, r}(T)\) and (A.3) holds when \({\mathfrak {E}} = \{\gamma _1, \gamma _2\}\). \(\square \)
Finally, we state and prove various versions of Hardy’s inequality.
Theorem A.3
Let \(1< p < \infty \) and \(\emptyset \ne {\mathfrak {E}} \subseteq \{\gamma _1,\gamma _2\}\). For \(0 \le s < \infty \) and \(i \in \{1,2\}\) such that \(\gamma _i \in {\mathfrak {E}}\), the mapping \(f \mapsto \omega _i^{-1} f\) is bounded (i) \(W^{s+1,p}(T) \cap W^{1,p}_{{\mathfrak {E}}}(T)\) to \(W^{s,p}(T)\), and (ii) \(W^{s+1,p}_{{\mathfrak {E}}}(T)\) to \(W^{s,p}_{{\mathfrak {E}}}(T)\), and there holds
Additionally, for \(0 \le s < 1\) and \(i \in \{1,2\}\), the mapping \(f \mapsto \omega _i^{-s} f\) is bounded \(W_{\gamma _i}^{s, p}(T)\) to \(L^p(T)\), and there holds
Proof
Let \(1< p < \infty \) be given.
Step 1: Inequalities (A.7) and (A.8) when \({\mathfrak {E}} \in \{ \{\gamma _1\}, \{\gamma _1,\gamma _2\} \}\). Thanks to the fundamental theorem of calculus, there holds
By density (Lemma A.1), (A.10) holds for a.e. \(\varvec{x} \in T\) for all \(f \in W_{{\mathfrak {E}}}^{1, p}(T)\). Lemma A.2 and (A.10) then show that the mapping \(f \mapsto \omega _i^{-1} f\) is bounded (i) from \(W^{s+1,p}(T) \cap W^{1,p}_{{\mathfrak {E}}}(T)\) to \(W^{s,p}(T)\), and (ii) from \(W^{s+1,p}_{{\mathfrak {E}}}(T)\) to \(W^{s,p}_{{\mathfrak {E}}}(T)\) provided that \(\gamma _i \in {\mathfrak {E}}\). Inequalities (A.7) and (A.8) now follow from (A.3) and (A.10).
Step 2: Inequalities (A.7) and (A.8) when \({\mathfrak {E}} = \{2\}\). Note that \(f \in W^{s+1, p}(T) \cap W^{1, p}_{\gamma _2}(T)\) if and only if \(g:= f \circ {\mathfrak {F}}_1 \in W^{s+1, p}(T) \cap W^{1, p}_{\gamma _1}(T)\) and \(f \in W^{s+1}_{\gamma _2}(T)\) if and only if \(g \in W^{s+1}_{\gamma _1}(T)\), where \({\mathfrak {F}}_1\) is defined in (6.3). Inequalities (A.7) and (A.8) then follow from Step 1.
Step 3: Inequality (A.9) with \(i=1\). Now let \(0 \le s < 1\). For \(sp = 1\), (A.9) follows immediately from the definition of the norm. In the case \(sp < 1\), the proof of Theorem 1.4.4.4 in [38] gives
Finally, let \(sp > 1\) and let \(f \in W^{s, p}_{\gamma _1}(T)\) be given. We denote by \({\tilde{f}} \in W^{s, p}(T)\) any extension of f to \({\mathbb {R}}^2\) satisfying \(\Vert {\tilde{f}}\Vert _{s, p, {\mathbb {R}}^2} \lesssim _{s, p} \Vert f\Vert _{s, p, T}\) (see e.g. [34] or [44, Theorem 8.4]). Thanks to Theorem 6.79, inequality (6.58), and Remark 6.80 of [44], there holds
Since \(f|_{\gamma _1} = {\tilde{f}}|_{\gamma _1} = 0\), we obtain
and so \(\Vert \omega _1^{-s} f\Vert _{p, T} \lesssim _{s, p} \Vert f\Vert _{s, p, T}\), which completes the proof of (A.9) for \(i=1\).
Step 4: Inequality (A.9) with \(i=2\). This can be reduced to the case \(i=1\) using similar arguments as in Step 2 using the mapping \({\mathfrak {F}}_1\) defined in (6.3). \(\square \)
Corollary A.4
Let \(1< p < \infty \), \(0 \le s < \infty \), \(1 \le i \le 2\), and \(r_1, r_2 \in {\mathbb {N}}_0\) with \(r_i \ge 1\). Then, for all \(f \in W^{s+1, p}_{\gamma _1, r_1}(T) \cap W^{s+1, p}_{\gamma _2, r_2}(T)\), there holds \(\omega _i^{-1} f \in W^{s, p}_{\gamma _i, r_i - 1}(T) \cap W^{s, p}_{\gamma _j, r_j}(T)\) and
where \(1 \le j \le 2\), \(j \ne i\).
Proof
Let \(f \in W^{s+1, p}_{\gamma _1, r_1+1}(T) \cap W^{s+1, p}_{\gamma _2, r_2}(T)\) be given. By definition, there holds \(\partial _1 f \in W^{s, p}_{\gamma _1, r_1}(T) \cap W^{s, p}_{\gamma _2, r_2}(T)\) since \(\textbf{t}_{\gamma _2} = [0, 1]^T\). Thanks to identity (A.10), which was shown in the proof of Theorem A.3 to hold for all \(f \in W^{1, p}_{\gamma _1}(T)\), the result for \(i = 1\) follows from Lemma A.2. The case \(i=2\) can be reduced to the case \(i=1\) using similar arguments as in the proof of Theorem A.3 using the mapping \({\mathfrak {F}}_1\) (6.3). \(\square \)
Equivalent Boundary Norm
We begin with a result that states necessary and sufficient conditions for a function defined on two faces \(\Gamma _i \cup \Gamma _j \subset \partial K\) to belong to \(W^{s, p}(\Gamma _i \cup \Gamma _j)\).
Lemma B.1
Let \(0 \le s < 1\), \(1< p < \infty \), and \(1 \le i < j \le 4\). Then, \(f \in L^p(\Gamma _i \cup \Gamma _j)\) satisfies \(f \in W^{s, p}(\Gamma _i \cup \Gamma _j)\) if and only if
-
(i)
\(f_i \in W^{s, p}(\Gamma _i)\) and \(f_{j} \in W^{s, p}(\Gamma _{j})\);
-
(ii)
if \(s > 1/p\), then \(f_i|_{\gamma _{ij}} = f_j|_{\gamma _{ij}}\); and
-
(iii)
if \(s = 1/p\), then \(I_{ij}^p(f_i, f_j) < \infty \),
where \({\mathcal {I}}_{ij}^p(\cdot ,\cdot )\) is defined in (2.2). Additionally,
Proof
Let \(0 \le s < 1\), \(1< p < \infty \), and \(1 \le i < j \le 4\) be given.
Step 1: \(f \in W^{s,p}(\Gamma _i \cup \Gamma _j)\implies \text {(i--iii)}\). Assume first that \(f \in W^{s, p}(\Gamma _i \cup \Gamma _j)\). Condition (i) follows from the definition of the norms, and in particular, \( \Vert f_i\Vert _{s, p, \Gamma _i}^p + \Vert f_j\Vert _{s, p, \Gamma _j}^p \le \Vert f\Vert _{s, p, \Gamma _i \cup \Gamma _j}^p\). If \(s > 1/p\), then the trace theorem shows that f has a well-defined trace on \(\gamma _{ij}\), and so (ii) holds.
We now show that condition (iii) is satisfied. There holds
for all \((\varvec{x}, \varvec{y}) \in \Gamma _i \times \Gamma _j\), where \(\varvec{F}_{ij}: T\rightarrow \Gamma _i\) and \(\varvec{F}_{ji}: T\rightarrow \Gamma _j\) are defined in (2.1), and so
where \(\varvec{u}_{r}:= (u_1, -u_2)\). Now let \(s = 1/p\). Applying the triangle inequality in conjunction with the above inequality, we obtain
Assume, for the moment, that the following holds.
Then, we obtain the following bound for \(I_{ij}^p(f_i, f_j)\) defined in (2.2):
Thus, f satisfies (iii) and we have shown that for all \(f \in W^{s, p}(\Gamma _i \cup \Gamma _j)\) there holds
Annular regions of integration (red) in the proof of (B.2), where \(\tilde{\varvec{z}}:= (z_1, z_2/2)\) and \(\tilde{\varvec{u}}:= (u_1, u_2/2)\) (Color figure online)
We now turn to the proof of (B.2). First note that B is well-defined on the half plane \({\mathbb {R}}_+^2 = \{\varvec{x} \in {\mathbb {R}}^2: x_2 > 0\}\) and is a continuous function with \(B(\varvec{x}) > 0\) for \(\varvec{x} \in {\mathbb {R}}_+^2\), and so it suffices to show (B.2) for \(\varvec{x} \in T\) with \(x_2 < 1/4\). Let \((\rho , \theta ) \in [0, \infty ) \times [0, 2\pi )\) denote polar coordinates centered at \(\varvec{x}_r\) and let \(A_{\varvec{x}}\) denote the following annulus centered at \(\varvec{x}_r\) (pictured in Fig. 2):
Then, one may readily verify that \(A_{\varvec{x}} \subset T\), and so
which completes the proof of (B.2).
Step 2: \(\text {(i--iii)} \implies f \in W^{s,p}(\Gamma _i \cup \Gamma _j)\). Now assume that \(f \in L^p(\Gamma _i \cup \Gamma _j)\) satisfies (i–iii). Thanks to the triangle inequality, there holds
where we used that \(|\varvec{x}_r - \varvec{y}| \ge |\varvec{x} - \varvec{y}|\). To bound the second term, we perform a simple change of variables.
For the first term, there holds
for all \(\varvec{x} \in T\), and so
Thanks to conditions (ii)-(iii), the function \(g = f_i \circ \varvec{F}_{ij} - f_j \circ \varvec{F}_{ji}\) belongs to \(W_{\gamma _2}^{s, p}(T)\) and applying (A.8) and the triangle inequality gives
and so
Then, the reverse inequality of (B.3) immediately follows from (B.1), and so \(f \in W^{s, p}(\Gamma _i \cup \Gamma _j)\) and the result follows. \(\square \)
On noting that \(f \in W^{s, p}(\partial K)\) if and only if \(f \in L^p(\partial K)\) and \(f|_{\Gamma _i \cup \Gamma _j} \in W^{s, p}(\Gamma _i \cup \Gamma _j)\) for all \(1 \le i < j \le 4\) since
the following result is an immediate consequence of Lemma B.1.
Corollary B.2
Let \(0 \le s < 1\) and \(1< p < \infty \). Then, \(f \in L^p(\partial K)\) satisfies \(f \in W^{s, p}(\partial K)\) if and only if
-
(i)
\(f_i \in W^{s, p}(\Gamma _i)\) for \(1 \le i \le 4\);
-
(ii)
if \(s > 1/p\), then \(f_i|_{\gamma _{ij}} = f_j|_{\gamma _{ij}}\) for all \(1 \le i < j \le 4\); and
-
(iii)
if \(s = 1/p\), then \(I_{ij}^p(f_i, f_j) < \infty \) for all \(1 \le i < j \le 4\).
Additionally, (2.3) holds.
Partial Fractions Decomposition
Lemma C.1
For all \(\beta \in {\mathbb {N}}_0^3\) with \(|\beta | \ge 2\), there holds
where \(\{ c_{\alpha , j} \}\) are suitable positive constants.
Proof
We proceed by induction on \(|\beta |\). The case \(|\beta | = 2\) is trivially true. Assume that (C.1) holds for all \(\beta \in {\mathbb {N}}_0^3\) with \(|\beta | = r \ge 2\). Let \(\beta \in {\mathbb {N}}_0^3\) with \(|\beta | = r+1\). If \(\beta _j = 0\) for some \(j \in \{1,2,3\}\), then (C.1) is trivially true, so assume that \(\beta _j > 0\) for \(1 \le j \le 3\). Then,
By assumption, each of the three terms above is of the form (C.1), which completes the proof. \(\square \)
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Parker, C., Süli, E. Stable Liftings of Polynomial Traces on Tetrahedra. Found Comput Math (2024). https://doi.org/10.1007/s10208-024-09670-x
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DOI: https://doi.org/10.1007/s10208-024-09670-x