Abstract
We reduce the boundedness of operators in Morrey spaces \(L_p^r\left( {\mathbb R}^n\right) \), its preduals, \(H^{\varrho }L_p ({\mathbb R}^n)\), and their preduals \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) to the boundedness of the appropriate operators in Lebesgue spaces, \(L_p({\mathbb R}^n)\). Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications in harmonic analysis.
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1 Introduction
Let \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) be the completion of \(D({\mathbb R}^n)\) in \(L_p^r({\mathbb R}^n)\), where
Then we have
where the second duality assertion is due to [1, 9, 14, 22, 30] and the first assertion is observed by [2] and proved by [22]. Roughly speaking in this paper we prove that the \(L_p({\mathbb R}^n)\)-boundedness of an operator T satisfying the condition
implies its boundedness in \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \). Therefrom, under some additional conditions with respect to T we get also the boundedness of T in \(H^{\varrho }L_p ({\mathbb R}^n)\) and \(L_p^r({\mathbb R}^n)\) by (1) and duality arguments. Our paper can be considered as an extension of the new approach given in [21] and [22] to a wider class of operators and to the vector-valued situation. Let us mention that the extension of operators of this type and related norm estimates have to be treated with greater care than in many related papers investigating mapping properties of operators in \(L_p^r({\mathbb R}^n)\). We refer to Remark 4.5 for the relation of our paper to the existing literature. In particular, we cannot expect an unique extension to Morrey spaces \(L_p^r({\mathbb R}^n)\). On the contrary it turned out that there are infinitely many possible extension operators (cf. [22, Remark 5.3]). Let us also mention that the vector-valued situation under consideration is crucial having in mind applications as a Michlin–Hörmander type theorem (and hence applications to Navier–Stokes equations cf. [26] and [21, Remark 4.3]), Littlewood–Paley theory for Morrey spaces and its preduals as well as for Lizorkin representations of Triebel–Lizorkin–Morrey spaces. The given results are partially contained in [20]. Condition (2) is due to Soria and Weiss [23] who transferred the boundedness of singular operators on Lebesgue spaces to the boundedness of these operators in some weighted Lebesgue spaces.
The paper is organized as follows. Basic definitions and preliminaries which are needed later on are collected in Sect. 2. Duality theory for vector-valued Morrey-type spaces is treated in Sect. 3. The main results can be found in Theorem 3.1 (preduals of Morrey spaces) and Theorem 3.3 (Morrey spaces as bidual spaces). In final Sect. 4 we prove our main results concerning the transference of mapping properties of operators satisfying condition (2) to vector-valued Morrey type spaces. The general theorem is presented in Sect. 4.1 (Theorem 4.3) following the method developed in [21] and [22]. As a consequence of our main theorem we obtain mapping properties for various classes of operators in vector-valued Morrey-type spaces. Section 4.2 is concerned with Calderón–Zygmund operators. Here we present also an alternative approach via weighted spaces (Theorem 4.9). Maximal operators of Hardy-Littlewood and Calderón–Zygmund type as well as related vector-valued inequalities are considered in Sect. 4.3. The final Sect. 4.4 is devoted to some classes of Fourier multipliers such as characteristic functions, smooth multipliers and Bochner–Riesz mulipliers at the critical index.
2 Definitions and Preliminaries
2.1 Notation
We use standard notation. Let \(\mathbb {N}\) be the collection of all natural numbers and \(\mathbb {N}_0 = \mathbb {N}\cup \{0 \}\). Let \({\mathbb R}^n\) be the Euclidean n-space, where \(n\in \mathbb {N}\). Put \(\mathbb {R}= \mathbb {R}^1\). Let \(S({\mathbb R}^n)\) be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on \({\mathbb R}^n\) and let \(S' ({\mathbb R}^n)\) be the space of all tempered distributions on \({\mathbb R}^n\). Let \(D({\mathbb R}^n) = C^\infty _0 ({\mathbb R}^n)\) be the collection of all infinitely differentiable complex-valued functions with compact support in \({\mathbb R}^n\), where the support of a function f is abbreviated by \(\mathrm {supp \,}(f)\). Moreover, denotes \(C({\mathbb R}^n)\) and \(\text {Lip}({\mathbb R}^n)\) the collection of all continuous and Lipschitz continuous, respectively, and bounded complex-valued functions defined on \({\mathbb R}^n\).
Furthermore, \(L_p ({\mathbb R}^n)\) with \(1\,\le \,p\,<\infty \), is the standard complex Banach space with respect to the Lebesgue measure, normed by
For a measurable subset M of \({\mathbb R}^n\) we similarly define \(L_p (M)\). Moreover, |M| stands for the Lebesgue measure of M and \(\chi _M\) for the characteristic function on M. As usual \(\mathbb {Z}\) is the collection of all integers; and \({\mathbb Z}^n\) where \(n\in \mathbb {N}\) denotes the lattice of all points \(m= (m_1, \ldots , m_n) \in {\mathbb R}^n\) with \(m_j\,\in \,\mathbb {Z}\). As usual, \(L_p^{\mathrm {loc}}({\mathbb R}^n)\) collects all equivalence classes of almost everywhere coinciding measurable complex locally p-integrable functions, hence \(f\,\in \,L_p (M)\) for any bounded measurable set M in \({\mathbb R}^n\). For any \(p\,\in \,(1,\infty )\) we denote by \(p'\) the conjugate index, namely, \(1/p\,+\,1/{p'}\,=\,1\). For Banach spaces X and Y and an operator \(T:X\,\rightarrow \,Y\)
means, that the operator is bounded, that is,
where the the constant c is independent of \(x\in X\). Let \(D({\mathbb R}^n)\hookrightarrow X\). A bounded operator \(\widetilde{T}\) acting in X, hence \(\widetilde{T}:X\hookrightarrow X\), is called an extension of T to X if it coincides on \(D({\mathbb R}^n)\) with T. We denote the Fourier transform of f on \(S({\mathbb R}^n)\) or \(S'({\mathbb R}^n)\) by \(\hat{f}\) and its inverse by \(\check{f}\) where the normalisation of \(\hat{f}\) does not matter for our estimates. The concrete value of constants may vary from one formula to the next, but remains the same within one chain of (in)equalities. Finally, \(A\cong B\) is an abbreviation that there are two constants c, \(C>0\) such that \(c A \le B \le C A\).
2.2 Morrey Spaces, Duals and Preduals
Definition 2.1
For \(1< p< \infty \) and \(-\frac{n}{p}\le r<0\) we define Morrey spaces as
with the norm
where \(Q_{JM}\equiv Q_{J,M}\equiv 2^{-J}\left( M\,+\,\left[ -1, 1\right] ^n\right) \) and \(B_R(x)\) denotes the ball with radius R centered at x.
Moreover, \(\mathop {{ L}^r_{p}}\limits ^{\circ }({\mathbb R}^n)\) denotes the closure of \(D({\mathbb R}^n)\) with respect to \(\left\| \cdot |{L}^r_{p}\left( \mathbb {R}^n\right) \right\| \).
Definition 2.2
Let \(1<p<\infty \) and \(-n <\varrho <-n/p\). Then the predual Morrey spaces \(H^{\varrho } L_p ({\mathbb R}^n)\) collects all \(h\in S'({\mathbb R}^n)\) which can be represented as
such that
Furthermore,
where the infimum is taken over all representations (3), (4).
Remark 2.3
The notation of \(H^{\varrho } L_p ({\mathbb R}^n)\) as a predual will be justified in the Theorem 3.1. By triangular and Hölder’s inequality (3) and (4) ensure that the convergence in (3) is unconditionally in \(L_u({\mathbb R}^n)\), where \(\varrho u =-n\). In particular it holds \(H^{\varrho } L_p ({\mathbb R}^n) \hookrightarrow L_u({\mathbb R}^n)\) and we have \(1<u<p\) (cf. [22, (3.10)]). Let \(L_p ({\mathbb R}^n, w_\alpha )\) with \(1< p <\infty \) and \(w_\gamma (x) = (1+|x|^2)^{\gamma /2}\), \(\gamma \in \mathbb {R}\), be the weighted Lebesgue spaces, normed by
Then it holds
with \(\alpha >n/{p'}\) (cf. [22, (3.5)]). Furthermore, \(D({\mathbb R}^n)\), \(S({\mathbb R}^n)\) are dense both in \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) and \(H^{\varrho } L_p ({\mathbb R}^n)\). \(H^{\varrho } L_p ({\mathbb R}^n)\) and \(L^r_{p}\left( \mathbb {R}^n\right) \) are Banach spaces and \(L_u({\mathbb R}^n)\hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow L_p (w_\alpha ,{\mathbb R}^n)\) for \(u=-n/r\) and \(\alpha <-n/p\) (cf. [22, Theorem 3.1]). The last embedding as well as (6) can be sharpened cf. (20) below.
Definition 2.4
Let \(1 < p <\infty \), \(-n <\varrho <-n/p\). Let \(H^{\varrho } L_p ({\mathbb R}^n)_F^\varepsilon \) be the following subspace of \(H^{\varrho } L_p ({\mathbb R}^n)\) defined as
Proposition 2.5
Let \(1 < p <\infty \), \(-n <\varrho <-n/p\). Then \(H^{\varrho } L_p ({\mathbb R}^n)_F^\varepsilon \) is dense in \( H^\varrho L_p ({\mathbb R}^n) \).
Proof
Let \(h\in H^{\varrho } L_p ({\mathbb R}^n) \) and \(\varepsilon >0\). Let \(h= \sum _{J\in \mathbb {Z}, M\in {\mathbb Z}^n} \lambda _{J,M} a_{J,M}\) in \(S'({\mathbb R}^n)\) such that \(\sum _{J\in \mathbb {Z}, M \in {\mathbb Z}^n} |\lambda _{J,M}| \le (1+\varepsilon /2) \Vert h \, | H^{\varrho } L_p ({\mathbb R}^n) \Vert \) with \(\mathrm {supp \,}a_{J,M} \subset {Q_{J,M}}\) and \( \Vert a_{J,M} \, | L_p ({\mathbb R}^n) \Vert \le 2^{-J\left( \frac{n}{p} + \varrho \right) }\). We define then \(h_{J,M}\equiv \lambda _{J,M} a_{J,M}\) for \(J\in \mathbb {Z}\), \(M\in {\mathbb Z}^n\) and obtain
Let
Then
Hence,
\(\square \)
2.3 Vector-Valued Morrey Spaces
Definition 2.6
Let \(1< p< \infty \), \(-\frac{n}{p}\le r<0\) and \(1< q<\infty \). Let \(L^r_{p}(\ell _q, \mathbb {R}^n)\) be the collection of all sequences of functions \(f_j\) belonging to \(L^r_{p}({\mathbb R}^n)\) such that
is finite. Moreover,
Furthermore, for \(\alpha \in \mathbb {R}\) we define the space \(L_{p}(\ell _{q}, w_{\alpha },{\mathbb R}^n)\) as \(L^r_{p}(\ell _{q},{\mathbb R}^n)\) using the norm of \(L_p (w_\alpha ,{\mathbb R}^n)\) instead the norm of \(L^r_{p}\left( \mathbb {R}^n\right) \). If \(\alpha =0\), we simply write \(L_{p}(\ell _{q},{\mathbb R}^n)\).
Definition 2.7
Then \(H^{\varrho } L_p (\ell _{q},{\mathbb R}^n)\) denotes the collection of all sequences of functions \(g_j\) belonging to \(H^{\varrho } L_p ({\mathbb R}^n)\) such that \(\left\| g_j(\cdot )|\ell _{q}\right\| \) is in \(H^{\varrho } L_p ({\mathbb R}^n)\). Moreover, \(H^{\varrho } L_p (\ell _{q},{\mathbb R}^n)_F^\varepsilon \) stands for the collection of all sequences of functions \(g_j\) belonging to \(H^{\varrho } L_p (\ell _{q},{\mathbb R}^n)\) such that \(\left\| g_j(\cdot )|\ell _{q}\right\| \) is in \(H^{\varrho } L_p ({\mathbb R}^n)_F^\varepsilon \).
3 Duals and Preduals: The Vector-Valued Case
3.1 Predual Spaces
The duality with respect to Morrey spaces is discussed in the scalar case in detail with complete proofs in [22]. Here we give complete proofs in the vector-valued case following their approach.
Theorem 3.1
Let \(1<p<\infty \), \(-\frac{n}{p}< r<0\), \(r+\varrho =-n\) and \(1<q<\infty \). Then the predual space of \(L_p^r(\ell _q,{\mathbb R}^n)\) is \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\). Moreover,
if, and only if, it can be uniquely represented as
for all \(f\equiv \{f_j \}\in L_{p'} (\ell _{q'}, w_{\alpha },{\mathbb R}^n) \hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\), \(\alpha >n/p\), where
Moreover, if \(\{g_j\}\in L_p^r(\ell _q,{\mathbb R}^n)\), then
where the supremum is taken over all \(f \equiv \{f_j\}\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) with \(\big \Vert f|H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\big \Vert \le 1\).
Proof
Let \(g\equiv \{g_j\}\in L_p^r(\ell _q,{\mathbb R}^n)\) and \(\{\tilde{f_j}\}\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) such that \(\left\| \left. \tilde{f_j}(\cdot )\right| \ell _{q'}\right\| \) is in \(H^{\varrho } L_{p'} ({\mathbb R}^n)_F^\varepsilon \). Hölder’s inequality yields
where \(\left\| \left. \tilde{f_j}(\cdot )\right| \ell _{q'}\right\| \) is represented as in Definition 2.4 and \(r+\varrho +n=0\). Therefore the operator \(T_{g}\) given by
is bounded on \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)_F^\varepsilon \). We get the (unique) continuous extension \(T_{g}:H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow \mathbb {R}\) by means of Proposition 2.5, where this extension is justified as in the linear case cf. (15) and (17) below. Let \(\{f_j\}\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\). By Proposition 2.5 there is furthermore a sequence \(\{f^k_j\}\) of \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)_F^\varepsilon \) such that \(\{f^k_j\}\) tends to \(\{f_j\}\) in \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) for \(k\rightarrow \infty \). For u such that \(\varrho u\,=\,-n\) by \(H^{\varrho } L_{p'} ({\mathbb R}^n)\hookrightarrow L_u({\mathbb R}^n)\) exists a subsequence such that \(\left\| \left. f^{k_l}_j(\cdot )-f_j(\cdot )\right| \ell _{q'}\right\| \rightarrow 0\) almost everywhere with respect to the Lebesgue measure in \({\mathbb R}^n\) for \(l\rightarrow \infty \). This implies \(f^{k_l}_j \rightarrow f_j\) almost everywhere for all \(j\in \mathbb {N}_0\) if \(l\rightarrow \infty \). The Lemma of Fatou yields then
Thus, for \(\varepsilon \searrow 0\) we obtain
for \(\{g_j\}\in L_p^r(\ell _q,{\mathbb R}^n)\) and \(\{f_j\}\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\). Hence, in particular, any \(\{g_j\}\in L_p^r(\ell _q,{\mathbb R}^n)\) induces a bounded linear functional on \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\).
Conversely, suppose that g is a bounded linear functional on \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) with the norm \(\left\| g\right\| \). Taking into account (6) the linear functional g induces a bounded linear functional on \(L_{p'} (\ell _{q'}, w_{\alpha },{\mathbb R}^n)\) for \(\alpha >n/p\) and therefore we have the representation formula
for some \(\{g_j\}\in L_{p}(\ell _{q},w_{-\alpha },{\mathbb R}^n)\) and for all \(\{f_j\}\in L_{p'}(\ell _{q'}, w_{\alpha },{\mathbb R}^n)\). Let \(\{\tilde{f_j}\}\in L_{p'}(\ell _{q'}, w_{\alpha },{\mathbb R}^n)\) with \(\mathrm {supp \,}\tilde{f_j} \subset Q_{J,M}\) for all \(j\in \mathbb {N}_0\). Then
With \(\frac{n}{p'} +\varrho = -\frac{n}{p} -r\) one obtains
Then one has by duality in \(L_{p'} (\ell _{q'},Q_{J,M})\) and (10)
Hereby, \(L_{p'} (\ell _{q'},Q_{J,M})\) is defined similarly as \(L^r_{p}(\ell _{q'},{\mathbb R}^n)\) using \(L_{p'}(Q_{J,M})\) instead of \(L^r_{p}\left( \mathbb {R}^n\right) \). Note also that an element of \(L_{p'} (\ell _{q'},Q_{J,M})\), say \(\{\tilde{f_j}\}\), is also in \(L_{p'}(\ell _{q'}, w_{\alpha },{\mathbb R}^n)\) if one extends \(\tilde{f_j}\), \(j\in \mathbb {N}_0\), outside of \(Q_{J,M}\) by zero. The last inequality proves \(\{g_j\}\in L_p^r(\ell _q,{\mathbb R}^n) \) and
\(\square \)
3.2 Dual Spaces
In the proof of the next theorem, which is a vector-valued extension of [22, Theorem 4.1, (4.5)], we benefit from the following general assertion.
Proposition 3.2
([4, p. 73] and [24, Lemma in Sect. 1.11.1]) Let \(\{ A_j \}_{j\in \mathbb {N}_0}\) be a sequence of complex Banach spaces and \(\{A_j'\}_{j\in \mathbb {N}_0}\) their respective duals. Moreover, we put
Then
Theorem 3.3
Let \(1<p<\infty \), \(-\frac{n}{p}< r<0\), \(r+\varrho =-n\) and \(1<q<\infty \). Then the dual space of \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) is \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\). Moreover, \(g\in \left( \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\right) '\) if, and only if, it can be uniquely represented as
for all \(f\equiv \{f_j \} \in L_{-\frac{n}{r}}(\ell _q,{\mathbb R}^n) \hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\), where
Moreover, if \(\{g_j\}\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\), then
where the supremum is taken over all \(f \equiv \{f_j\} \in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with \(\left\| f\left| L^r_{p}(\ell _q,\mathbb {R}^n)\right. \right\| \le 1\).
Proof
It follows from (9) that any \(\{g_j\}\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) induces a bounded linear functional on \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\).
Conversely, suppose g is a bounded linear functional on \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with norm \(\left\| g\right\| \). We observe that
where \(f^j_{JM}\equiv f_j \chi _{Q_{JM}}\), \(\mu _J({\mathrm d}x)\equiv 2^{J(n +pr)}\) and
This shows that \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) is isomorphic to a closed subspace of \(c_0\left( L_p(\ell _q,\mu _J,Q_{JM})\right) \) analogously to the scalar-valued case in [22, (4.18)–(4.20)]. More precisely, we have a linear, surjective and isometric map \(I:\{f_j\}\mapsto \{f^j_{JM}\}\) from \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) onto the closed subspace \(\{\{f^j_{JM}\} | \{f_j\}\in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\}\) of \(c_0\left( L_p(\ell _q,\mu _J,Q_{JM})\right) \) and
Hahn-Banach’s theorem yields \(g\in \left( \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\right) '\) if, and only if, \(g\in \big (c_0\big (L_p(\ell _q,\mu _J,Q_{JM})\big )\big )'\) and by Proposition 3.2 we have the representation
for any \(\{f_j\}\in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with \(\{ g^j_{JM} \}\in \ell _1\left( L_{p'}(\ell _{q'},\mu _J,Q_{JM})\right) \), where
Moreover, Hahn-Banach’s theorem implies that
Using Lebesgue’s dominated convergence theorem we deduce from (12) (cf. (13) for an integrable majorant) the representation
for \(\{f_j\}\in L_{-\frac{n}{r}}(\ell _q,{\mathbb R}^n)\). Let \(\varepsilon >0\). For \(h^j_{JM}\equiv g^j_{JM} \chi _{Q_{JM}} 2^{J(n +pr)}\) we obtain
Therefore \(\{\lambda _{JM}\}_{J,M} \in \ell _1\) and for an appropriate choice of \(g^j_{JM}\) we obtain also \(\left\| \lambda |\ell _1\right\| \le (1+\varepsilon ) \left\| g\right\| \). For \(a^j_{JM}\) given by \(h^j_{JM}=\lambda _{JM} a^j_{JM}\) we have then \(\left\| \left. a^j_{JM} \right| L_{p'}(\ell _{q'},{\mathbb R}^n)\right\| \le 2^{J\left( \frac{n}{p}+r\right) }\) with \(\mathrm {supp \,}\left( a^j_{JM}\right) \subset {Q_{JM}}\). Finally, it holds \(\left\{ \sum _{J\in \mathbb {Z},M\in \mathbb {Z}^n} \lambda _{JM} a^j_{JM}\right\} _j\in H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) and
Indeed, we have
using \(b_{JM} \equiv \left( \sum _{j\in \mathbb {N}_0} \left| a^j_{JM}\right| ^{q'} \right) ^\frac{1}{q'}\) with \(\mathrm {supp \,}\left( b_{JM}\right) \subset {Q_{JM}}\) and \(\left\| b_{JM}|L_{p'}({\mathbb R}^n)\right\| \le 2^{J\left( \frac{n}{p}+r\right) }=2^{-J\left( \frac{n}{p'}+\varrho \right) }\). Finally,
By the same argumentation we obtain also
Together with \(\{f_j\}\in L_{-\frac{n}{r}}(\ell _q,{\mathbb R}^n)\) and Hölder’s inequality
is an integrable majorant. Moreover, we observe \(L_{-\frac{n}{r}}(\ell _q,{\mathbb R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\). Indeed, for \(\{f_j\}\in L_{-\frac{n}{r}}(\ell _q,{\mathbb R}^n)\) there is a sequence \(\{f^k_j\}_j\) tending to \(\{f_j\}\) in \(L_{-\frac{n}{r}}(\ell _q,{\mathbb R}^n)\) as \(k\rightarrow \infty \) with \(f_j^k\in D({\mathbb R}^n)\) and \(f_j^k=0\) for \(j>k\) (and \(f_j^k \nearrow f_j\) as \(k\rightarrow \infty \)) which also implies \(\{f^k_j\}_j\rightarrow \{f_j\}\) in \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) as \(k\rightarrow \infty \) by \(L_{-\frac{n}{r}}({\mathbb R}^n)\hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \). \(\square \)
4 Mapping Properties of Operators
4.1 The Main Theorem
Next we extend the approach developed in [21] and [22] to a wider class of operators and to vector-valued spaces.
Proposition 4.1
Let \(1<p<\infty \), \(-\frac{n}{p}< r<0\) and \(1<q<\infty \). Then \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) coincides with the completion of finite sequences of continuous compactly supported functions. More precisely, it holds
Proof
Let \(\{f_j\}\in \overline{C_0(\ell _q, \mathbb {R}^n)}^{\Vert \cdot | L^r_{p}\left( \mathbb {R}^n\right) \Vert }\). Let \(\varepsilon >0\). Then there exists a sequence \(\{g_j\}\) with \(g_j\in C({\mathbb R}^n)\) compactly supported with \(g_j=0\) for \(|j|>k\) and some \(k\in \mathbb {N}\) such that \(\left\| f_j -g_j|L^r_{p}(\ell _q,\mathbb {R}^n)\right\| <\varepsilon \). Let \(x\in \mathbb {R}^n\), \(R>0\). Let \(\bar{R}>1\) such that \(\mathrm {supp \,}\sum _{j=0}^k |g_j|^q \subset B_{\bar{R}-1}(0)\). Let \(y\in {\mathbb R}^n\) with \(|y|<1\). Moreover, for \(R\ge \bar{R}\)
whenever
which holds by the uniform continuity of \(g_j\), \(j=0,\ldots ,k\), for \(|y|<\delta =\delta (\varepsilon ,\bar{R},r,g_0,\ldots ,g_k)\), where c is a constant depending on n. Furthermore, for \(R< \bar{R}\) again by (14)
Let \(\psi \in D({\mathbb R}^n)\) with \(\mathrm {supp \,}\psi \subset B_1(0)\), \(\int _{{\mathbb R}^n} \psi (y){\mathrm d}y=1\), \(0\le \psi \le 1\) and \(\psi _l(\cdot )\equiv l^n\psi (l\cdot )\), \(l\in \mathbb {N}\). Then it holds \(\left\| \{g_j*\psi _l -g_j\}_j|L^r_{p}(\ell _q,\mathbb {R}^n)\right\| <\varepsilon \) for l sufficient large where \(g_j*\psi _l \in D({\mathbb R}^n)\), \(j\in \mathbb {N}_0\). Indeed, by means of Minkowski’s inequality and the properties of \(\psi _l\) we find
where l is sufficiently large (depending on \(\varepsilon \)). \(\square \)
Remark 4.2
In the last Proposition we adapted the proof in scalar-valued case \(L^r_{p}\left( \mathbb {R}^n\right) \) given in [30, Proposition 3] to the vector-valued situation.
Theorem 4.3
Let \(1<p<\infty \), \(-\frac{n}{p}< r<0\), \(r+\varrho =-n\), \(1<q<\infty \) and let \(\left\{ T_j\right\} _{j\in \mathbb {N}_0}\) be a sequence of operators with the following properties:
-
(i)
\(T_j:D({\mathbb R}^n)\rightarrow C({\mathbb R}^n)\), \(j\in \mathbb {N}_0\), and \(T_j\), \(j\in \mathbb {N}_0\), are
-
(a)
either linear or
-
(b)
$$\begin{aligned}&\left( T_j (f_1 +f_2)\right) (y) \le (T_j f_1)(y)+(T_j f_2)(y),\nonumber \\&(T_j f)(y) = \left( T_j (-f)\right) (y) ,\ T_j 0=0 \end{aligned}$$(15)
for f, \(f_1\), \(f_2 \in D({\mathbb R}^n)\) and \( y\in {\mathbb R}^n\);
-
(a)
-
(ii)
we have
$$\begin{aligned} |(T_j f)(y)|\le c_1 \int _{\mathbb {R}^n} \frac{|f(z)|}{|y-z|^n} dz \end{aligned}$$(16)for all \(f\in D({\mathbb R}^n)\) and all \(y\notin \mathrm {supp \,}(f)\), where \(c_1\) does not depend on \(j\in \mathbb {N}_0\), f and y;
-
(iii)
there is a constant \(c_2\) such that
$$\begin{aligned} \left\| T_j f_j|L_{p}(\ell _q, {\mathbb R}^n)\right\| \le c_2 \left\| f_j|L_{p}(\ell _q, {\mathbb R}^n)\right\| \end{aligned}$$for all \(\{f_j\}_{j\in \mathbb {N}_0}\subset D({\mathbb R}^n)\).
Then, the following statements hold true.
-
(1)
There are unique continuous and bounded extensions \(\widetilde{T_j}\) of \(T_j\) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) for \(j\in \mathbb {N}_0\) such that
$$\begin{aligned} \left\{ \widetilde{T_j}\right\} _{j\in \mathbb {N}_0}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n). \end{aligned}$$ -
(2)
If \(T_j\) are linear for \(j\in \mathbb {N}_0\), then the dual operators of the unique linear and bounded extensions \(\widetilde{T_j}\) of \(T_j\) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \), \(\widetilde{T_j}': H^{\varrho } L_{p'} ({\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} ({\mathbb R}^n)\), satisfy
$$\begin{aligned} \left\{ \widetilde{T_j}'\right\} _{j\in \mathbb {N}_0}: H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n). \end{aligned}$$If the extensions of \(T_j\) to \(L_p({\mathbb R}^n)\) due to assumption (iii) are formally self-adjoint for all \(j\in \mathbb {N}_0\), then \(\widetilde{T_j}'\) are the unique linear and bounded extensions of \(T_j\) acting in \(H^{\varrho } L_{p'} ({\mathbb R}^n)\).
-
(3)
If \(T_j\) are linear for \(j\in \mathbb {N}_0\), then there are linear and bounded extensions \(\widetilde{T_j}\) of \(T_j\) to \(L^r_{p}\left( \mathbb {R}^n\right) \) such that
$$\begin{aligned} \left\{ \widetilde{T_j}\right\} _{j\in \mathbb {N}_0}: L^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n). \end{aligned}$$
Proof
Step 1 We start showing Assertion (1).
At first we will show that \(\left\{ T_j\right\} _{j\in \mathbb {N}_0}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n)\). Let \(\{f_j\}_{j=0}^\infty \in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with \(f_j\in D({\mathbb R}^n)\) for all j. Let \(x\in \mathbb {R}^n\) and \(R>0\). We decompose
where \(f_j^0\equiv \varphi _0 f_j\) and \(f_j^i\equiv \varphi _i f_j\) for \(i,j\in \mathbb {N}\) with \(\{\varphi _i\}_{i\in \mathbb {N}_0}\subset D({\mathbb R}^n)\) such that
and
By means of (iii) we obtain
Let \(i \in \mathbb {N}\) and \(y\in B_R(x)\). It follows from (16) that
Hölder’s inequality yields
By subadditivity of the operators we obtain
where c does not depend on \(\{f_j\}_j\). We get the unique continuous extension \(T:\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n)\) of \(\{T_j \}_j\) whenever \(T_j\) are linear. If \(T_j\) fulfills (15), then we have for \(f_1\), \(f_2\in D({\mathbb R}^n)\), \(y\in {\mathbb R}^n\), \(j\in \mathbb {N}_0\)
and hence for \(\{f_j\}\), \(\{\widetilde{f_j}\} \in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with \(f_j\), \(\widetilde{f_j}\in D({\mathbb R}^n)\) for all \(j\in \mathbb {N}_0\)
Therefore, \(\{T_j \}_j\) is (Lipschitz-)continuous and moreover we get the unique continuous extension \(T:\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n)\) of \(\{T_j \}_j\) using (17) in the same way as in the linear case.
Step 2 It remains to justify that also \(T: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\). By means of a density argument we may assume that \(\{f_j\}_j\in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with \(f_j\in D({\mathbb R}^n)\) for all \(j\in \mathbb {N}_0\) and a k such that \(f_j=0\) for \(|j|>k\). There is an \(\bar{R}\) such that \(\mathrm {supp \,}f_j \subset B_{\bar{R}}(0)\) for all \(j\in \mathbb {N}_0\). Then
using (16) and triangle inequality. Here the constant c depends on \(\{f_j\}_j\). Let \(R\ge 2 \bar{R}\). Then one has for cubes \(Q_{JM}\) with \(Q_{JM} \subset \{ x\in {\mathbb R}^n: \, |x| >R \}\),
Using in addition \(1<p<\infty \) we obtain
if \(-J\in \mathbb {N}\). Let \(\psi _R\in D({\mathbb R}^n)\) be a smooth cut-off function with \(\psi _R (x) =1\) if \(|x| \le R\). Then \(\psi _R \, T_j f_j \in C({\mathbb R}^n)\) compactly supported for \(0\le j\le k\) (by triangle inequality using \(T_j:D({\mathbb R}^n)\rightarrow L_\infty ({\mathbb R}^n)\)) and it follows
with \(1<p<\infty \) and \(0 < \frac{n}{p} +r <\frac{n}{p}\). Here one should mention that cubes which are not completely inside of \(\{ x\in {\mathbb R}^n: \, |x| >R \}\) are treated analogously and that \(T_j f_j=0\) for \(j>k\) by \(T_j 0=0\). Hence \(\{T_j f_j\}\in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) by Proposition 4.1 and therefore \(T: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) by the unique extension of \(\{T_j\}_j\) to \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\). Futhermore, we observe that the projection of T to its \(k-th\) component (\(k\in \mathbb {N}_0\)) coincides with \(\widetilde{T_k}\) where \(\widetilde{T_k}:\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow \overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) is the unique continuous and bounded extension of \(T_k\). This yields Assertion (1).
Step 3 Finally, Assertions (2) and (3) follow by duality (Theorems 3.1 and 3.3). As for the abstract background of duality one may consult [28, pp. 112/113] and [18, pp. 35/36]. We get firstly \(\widetilde{T_j}':H^{\varrho } L_{p'} ({\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} ({\mathbb R}^n)\) and \(\left( \widetilde{T_j}'\right) ':L^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \) for all \(j\in \mathbb {N}_0\). Moreover, by the linearity of \(T= \{\widetilde{T_j}\}_{j\in \mathbb {N}_0}:\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) duality also implies \(T': H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) as well as \((T')': L^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n)\). The projection of \(T'\) to its \(k-th\) component (\(k\in \mathbb {N}_0\)) coincides with \(\widetilde{T_k}'\). Indeed, let \(f_j\equiv g_j\equiv 0\) for all \(j\ne k\) and let \(f_k, g_k\in D({\mathbb R}^n)\). By means of the definition of \(T'\) and \(\widetilde{T_k}'\) we have
for \(\alpha <-n/p\). Analogously, using \(H^{\varrho } L_{p'} ({\mathbb R}^n)\hookrightarrow L_{-n/\varrho }({\mathbb R}^n)\) we deduce that the projection of \((T')'\) to its \(k-th\) component (\(k\in \mathbb {N}_0\)) coincides with \(\left( \widetilde{T_k}'\right) '\) on \(D({\mathbb R}^n)\).
Moreover, we assume that the extensions of \(T_j\) to \(L_p({\mathbb R}^n)\) due to assumption (iii) are formally self-adjoint for all \(j\in \mathbb {N}_0\). Then we obtain
for all \(f,g\in D({\mathbb R}^n)\) and \(\alpha <-n/p\). Therefore, \(\widetilde{T_j}'g= T_j g\) almost everywhere for all \(g\in D({\mathbb R}^n)\) and \(j\in \mathbb {N}_0\) which means that \(\widetilde{T_j}'\) are extensions of \(T_j\) to \(H^{\varrho } L_{p'} ({\mathbb R}^n)\). Moreover, the biduals \(\widetilde{T_j}'' = \left( \widetilde{T_j}'\right) '\), \(j\in \mathbb {N}_0\), are extensions of \(T_j\) to \(L^r_{p}\left( \mathbb {R}^n\right) \) by
for all \(f,g\in D({\mathbb R}^n)\), \(u=-{n}/{\varrho }\) and \(j\in \mathbb {N}_0\). Therefore, \(\widetilde{T_j}'' = T_j\) on \(D({\mathbb R}^n)\) for \(j\in \mathbb {N}_0\). \(\square \)
Remark 4.4
The extension in Part 3 of the theorem is not unique. There exist infinitely many extensions of \(T_j\) acting in \(L^r_p({\mathbb R}^n)\). This can be seen following the same arguments as in [22, Remark 5.3]. Assumption (iii) can be replaced by
if \(T_j=T\) for all j, T is linear and if q is between 2 and p (including 2 and p) which holds by the fact that the \(L_p\)-boundedness of a linear operator implies the \(L_{p}(\ell _q,{\mathbb R}^n)\)-boundedness for \(q\in [p,2]\) for \(p\le 2\) and \(q\in [2,p]\) for \(p>2\) cf. [10, Corollary 4.5.4].
Remark 4.5
There are a lot of papers dealing with singular integrals in Morrey spaces. However, its well-definedness on the Morrey-type spaces under consideration as well as the norm estimates in these spaces have to be treated with greater care than usually done. On the one-hand one has to investigate how to extend singular integrals to Morrey spaces and on the other hand the estimates (16) are not available in general for functions belonging to Morrey spaces. Let us emphasize that we used (16) just for functions of \(D({\mathbb R}^n)\). The question if the estimate (16) holds for some singular integrals also for all \(f\in L^r_{p}\left( \mathbb {R}^n\right) \) leads to an investiation of its maximal truncated versions (cf. [27, Proposition 2.25, Remark 2.26] as well as Sects. 4.2.2 and 4.3). Indeed, for these reasons in many papers one can only find the weaker mapping property \(T:\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \) (see, for example [5, 17], and [6, 12, 16] for operators satisfying (16)). In this sense our results on \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) and \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) are new (including even the boundedness of the Hardy-Littlewood maximal operator), in particular with respect to their generality. Note that results for Calderón–Zygmund operators in \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) (scalar case) have been proved already in [21] and [22]. Moreover, some results in \(H^{\varrho } L_{p'} ({\mathbb R}^n)\) and in \(H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) and \(L^r_{p}(\ell _q,\mathbb {R}^n)\) seem to be new. The paper [2] made the important observation that the bidual of the completion of \(D({\mathbb R}^n)\) with respect to the Morrey norm coincides with the Morrey space itself (cf. (1)) and provided the basis of our investigations. To overcome the above mentioned problems investigating Calderón–Zygmund operators in \(L_p^r({\mathbb R}^n)\) they considered Muckenhoupt weighted characterizations of Morrey spaces and their preduals. However, this approach has also some weak points with respect to norm estimates since it does not take into account that the operator norm of classical operators of harmonic analysis (as the Hilbert transform) in Muckenhoupt weighted spaces usually depends on the Muckenhoupt weight.
We want to refer also to a less known forerunner result which can be found in [3]. There a solution for the above mentioned difficulties has been given for some Calderón–Zygmund operators in \(H^{\varrho }L_p ({\mathbb R}^n)\).
4.2 Calderón–Zygmund Operators
4.2.1 Duality Approach
Definition 4.6
We define Calderón–Zygmund operators with homogeneous kernels with degree \(-n\) setting,
where \(f\in S({\mathbb R}^n)\) and \(\Omega \in L_\infty ({\mathcal S}^{n-1})\) with zero integral and \({\mathcal S}^{n-1}\) denotes the unit sphere.
Corollary 4.7
Let \(1<p<\infty \), \(-\frac{n}{p}\le r<0\), \(-n < \varrho < -\frac{n}{p'}\), \(1< q<\infty \). Then the following statements hold true.
-
(1)
There are unique linear and bounded extensions of \(T^\Omega \) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) and to \(H^{\varrho } L_{p'} ({\mathbb R}^n)\) denoted again by \(T^\Omega \) such that
$$\begin{aligned}&\left\{ T^\Omega \right\} _{j\in \mathbb {N}_0}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\quad \text {and}\quad \\&\left\{ T^\Omega \right\} _{j\in \mathbb {N}_0}: H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n). \end{aligned}$$ -
(2)
There are infinitely many linear and bounded extensions of \(T^\Omega \) to \(L^r_{p}\left( \mathbb {R}^n\right) \) denoted again by \(T^\Omega \) such that
$$\begin{aligned} \left\{ T^\Omega \right\} _{j\in \mathbb {N}_0}: L^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n). \end{aligned}$$
Proof
We observe that \(T^\Omega : D({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\). Indeed, by the same arguments as in [21, proof of Step 2 of Theorem 1.1, p. 8] we get the mapping properties \(T^\Omega : W_p^k({\mathbb R}^n)\hookrightarrow W_p^k({\mathbb R}^n)\) for Sobolev spaces which lead to the above assertion by means of Sobolev type embeddings. Moreover, for \(\Omega \in L_\infty ({\mathcal S}^{n-1})\) it holds \(T^\Omega :L_p(\ell _q,{\mathbb R}^n)\hookrightarrow L_p(\ell _q,{\mathbb R}^n)\) by [7]. Now we obtain \(T^\Omega : L^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n)\) for \(\Omega \in L_\infty ({\mathcal S}^{n-1})\) applying Theorem 4.3. Note that the dual of the extension of \(T^{\Omega (-\cdot )}\) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) coincides with \(T^{\Omega }\) on \(D({\mathbb R}^n)\) by the same arguments as in (19). \(\square \)
4.2.2 Alternative Approach Using Some Muckenhoupt Weights
The following alternative method due to Triebel [27, Sect. 2.5.3, Proposition 2.25, Remark 2.26] yields extensions of Calderón–Zygmund operators which are bounded in \(L^r_{p}\left( \mathbb {R}^n\right) \). He studied the boundedness of \(T^{\Omega }\) with \(\Omega \in C^1({\mathcal S}^{n-1})\). Here we generalize his approach to some non-convolution type Calderón–Zygmund operators. At first we observe that Morrey spaces \(L^r_{p}\left( \mathbb {R}^n\right) \) are continuously embedded into some Muckenhoupt weighted \(L_p\)-spaces. Recall that \(w_\alpha (\cdot )=(1+|\cdot |^2)^\frac{\alpha }{2}\), and that \(L_p ({\mathbb R}^n, w_\alpha )\) be the corresponding weighted \(L_p\)-space, normed as in (5).
Proposition 4.8
([27, Proposition 2.10]) Let \(1<p<\infty \), \(-\frac{n}{p}\le r<0\), \(-n<\alpha \,p<-n-rp\). Then it holds
Proof
Let \(f\in L^r_{p}\left( \mathbb {R}^n\right) \). Then (20) follows from
\(\square \)
Theorem 4.9
Let \(1<p<\infty \), \(-\frac{n}{p}< r<0\), \(-n < \varrho < -\frac{n}{p'}\). Let T be an operator with domain \(D({\mathbb R}^n)\) satisfying
where the constant \(c_1\) is independent of \(f\in D({\mathbb R}^n)\) and
almost everywhere for all \( f\in D({\mathbb R}^n)\), where the function \(K(\cdot ,\cdot )\) defined \({\mathbb R}^n\times {\mathbb R}^n\setminus \{(x,x):x\in {\mathbb R}^n\} \) satisfies the conditions \(|K(x,y)|\le c_2 |x-y|^{-n}\) and
Then the following statements hold true.
-
(1)
There are linear and bounded extensions of T to \(L^r_{p}\left( \mathbb {R}^n\right) \).
-
(2)
There is an unique linear and bounded extension of T to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) and to \(H^{\varrho } L_{p'} ({\mathbb R}^n)\).
Proof
By [11, Corollary 9.4.7] there is an unique linear and bounded extension \(\widetilde{T}\) of T to \(L_p (w_\alpha ,{\mathbb R}^n)\) with \(-n<\alpha \,p< n(p-1) \). Therefore, Proposition 4.8 yields \(\widetilde{T}:L^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow L_p (w_\alpha ,{\mathbb R}^n)\). We have even
by [11, Theorem 9.4.6]. Together with (21) we see that
almost everywhere for all \(f\in L_p (w_\alpha ,{\mathbb R}^n)\) by [11, Theorem 2.1.14]. Now (16) holds for all \(f\in L^r_{p}\left( \mathbb {R}^n\right) \) with \(y\notin \mathrm {supp \,}f\). As in Step 1 of the proof of Theorem 4.3 we obtain \(\widetilde{T}:L^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \) that is Assertion (1). In particular, \(\widetilde{T}:\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \). For \(\alpha =0\) by [11, Corollary 9.4.7] we have especially
for all \(f\in D({\mathbb R}^n)\) where the constant c does not depend on f. Hence, \(T:D({\mathbb R}^n)\rightarrow L_{-{n}/{r}}({\mathbb R}^n)\). Because of the embedding \(L_{-{n}/{r}}({\mathbb R}^n) \hookrightarrow L^r_{p}\left( \mathbb {R}^n\right) \) and the density of \(D({\mathbb R}^n)\) in \(L_{-{n}/{r}}({\mathbb R}^n)\) we even have \(T: D({\mathbb R}^n) \rightarrow \overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \). Indeed, let \(f\in D({\mathbb R}^n)\). Then \(Tf\in L_{-{n}/{r}}({\mathbb R}^n)\) and thus there is a sequence of functions of \(D({\mathbb R}^n)\) which tends to Tf in \(L_{-{n}/{r}}({\mathbb R}^n)\) and hence in \(L^r_{p}\left( \mathbb {R}^n\right) \) which shows \(Tf \in \overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \). Thus, \(\widetilde{T}:\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow \overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \). The adjoint kernel of K(x, y) given by \(\overline{K(y,x)}\) also satisfies the required assumptions on the kernel. Hence, its corresponding operator is also bounded in \(L_{p}({\mathbb R}^n)\) (cf. [11, Definition 8.1.2]) but its dual coincides by the same arguments as in Step 3 of the proof of Theorem 4.3 with the operator T (with the kernel K(x, y)) on \(D({\mathbb R}^n)\) which implies Assertion (2).
Remark 4.10
Let us point out that for this method the embedding of \(L^r_{p}\left( \mathbb {R}^n\right) \) in some Muckenhoupt weighted space is crucial for extending the domain of the considered Calderón–Zygmund operators to \(L^r_{p}\left( \mathbb {R}^n\right) \). Recall the fact that the Hilbert transform is acting in \(L_p (w_\alpha ,{\mathbb R}^n)\) if, and only if, \(w_\alpha \) is a Muckenhoupt weight. Moreover, we needed
almost everywhere for all \( f\in L^r_{p}\left( \mathbb {R}^n\right) \). This is a rather deep result in comparison to the \(L_p\)-boundedness which we require in Theorem 4.3. Finally, let us emphasize again that the extension in Part 1 is by no means unique.
4.3 Vector-Valued Maximal Inequalities and Maximal Calderón–Zygmund Operators
Definition 4.11
We define maximal Calderón–Zygmund operators with homogeneous kernels with degree \(-n\) by setting
where \(f\in \bigcup _{1\le p<\infty } L_p({\mathbb R}^n)\) and \(\Omega \in L_\infty ({\mathcal S}^{n-1})\) with zero integral and \({\mathcal S}^{n-1}\) denotes the unit sphere. As usual, the Hardy-Littlewood maximal operator M is given by
Remark 4.12
If \(f\in \bigcup _{1\le p<\infty } L_p({\mathbb R}^n)\) then
is bounded for each \(\varepsilon >0\) and \(y\in {\mathbb R}^n\) by Hölder’s inequality. Hence \((T^\Omega _* f)(y)\) is well-defined for all \(y\in {\mathbb R}^n\), but might be infinite.
Corollary 4.13
Let \(1<p<\infty \), \(-\frac{n}{p}\le r<0\), \(-n < \varrho < -\frac{n}{p'}\), \(1< q<\infty \). Then
Moreover, if \(\Omega \in C^1({\mathcal S}^{n-1})\), then
Proof
At first we show that \(M:D({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\). Let \(f\in D({\mathbb R}^n)\) and \(f_h(\cdot )\equiv f(\cdot +h)\) for \(h\in {\mathbb R}^n\). By sublinearity of M we obtain
It follows that
where L is the Lipschitz constant of f and \(x\in {\mathbb R}^n\) (we even showed \(M:\text {Lip}({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\) with the arguments due to [15, Remark 2.2]). A version of Cotlar’s inequality leads to the estimate
for \(x\in {\mathbb R}^n\) (cf. [8, Lemma 5.15]). As above we obtain
Together with \(T^\Omega : D({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\) (cf. proof of Corollary 4.7) it follows from \(M:\text {Lip}({\mathbb R}^n) \rightarrow \text {Lip}({\mathbb R}^n)\) also that \(T^\Omega _*:D({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\). Moreover, we claim that (16) holds also for M. Indeed, let \(f\in D({\mathbb R}^n)\) with \(y\notin \mathrm {supp \,}(f)\). Then there exists an \(i\in \mathbb {Z}\) such that \(B_{2^i}(y)\cap \mathrm {supp \,}f = \emptyset \). Let \({f}^j\equiv \chi _{{B_{2^{j+1}}(y)}\setminus {B_{2^{j}}(y)}}{f}\) for \(j\ge i\). Hence,
Now Theorem 4.3 implies the existence of unique continuous and bounded extensions of M to \(\overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) and of \(T^\Omega _*\) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \). Since (16) also holds for M and all \(f\in L^r_{p}\left( \mathbb {R}^n\right) \) in place of all \(f\in D({\mathbb R}^n)\) we achieve at (22) as in Step 1 of the proof of Theorem 4.3. Moreover, \(T^\Omega _*\) is also well-defined on \(L_p (w_\alpha ,{\mathbb R}^n)\) if \(-n<\alpha \,p<n(p-1)\) by [11, Theorem 9.4.6] (and not only on \(\bigcup _{1\le p<\infty } L_p({\mathbb R}^n)\)) and hence on \(L^r_{p}\left( \mathbb {R}^n\right) \) by Proposition 4.8. Thus, (16) also holds for \(T^\Omega _*\) and all \(f\in L^r_{p}\left( \mathbb {R}^n\right) \) in place of all \(f\in D({\mathbb R}^n)\). This yields (23). \(\square \)
4.4 Fourier Multipliers
4.4.1 Multipliers Generated by Characteristic and Smooth Functions
Corollary 4.14
Let \(1<p<\infty \), \(-\frac{n}{p}\le r<0\), \(-n < \varrho < -\frac{n}{p'}\), \(1< q<\infty \). Let \(\{I_j\}_{j\in \mathbb {N}_0}\) be a sequence of intervals on the real line, finite or infinite, and let \(\{ S_j\}_j\) be the sequence of operators defined by
Moreover, let \(\psi \in S\left( \mathbb {R}^n\right) \) with \(\psi (0)=0\). We define
Then the following statements hold true.
-
(1)
There are unique linear and bounded extensions of \(S_j\) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) and to \(H^{\varrho } L_{p'} ({\mathbb R}^n)\) denoted again by \(S_j\) and satisfying the mapping properties
$$\begin{aligned}&\left\{ S_j\right\} _{j\in \mathbb {N}_0}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R})\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R})\quad \text {and}\quad \\&\left\{ S_j\right\} _{j\in \mathbb {N}_0}: H^{\varrho } L_{p'} (\ell _{q'},\mathbb {R})\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},\mathbb {R})\end{aligned}$$ -
(2)
There are infinitely many linear and bounded extensions of \(S_j\) to \(L^r_{p}\left( \mathbb {R}^n\right) \) denoted again by \(S_j\) such that
$$\begin{aligned} \left\{ S_j\right\} _{j\in \mathbb {N}_0}: L^r_{p}(\ell _q,\mathbb {R})\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}). \end{aligned}$$ -
(3)
We have the mapping properties
$$\begin{aligned}&\left\{ \tilde{S_j}\right\} _{j\in \mathbb {Z}}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n), \quad \\ {}&\left\{ \tilde{S_j}\right\} _{j\in \mathbb {Z}}: H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n), \end{aligned}$$and
$$\begin{aligned} \left\{ \tilde{S_j}\right\} _{j\in \mathbb {Z}}: L^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n)~. \end{aligned}$$Here we used the notation \(\ell _q=\ell _q(\mathbb {Z})\).
Proof
Part (1) and part (2) are consequences of Theorem 4.3 (see also Corollary 4.7). The required \(L_{p}(\ell _q,{\mathbb R}^n)\)-boundedness follows from [8, Corollary 8.2]. Alternatively, it suffices the \(L_{p}(\ell _q,{\mathbb R}^n)\)-boundedness of the Hilbert transform \((H f)(y)\equiv \frac{1}{\pi } \lim _{\varepsilon \searrow 0} \int _{|z-y|\ge \varepsilon } \frac{f(z)}{y-z} {\mathrm d}z, f\in S({\mathbb R}^n)\) (see e.g., [10, Corollary 4.6.3]) This can be seen using the formula
where \(I_j=(a_j,b_j)\) (with the obvious modifications if the interval is unbounded) and where \(M_{a} f (\cdot )\equiv e^{2\pi i a\cdot } f (\cdot )\) [8, (3.9)]. Now the desired result follows from Theorem 4.3 (for \(n=1\)) taking into account also that the dual of the extension of the multiplier generated by \(-I_j\) coincides with \(S_j\) on \(D({\mathbb R}^n)\) by the same arguments as for (19).
Moreover, we observe that
This follows, for example, from [8, (8.1), p. 158]. The needed Hörmander condition is fulfilled by (26). Indeed, we have
(cf. [8, p. 161]). Hölder’s inequality yields
and furthermore using (25)
for \(|x|\ge 2 |y|\). Using (24) we find \(\left\{ \tilde{S_j}\right\} _{j\in \mathbb {Z}}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) and \(\left\{ \tilde{S_j'}\right\} _{j\in \mathbb {Z}}: H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\) by means of Theorem 4.3. Here we have to show that in particular assumption (16) is fulfilled. If \(\hat{\Psi }\equiv \psi \) and \(\Psi _j(\cdot )\equiv 2^{jn}\Psi (2^j \cdot )\), then \(\hat{\Psi }_j=\psi _j\) and \(\tilde{S_j} f=\Psi _j*f \in C^\infty ({\mathbb R}^n) \cap S'({\mathbb R}^n)\) by \(L^r_{p}\left( \mathbb {R}^n\right) \hookrightarrow S'({\mathbb R}^n)\). In particular, \(\tilde{S_j} f=\Psi _j*f\) makes sense pointwise for all \(f\in L^r_{p}\left( \mathbb {R}^n\right) \). Furthermore,
for all \(x\in {\mathbb R}^n\) and all \(f\in L^r_{p}\left( \mathbb {R}^n\right) \) with \(x\notin \mathrm {supp \,}(f)\) where \(\left\| \left\{ \Psi _j (\cdot )\right\} |\ell _2\right\| \le c |\cdot |^{-n}\) (cf. [8, p. 161]). This implies (16). We note that dual of the extension of the multiplier of \(\psi _j(-\cdot )\) coincides with \(\tilde{S_j}\) on \(D({\mathbb R}^n)\) by the same arguments as for (19). Hence,
We obtain
with the same norm estimates as in Theorem 4.3. Hereby we emphasize that the operator \(\tilde{S_j}\) is well-defined on \(S'({\mathbb R}^n)\), in particular on \(L^r_{p}\left( \mathbb {R}^n\right) \). Moreover, (16) holds for \(f\in L^r_{p}\left( \mathbb {R}^n\right) \) in place of \(f\in D({\mathbb R}^n)\). \(\square \)
Remark 4.15
The vector-valued Fourier multiplier assertion proved Corollary 4.14 paves the way to introduce predual Morrey versions \(H^\varrho A^s_{p,q}({\mathbb R}^n)\) of the Besov–Triebel–Lizorkin spaces \(A^s_{p,q}({\mathbb R}^n)\). In particular it implies the independence of admitted resolutions of unity. One replaces the \(L_p({\mathbb R}^n)\)-norm in the definition of \(A^s_{p,q}({\mathbb R}^n)\) by the \(H^{\varrho } L_{p'} ({\mathbb R}^n)\)-norm in order to define \(H^\varrho A^s_{p,q}({\mathbb R}^n)\). The vector-valued Fourier multiplier assertion in Corollary 4.14 is also the key ingredient to obtain as in [25, Sect. 2.3.3] the density of \(S({\mathbb R}^n)\) in \(H^\varrho A^s_{p,q}({\mathbb R}^n)\). Moreover as in [25, Sect. 2.11.2] one can show using our vector-valued duality assertions (Theorems 3.1 and 3.3) also that the dual of \(H^\varrho A^{-s}_{p',q'}({\mathbb R}^n)\) is \(L^r A^s_{p,q}({\mathbb R}^n)\) and furthermore that the dual of the completion of \(S({\mathbb R}^n)\) with respect to \(L^r A^s_{p,q}({\mathbb R}^n)\) is \(H^\varrho A^{-s}_{p',q'}({\mathbb R}^n)\). Here \(L^r A^s_{p,q}({\mathbb R}^n)\) stands for the morreyfied versions of \(A^s_{p,q}({\mathbb R}^n)\) which are defined by replacing the \(L_p({\mathbb R}^n)\)-norm in the definition of \(A^s_{p,q}({\mathbb R}^n)\) by the \(L^r_p({\mathbb R}^n)\)-norm.
Moreover, in the one-dimensional case (\(n=1\)) Corollary 4.14 implies also Lizorkin representations of the Triebel–Lizorkin–Morrey spaces. \(L^r A^s_{p,q}(\mathbb {R})\) (cf. [25, Sect. 2.5.4]) using in addition Nikol’skij inequalities for Morrey spaces published in [20, Theorem 2.2.9, Theorem 2.2.20]. We want to mention that the spaces \(L^r A^s_{p,q}({\mathbb R}^n)\) are studied, in particular, in [13, 19, 20, 26, 29].
4.4.2 Strongly Singular Integrals
Definition 4.16
Let \(0<b<1\) and let \(\varphi \) be a smooth cut-off function with \(\varphi =1\) on \(\{|\xi |\ge 1 \}\) and \(\varphi =0\) on \(\{|\xi |\le 1/2 \}\). If \(f\in S({\mathbb R}^n)\), then we define strongly singular integrals as
(cf. [23, p. 192]).
Corollary 4.17
Let \(1<p<\infty \), \(-\frac{n}{p}\le r<0\), \(-n < \varrho < -\frac{n}{p'}\) and let \(q\in [p,2]\) for \(p\le 2\) and \(q\in [2,p]\) for \(p>2\). Then the following statements hold true.
-
(1)
There are unique linear and bounded extensions of \(T_b\) to \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) and to \(H^{\varrho } L_{p'} ({\mathbb R}^n)\) denoted again by \(T_b\) and satisfying
$$\begin{aligned}&\left\{ T_b\right\} _{j\in \mathbb {N}_0}: \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\quad \text {and}\quad \\ {}&\left\{ T_b\right\} _{j\in \mathbb {N}_0}: H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\hookrightarrow H^{\varrho } L_{p'} (\ell _{q'},{\mathbb R}^n)\end{aligned}$$ -
(2)
There are infinitely many linear and bounded extensions of \(T_b\) to \(L^r_{p}\left( \mathbb {R}^n\right) \) denoted again by \(T_b\) such that
$$\begin{aligned} \left\{ T_b\right\} _{j\in \mathbb {N}_0}: L^r_{p}(\ell _q,\mathbb {R}^n)\hookrightarrow L^r_{p}(\ell _q,\mathbb {R}^n). \end{aligned}$$
Proof
\(T_b\) satisfies (16) by [23, p. 192], see also [8, Chapter 5, Sect. 6.8]. The strongly singular integrals \(T_b\) are bounded on \(L_{p}\left( \mathbb {R}^n\right) \), \(1<p<\infty \), by [8, Sect. 6.8] and the references given there. Moreover, \(T_b:W^k_p ({\mathbb R}^n)\hookrightarrow W^k_p ({\mathbb R}^n)\) for all \(k \in \mathbb {N}\) using the lift operator (which is the Fourier multiplier corresponding to \((1+|\cdot |^2)^{\sigma /2}\) for \(\sigma \in \mathbb {R}\)) and, in particular, it follows that \(T_b f\in C^\infty \) for \(f\in D({\mathbb R}^n)\) by well-known Sobolev embeddings. Having in mind Remark 4.4 we obtain the assertion by Theorem 4.3. \(\square \)
4.4.3 Bochner–Riesz Multipliers
Definition 4.18
Let \(\lambda >0 \) and let \(f\in S({\mathbb R}^n)\). We define Bochner–Riesz multipliers as
It is well-known that \(B^\lambda f\) can be reformulated as
where \(J_\alpha \) stands for the Bessel function (cf. [8, Lemma 8.18] or [10, (10.2.1)]).
Corollary 4.19
If \(\lambda \ge (n-1)/2\) then the statements of Corollary 4.17 hold with \(B^\lambda \) in place of \(T_b\).
Proof
Let \(\lambda = (n-1)/2\) be the critical index. Using (27) as well as the estimate \(J_{n/2+\lambda }(|x|)\le {c}{|x|^{-1/2}}\) (see e.g., [10, Appendix B.6]) we see that \(B^\lambda \) satisfies (16). The \(L_p\)-boundedness of \(B^\lambda \) at the critical index for \(1<p<\infty \) is known (cf. [8, Theorem 8.15]). We also have \(B^\lambda : D({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\) by the same arguments as in [21, proof of Step 2 of Theorem 1.1, p. 8] taking into account the convolution structure of \(B^\lambda \). Thus the assertion for \(\lambda = (n-1)/2\) is a consequence of Theorem 4.3. Let \(\lambda > (n-1)/2\). Then \(|(B^\lambda f)|\) can be dominated pointwise by the Hardy-Littlewood maximal function Mf for \(f\in D({\mathbb R}^n)\) (cf. [10, Exercise 10.2.8]). Hence,
where c does not depend on \(\{f_j\}_{j=0}^\infty \in \overset{\circ }{L}{}^r_{p}(\ell _q,\mathbb {R}^n)\) with \(f_j\in D({\mathbb R}^n)\) for all j. As above we have \(B^\lambda : D({\mathbb R}^n)\rightarrow \text {Lip}({\mathbb R}^n)\). As in the proof of Theorem 4.3 we find an unique extension of \(B^\lambda \) denoted again as \(B^\lambda \) such that
Hereby, we mention that the constant in (18) is allowed to depend on the fixed sequence of functions. The proof of the reaming parts of the corollary for \(\lambda > (n-1)/2\) follows the same lines as in the proof of Theorem 4.3. \(\square \)
References
Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(6), 1629–1663 (2004)
Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50, 201–230 (2012)
Alvarez, J.: Continuity of Calderón–Zygmund type operators on the predual of a Morrey space. In: Clifford Algebras in Analysis and Related Topics, pp. 309–319. CRC Press, Boca Raton (1996)
Cambridge tracts in mathematics 120. Cambridge University Press, Cambridge (1996)
Di Fazio, G., Ragusa, M.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112, 241–256 (1993)
Ding, Y., Yang, D., Zhou, Z.: Boundedness of sublinear operators and commutators on Morrey spaces. Yokohama Math. J. 46(1), 15–27 (1998)
Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)
Edmunds, D. E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge Tracts in Mathematics, vol 120. Cambridge: Cambridge University Press, Cambridge (1996)
Gogatishvili, A., Mustafayev, R.C.: New pre-dual space of Morrey space. J. Math. Anal. Appl. 397(2), 678–692 (2013)
Grafakos, L.: Classical and modern Fourier analysis. Pearson/Prentice Hall, Upper Saddle River (2004)
Grafakos, L.: Modern Fourier analysis. Graduate texts in mathematics, vol. 250, 2nd edn. Springer, New York (2009)
Guliyev, V.S., Aliyev, S.S., Karaman, T., Shukurov, P.S.: Boundedness of sublinear operators and commutators on generalized morrey spaces. Integr. Equ. Oper. Theory 71, 327–355 (2011)
Haroske, D.D., Skrzypczak, L.: Continuous embeddings of Besov–Morrey function spaces. Acta Math. Sin. Engl. Ser. 28(7), 1307–1328 (2012)
Kalita, E.A.: Dual Morrey spaces. Dokl. Akad. Nauk 361 (1998), 447–449 (Russian); Engl. transl. Dokl. Math. 58, 85–87 (1998)
Kinnunen, J.: The Hardy-Littlewood maximal function of a Sobolev function. Isr. J. Math. 100, 117–124 (1997)
Mustafayev, R.C.: On boundedness of sublinear operators in weighted Morrey spaces. Azerb. J. of Math. 2, 66–79 (2012)
Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)
Pietsch, A.: History of Banach spaces and linear operators. Birkhäuser, Boston (2007)
Rosenthal, M.: Local means, wavelet bases and wavelet isomorphisms in Besov–Morrey and Triebel–Lizorkin–Morrey spaces. Mathematische Nachrichten 286(1), 59–87 (2013)
Rosenthal, M.: Mapping properties of operators in Morrey spaces and wavelet isomorphisms in related Morrey smoothness spaces. PhD-Thesis, Jena (2013)
Rosenthal, M., Triebel, H.: Calderón–Zygmund operators in Morrey spaces. Rev. Mat. Comp. 27, 1–11 (2013)
Rosenthal, M., Triebel, H.: Morrey spaces, their duals and preduals. Rev. Mat. Complut. 28, 1–30 (2014). doi:10.1007/s13163-013-0145-z
Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43(1), 187–204 (1994)
Triebel, H.: Interpolation theory. Function spaces. Differential operators. Deutscher Verlag des Wissenschaften, Berlin (1978)
Triebel, H.: Theory of function spaces. Birkhäuser, Basel (1983)
Triebel, H.: Local function spaces, heat and Navier–Stokes equations. European Mathematical Society, Zürich (2013)
Triebel, H.: Hybrid function spaces, heat and Navier–Stokes equations. European Mathematical Society, Zürich (2015)
Yosida, K.: Functional analysis, 6th edn. Springer, Berlin (1980)
Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato meet Besov. Lizorkin and Triebel. Springer, Heidelberg (2010)
Zorko, C.T.: Morrey spaces. Proc. Am. Math. Soc. 98, 586–592 (1986)
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Communicated by Loukas Grafakos.
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Rosenthal, M., Schmeisser, HJ. On the Boundedness of Singular Integrals in Morrey Spaces and its Preduals. J Fourier Anal Appl 22, 462–490 (2016). https://doi.org/10.1007/s00041-015-9427-9
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DOI: https://doi.org/10.1007/s00041-015-9427-9