Abstract
The purpose of this article is to extend to \(\mathbb {R}^{n}\) known results in dimension 2 concerning the structure of a Hilbert space with reproducing kernel of the space of Herglotz wave functions. These functions are the solutions of Helmholtz equation in \(\mathbb {R} ^{n}\) that are the Fourier transform of measures supported in the unit sphere with density in \(L^{2}(\mathbb {S}^{n-1})\). As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in \(\mathbb {R}^{n}\) belonging to weighted Sobolev type spaces \(\mathcal {H}^{p}\) having in a non local norm that involves radial derivatives and spherical gradients. We calculate the reproducing kernel of the Herglotz wave functions and study in \(\mathcal {H}^{p}\) and in mixed norm spaces, the continuity of the orthogonal projection \(\mathcal {P}\) of \(\mathcal {H}^{2}\) onto the Herglotz wave functions.
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1 Introduction and Preliminaries
Consider the Fourier extension operator
where \(\phi \in L^{2}(\mathbb {S}^{n-1})\), \(d\sigma \) is the Lebesgue measure in \(\mathbb {S}^{n-1}\) and \(\widehat{\cdot }\) denotes the Fourier transform in \(\mathbb {R}^{n}.\)
We have that \(W\phi \) is an entire solution (a solution in \(\mathbb {R} ^{n})\) of the Helmholtz equation
The functions \(u=W\phi \) with \(\phi \in L^{2}(\mathbb {S}^{n-1})\) called Herglotz wave functions are relevant in analysis and in particular are extensively used in scattering theory. Hartman and Wilcox in [10] proved the familiar characterization of the Herglotz functions as the entire solutions of the Helmholtz equation satisfying
The operator W is the transpose of the restriction operator for the Fourier transform, namely the operator \(Rf=\widehat{f}_{\mid _{\mathbb {S} ^{n-1}}}\) defined in the Schwartz space.
The restriction problem of Stein–Tomas asks for the values of p and q such that
The best known result for \(q=2\) is given in the Stein–Tomas theorem:
Theorem 1
(Stein–Tomas) If \(f\in L^{p}(\mathbb {R}^{n})\) with \(1\le p\le \frac{2(n+1)}{n+3}\) then
Or equivalent, if \(f\in L^{2}(\mathbb {S}^{n-1})\)
for \(q\ge \frac{2(n+1)}{n-1}\).
In [2], it was proved that the extension operator is an isomorphism of \(L^{2}(\mathbb {S}^1)\) onto the space \(\mathcal {W}^{2}\) consisting of all entire solutions of Helmholtz equation with radial and angular derivatives satisfying
This gave a new characterization of the space \(\mathcal {W}^{2}\) of Herglotz wave functions in \(\mathbb {R}^2\) as a Hilbert space with reproducing kernel. Also, for \(1<p<4/3\), it was proved that the orthogonal projection \(\mathcal {P}\) of \(\mathcal {H}^{2}\) onto \(\mathcal {W}^{2}\), can’t be extended as a bounded operator on \(\mathcal {H}^{p}\), the p-version of \(\mathcal {H}^{2}\). Then in [4], Barceló, Bennet and Ruiz proved that \(\mathcal {P}\) can’t be extended as a bounded for any \(p>1\) except for \(p\ne 2.\) However they obtained a positive result for \(4/3<p<4\), considering mixed norm spaces \(\mathcal {H}^{p,2}\), defined by
In this article, we will define Banach spaces \(\mathcal {H}^{p}\) and \(\mathcal {W}^{p}\) in \(\mathbb {R}^{n},\) \(n\ge 3,\) generalizing the mentioned spaces in [2]. \(\mathcal {H}^{p}\) will consist of all functions belonging to \(L^{p}\Big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\Big )\) jointly with their radial derivative and the euclidean norm of their spherical gradient. Then \(\mathcal {W}^{p}\) will be the closed subspace of all solutions in \(\mathcal {H}^{p}\) of the Helmholtz equation \(\Delta u+u=0\) on the Euclidean space \(\mathbb {R}^{n}\) for \(n\ge 3\) . We will construct and study the reproducing kernel for \(\mathcal {W} ^{2},\) which as for \(n=2,\) turns out to be the space of all Herglotz wave functions and it is characterized as the space of all the entire solutions of the Helmholtz equation satisfying
where \(\nabla _{S}\) denotes the spherical gradient.
In Sect. 2 we will study the space \(\mathcal {W}^{2}\). We will show that this is precisely the space of all Herglotz wave functions and we will calculate its reproducing kernel as a subspace of \(\mathcal {H}^{2}\). In Sect. 3 we consider the spaces \(\mathcal {H}^{p}\) and \(\mathcal {W}^{p}\) for exponents \(p>1\). We will prove that these are Banach spaces and we will show that the reproducing kernel of \(\mathcal {H}^{2}\) has also reproducing properties for \(\mathcal {W}^{p}\). Finally, in Sect. 4 we study the continuity properties of the orthogonal projection \(\mathcal {P}\) of \(\mathcal {H}^{2}\) onto \(\mathcal {W}^{2}\) in mixed-normed spaces \(\mathcal {H}^{p,2}\) extending the results in [4] for \(n=2\). Then we consider the continuity of \(\mathcal {P}\) in \(\mathcal {H}^{p}\). As in \(n=2\) this continuity is related to the boundedness in \(L^{p}\Big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\Big )\) of a singular operator T acting on vector fields and given by
where \(\mathbf {P}_z\) denotes the orthogonal projection in the direction of z and \(J_{n/2+1}\) is the Bessel function of the first kind. Finally we give a non-boundedness result of T in \(\mathbb {R}^3\).
Throughout paper we will use the following notations and results: \(\mathcal {B} _{R}\subset \mathbb {R}^n\) denotes the open ball with center at the origin and radius R, \(\mathcal {B}=\mathcal {B}_{1}\), and \(\mathbb {S}^{n-1}\) is the \((n-1)-\) dimensional unit sphere with surface area \(\sigma _{n}=\frac{2\pi ^{n/2}}{\Gamma (n/2)}\). \(\Delta _{S}\) denotes the Laplacian on \(\mathbb {S}^{n-1} \), that is, the Laplace Beltrami operator on \(\mathbb {S}^{n-1}\) and \(\nabla _{S}\) will be the spherical gradient. The conjugate exponent of p will be denoted by \(p'\).
Throughout this article c and C will denote generic positive constants that may change in each occurrence.
As usual, if \(\mu \) is a Borel measure in \(\mathbb {R}\), \(M_\mu f\) will denote the Hardy–Littlewood maximal function of a locally integrable function f on \(\mathbb {R}\):
where the supremum is taken over intervals \(I\subset \mathbb {R}\). Let w be a weight in \(\mathbb {R}\), namely a non-negative function in \(L^{1}_{loc}(\mu )\). By \(A_p(\mu )\) we will denote the Muckenhoupt classes. We say that w is an \(A_p(\mu )\) weight (\(w\in A_p(\mu )\)) if
for \(1<p<\infty \) and
when \(p=1,\) where C is always independent of I.
We have \(A_p(\mu )\subset A_q(\mu )\), \(1\le p<q\), in particular, \(A_1(\mu )\subset A_2(\mu )\), see [8].
We denote by \(J_{\nu }\) the Bessel functions of the first kind of order \(\nu \)
The Bessel functions satisfy the following recurrence formulas:
-
\(\left( R1\right) \) \(J_{\nu -1}(z)-J_{\nu +1}(z)=2J_{\nu }^{'}(z)\).
-
\(\left( R2\right) \) \(J_{\nu -1}(z)+J_{\nu +1}(z)=\frac{2\nu }{z}J_\nu (z)\).
Also, we have that
We will use the following estimates for Bessel functions.
-
\(\left( D1\right) \) For any \(\nu >-1/2\) and \(z\in \mathbb {C}\),
$$\begin{aligned} |J_{\nu }(z)|\le \frac{\left( \frac{|z|}{2}\right) ^{\nu }}{\Gamma (\nu +1)}\,e^{|Im\,z|}. \end{aligned}$$For integer \(n\ge 0\) we have
$$\begin{aligned} |J_{n}(z)|\le \frac{|z|^{n}}{n!2^{n}}\,e^{\frac{|z|^{2}}{4}}. \end{aligned}$$ -
\(\left( D2\right) \) For \(\nu \ge 1/2\) and \(0<r\le 1\),
$$\begin{aligned} |J_{\nu }(r)|\le C\left( \frac{r}{2}\right) ^{\nu }\frac{1}{\Gamma (\nu +1)}. \end{aligned}$$ -
\(\left( D3\right) \) For \(\nu \ge 1/2\), \(\alpha _0>0\), and \(1\le \nu \;\mathrm {sech}\;\alpha \le \nu \;\mathrm {sech}\;\alpha _0\),
$$\begin{aligned} |J_{\nu }(\nu \;\mathrm {sech}\;\alpha )|\le C\frac{e^{-\nu (\alpha -\mathrm {tanh}\;\alpha )}}{\nu ^{1/2}}. \end{aligned}$$ -
\(\left( D4\right) \) If \(z=r\in \mathbb {R}\), then
$$\begin{aligned} |J_{\nu }(r)|&\le C\frac{1}{\nu }\quad \,0\le r\le \nu /2,\,\nu \ge 1,\\ |J_{\nu }(r)|&\le Cr^{-1/3}\quad \,r\ge 1,\,\nu \ge 0,\\ |J_{n}(r)|&\le C_{n}r^{-1/2}\quad \!r>0,\,n\in \mathbb {Z}. \end{aligned}$$
A known asymptotic formula for Bessel functions is
as \(r\rightarrow \infty \). In particular,
The proof of following lemma can be found in [5].
Lemma 1
Let \(\nu >0\), \(p\ge 1\) and \(a\ge 1\), then there exists a constant C depending only on p and a, such that
where \(\nu ^{\frac{2}{3}}\le 2^K\le 2\nu ^{\frac{2}{3}}\).
The following lemma [9, p. 675] is useful in this paper.
Lemma 2
Let \(\nu (m)=m+\frac{n-2}{2}\). Then
for all \(m\ge 1\) if \(n=3\) and for all \(m\ge 0\) if \(n\ge 4\).
The space of all surface spherical harmonics of degree m will be denoted by \(\mathcal {Y}_{m}\). In addition, \(\{Y_{m}^{j}:m\in \mathbb {N},j=1,\ldots ,d_{m}\}\) will always denote a basis of real valued spherical harmonics for \(L^{2}(\mathbb {S}^{n-1})\), where
Theorem 2
(Spherical Harmonic Addition Theorem) Let \(\{Y_m^j\}\), \(j=1,\ldots ,d_m\) be an orthonormal basis for \(\mathcal {Y}_m\). Then
where \(Z_m(\xi ,\eta )= \sum _{j=1}^{d_m}\overline{Y_m^j(\xi )}Y_m^j(\eta )\) are called zonal harmonics of degree m, \(P_m\) is the Legendre polynomial of degree m and \(\sigma _{n}\) is the total surface area of \(\mathbb {S}^{n-1}\).
The following lemma is known as the Addition Theorem of the Bessel functions (see [12, Lemma 2, p. 121]).
Lemma 3
If \(x=r\xi \), \(y=s\theta \), we have
where
We have that
that is,
An identity that relates the eigenvalues of the spherical Laplacian with the norm \(L^{2}(\mathbb {S}^{n-1})\) of the spherical gradient for some spherical harmonic \(Y_{k}\) of degree k is given by
which implies that the norms \(\Vert (-\Delta _{S})^{1/2}u\Vert _{L^{2} (\mathbb {S}^{n-1})}\) and \(\Vert |\nabla _{S}u|\Vert _{L^{2}(\mathbb {S}^{n-1})}\) are equivalent.
A classical result due to Bakry (see [3]), valid for any Riemannian manifold with non-negative Ricci curvature and in particular for the sphere, is the following.
Theorem 3
(Bakry) If \(1<p<\infty \), there exist constants \(c_{p}\) and \(C_{p}\) such that
for all \(u\in C_{c}^{\infty }(\mathbb {S}^{n-1})\).
Definition 1
-
(i)
For \(1\le p<\infty \), we denote by \(\mathcal {H} ^{p}\) the space of all \(u\in \mathcal {D}^{\prime }(\mathbb {R}^{n})\) such that u, \(\frac{\partial u}{\partial r}\) and \(|\nabla _{S}u|\in L_{loc}^{1}(\mathbb {R}^{n})\)
$$\begin{aligned} \left\| u\right\| _{\mathcal {H}^{p}}&=\left\{ \int _{\mathbb {R}^{n} }\left( {\left| u(x)\right| }^{p}+{\left| \frac{\partial u}{\partial r}(x)\right| }^{p}+{\left| \nabla _{S}u(x)\right| } ^{p}\right) \frac{dx}{{\langle x\rangle }^{3}}\right\} ^{1/p}\end{aligned}$$(13)$$\begin{aligned}&=\left( {\left\| u\right\| }_{L^{p}(\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}})}^{p}+{\left\| \frac{\partial u}{\partial r}\right\| }_{L^{p}(\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}})} ^{p}+{\left\| |\nabla _{S}u|\right\| }_{L^{p}(\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}})}^{p}\right) ^{1/p}, \end{aligned}$$(14)where \(\langle x\rangle :=1/(1+|x|^{2})^{1/2}.\)
-
(ii)
We denote by \(\mathcal {W}^{p}\) the space of all functions \(u\in \mathcal {H}^{p}\) satisfying the Helmholtz equation (2) in \(\mathbb {R}^{n}\).
Remark 1
-
(1)
\(C^{\infty }(\mathbb {R}^{n})\cap \mathcal {H}^{p}\) is dense in \(\mathcal {H}^{p}\) and the elements of \(\mathcal {H}^{p}\) belong locally to a weighted Sobolev space in \(\mathbb {R}^{n}\).
-
(2)
By Theorem 3, we can define in \(\mathcal {H}^{p}\) the equivalent norm \({\left\| \mathbf {\centerdot \centerdot }\right\| }_{\mathcal {H} ^{p}}^{\Delta ^{\frac{1}{2}}}\) given by
$$\begin{aligned} {\left\| u\right\| }_{\mathcal {H}^{p}}^{\Delta ^{\frac{1}{2}}}=\left( {\left\| u\right\| }_{L^{p}\big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\big )}^{p}+{\left\| \frac{\partial u}{\partial r}\right\| } _{L^{p}\big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\big )}^{p}+{\Vert (-\Delta _{S})^{1/2}u\Vert }_{L^{p}\big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\big )}^{p}\right) ^{1/p}. \end{aligned}$$Throughout this article will be exchanging these norms as needed.
2 The Fourier Extension Operator in \(L^{2}(\mathbb {S}^{n-1})\) and \(\mathcal {W}^2\)
In this section we prove that the space \(\mathcal {W}^{2}\) is precisely the space of all Herglotz wave functions.
Lemma 4
If \(Y_{m}\) is a spherical harmonic and \(F_m:=WY_m\), then
-
(i)
\(F_m(x)=(2\pi )^{\frac{1}{2}} i^{m}r^{-(n-2)/2}J_{\nu (m)}(r)Y_{m}(\xi )\), \(x=r\xi \).
-
(ii)
\(\{F_{m}^{j}(r\xi ):=(2\pi )^{\frac{1}{2}}i^{m}r^{-(n-2)/2}J_{\nu (m)}(r)Y_{m}^j(\xi )\}_{m,j},m=0,1,\ldots ,j=1,2,\ldots ,d_{m}\) is an orthogonal family and
$$\begin{aligned} {\Vert F_{m} \Vert }_{\mathcal {H}^2}=\sqrt{2}+O\left( \frac{1}{m^2}\right) . \end{aligned}$$(15) -
(iii)
If \(f=\sum _{m,j}a_{mj}Y_m^j\in L^{2}(\mathbb {S}^{n-1})\) and \(u=\sum _{m,j}a_{mj}F_m^j\in \mathcal {W}^{2}\), then
$$\begin{aligned} \left\| u\right\| _{\mathcal {H}^2}\sim \left\| f\right\| _{L^{2}(\mathbb {S}^{n-1})}, \end{aligned}$$(16)and the series of u converges absolutely and uniformly on compact subsets of \(\mathbb {R}^{n}\).
Proof
-
(i)
Is a direct consequence of the Funk–Hecke’s formula (see [11, p. 37]) with \(x=r\xi \),
$$\begin{aligned}&\int _{\mathbb {S}^{n-1}}\exp {(-ix\cdot w)} Y_{m}(w)d\sigma (w)\nonumber \\&\quad =(2\pi )^{n/2}(-i)^{m}r^{-(n-2)/2}J_{\nu (m)}(r)Y_{m}(\xi ). \end{aligned}$$(17) -
(ii)
By the Lemma 2, (11) and the recursion formula (R1) we have that
$$\begin{aligned} {\Vert F_{m} \Vert }^2_{\mathcal {H}^2}&=\int _{\mathbb {R}^n} \left( {\left| F_{m}(x)\right| }^2+{\left| \frac{\partial F_{m}}{\partial r}(x) \right| }^{2}+{\left| \nabla _S F_{m}(x) \right| }^2 \right) \frac{dx}{{\langle x \rangle }^3}\nonumber \\&\quad =\,2+O\left( \frac{1}{m^2}\right) . \end{aligned}$$(18)Then
$$\begin{aligned} {\Vert F_{m} \Vert }_{\mathcal {H}^2}=\sqrt{2}+O\left( \frac{1}{m^2}\right) . \end{aligned}$$The orthogonality follows from the orthogonality of the spherical harmonics in \(L^{2}(\mathbb {S}^{n-1})\).
-
(iii)
By \(\left( ii\right) \) it follows that \(\left\| \mathbf {\centerdot \centerdot }\right\| _{\mathcal {H}^2}\sim \left\| \mathbf {\centerdot \centerdot }\right\| _{L^{2}(\mathbb {S}^{n-1})}\). Furthermore, using the recurrence formula (R1) for Bessel functions and the estimate (D1), it follows that the series for u converges absolutely and uniformly on compact subsets of \(\mathbb {R}^{n}\) \(\square \)
Theorem 4
The operator W is a topological isomorphism of \(L^{2}(\mathbb {S}^{n-1})\) onto \(\mathcal {W}^{2}\).
Proof
By Lemma 4, to prove that \(\left\| Wf\right\| _{\mathcal {H}^2}\sim \left\| f\right\| _{L^{2}(\mathbb {S}^{n-1})}\) it suffices to show that \(Wf=\sum _{m,j}a_{mj}F_m^j\) for any \(f=\sum _{m,j}a_{mj}Y_m^j\in L^{2}(\mathbb {S}^{n-1})\). Notice that if \(f_n\) converges to f in \(L^{2}(\mathbb {S}^{n-1})\) then \(Wf_n\) converges uniformly to Wf uniformly on compact sets of \(\mathbb {R}^n .\) Let \(L_0^2\) be the linear span of \(\{Y_m^j\}\) and \(W'=W\mid _{L_0^2}\). If \(\phi \) is a finite sum \(\sum _{m,j} a_{mj}Y_m^j\in L_0^2\) then \(W\phi =\sum _{m,j}a_{mj}F_m^j\) and by Lemma 4(iii) we have that \(\left\| W\phi \right\| _{\mathcal {H}^2}\sim \left\| \phi \right\| _{L^{2}(\mathbb {S}^{n-1})}\). Moreover, \(W'\) can be extended to a continuous operator from \(L^{2}(\mathbb {S}^{n-1})\) into \(\mathcal {W}^{2}\) so that \(W'\big (\sum _{m,j} a_{mj}Y_m^j\big )=\sum _{m,j}a_{mj}F_m^j\) converges uniformly on compact subsets of \(\mathbb {R}^n\).
Now let \(f=\sum _{m,j} a_{mj}Y_m^j\in L^{2}(\mathbb {S}^{n-1})\) and \(\phi _m=\sum _{k,j,k\le m} a_{kj}Y_k^j\). Then \(W(\phi _n)=W'(\phi _n)\rightarrow Wf\) uniformly on compact subsets and \(Wf=W'f\). Thus, \(\left\| Wf\right\| _{\mathcal {H}^2}\sim \left\| f\right\| _{L^{2}(\mathbb {S}^{n-1})}\).
It remains to prove that \(\mathcal {W}\) is onto.
Let \(u\in \mathcal {W}^2\), we have that \(u\in C^{\infty }(\mathbb {R}^n)\), so, for r fixed, consider the Fourier series in spherical harmonics of \(u(r\xi )\), that is,
with
Thus, we can apply term by term the Helmholtz operator in polar coordinates
to the representation of u. We obtain
and using the orthogonality of the spherical harmonics we have that
for each \(m\in \mathbb {N}\cup \{0\},j=1,\ldots ,d_m\), that is, the function \(A_{mj}(r)\) satisfies the Bessel equation of order \(\nu (m)\). Then, \(A_{mj}\) can be written as a linear combination,
where \(N_{\nu (m)}(r)\) is the Neumann function of order \(\nu (m)\). Since \(N_{\nu (m)}(r)\) has a singularity at \(r=0\) and \(A_{mj}(r)\) is bounded, it follows that \(b_{mj}=0\) for all m, j; therefore, \(A_{mj}(r)=a_{mj}J_{\nu (m)}(r)\).
We see that \(\sum _{m,j}\left| a_{mj} \right| ^{2}\le C{\Vert u\Vert }_{\mathcal {H}^2}\), so taking \(\phi =\sum _{m,j}a_{mj}Y_m^j\), we conclude that \(\phi \in L^2(\mathbb {S}^{n-1})\) and \(u=W\phi \). \(\square \)
Now we will construct the reproducing kernel for \(\mathcal {W}^{2}\) as a subspace of the Hilbert space \(\mathcal {H}^{2}\). Before, we observe that family \(\{\beta _{m}^{-1}F_{m}^{j}\}\) is an orthonormal basis for \(\mathcal {W}^{2}\), where \(\beta _{m}=\left\| F_{m}^{j} \right\| _{\mathcal {H}^{2}}\).
Let
where \(x=r\xi \), \(y=s\theta \). Using directly the Addition Theorem 2 we have
By the estimate (D1) for Bessel functions we can prove that the series that define \(\mathcal {K}(x,y)\) converges absolutely and uniformly on compacts subsets of \(\mathbb {R}^{n}\times \mathbb {R}^{n}\). Since \(Z_{m} (\xi ,\theta )\) is real then \(\mathcal {K}(x,y)\) is symmetric.
The orthogonal projection of \(\mathcal {H}^{2}\) onto \(\mathcal {W}^{2}\) is given by
with convergence in \(\mathcal {W}^{2}\) and also pointwise.
For \(x\in \mathbb {R}^{n}\) fixed, we have
The function \(\mathcal {K}(x,y)\) is the reproducing kernel for the space \(\mathcal {W}^{2}\).
The following lemma shows that after a topological isomorphism of \(\mathcal {W}^{2}, \) the kernel \(\mathcal {K}(x,y)\) has a closed form.
We call \(\mathcal {M}\) a multiplier on the sphere \(\mathbb {S}^{n-1}\) defined by a complex sequence \(\{\mu _m\}\) to the operator
for any finite sum \(\sum _{m,j}a_{mj}Y_{m}^{j}(\xi )\).
Lemma 5
Let \(\mathcal {M}\) be the multiplier on the sphere \(\mathbb {S}^{n-1}\) defined by the sequence \(\{\beta _m^2\}\). Then, \(\mathcal {M}\) is a topological isomorphism of \(\mathcal {W}^{2}\) onto itself, where here
Moreover, the kernel function of the composition \(\mathcal {M}\circ \mathcal {P}\) is
namely \((\mathcal {M}\circ \mathcal {P})u(x)=\langle u,\widetilde{\mathcal {K}}(x,\cdot ) \rangle _{\mathcal {H}^{2}}\).
Proof
Since \(c\le \beta _m^2\le C\) for some constants \(c,C>0\), then it is clear that \(\mathcal {M}\) is a topological isomorphism of \(\mathcal {W}^{2}\) onto itself. In particular, by (19), (8) and (9), we have that
where \(\mathcal {M}\) may be thought of as acting on \(\xi \) or on \(\theta \).
In \(\mathcal {H}^2\), the kernel \(\widetilde{\mathcal {K}}(x,y)\) defines a continuous operator \(\widetilde{\mathcal {P}}\) on \(\mathcal {H}^2\) given by
Let \(\mathcal {H}_0\) be the linear span of the set \(\{A(r)Y_m^j(\xi ): A\in C_c^\infty (0,\infty )\}_{m,j}\). We can prove that \(\widetilde{\mathcal {P}}=\mathcal {M}\circ \mathcal {P}\) in \(\mathcal {H}_0\). Since \(\mathcal {H}_0\) is dense in \(\mathcal {H}^2\), we conclude that \(\widetilde{\mathcal {P}}=\mathcal {M}\circ \mathcal {P}\). \(\square \)
Below we will need to study the continuity of the multiplier \(\mathcal {M}\) in \(L^p(\mathbb {S}^{n-1}).\) For this we will use the next two results by Strichartz and Bonami–Clerc proved in [13] and [6], respectively.
Theorem 5
Let m(x) be a function of a real variable satisfying
If \(m_{j}=m(j)\) then \(\{m_{j}\}\in \mathcal {M}_{p}(\mathbb {S}^{n-1})\) for
where \(\mathcal {M}_{p}(\mathbb {S}^{n-1})\) denotes the space of all \(L^{p}- \)multipliers on the sphere \(\mathbb {S}^{n-1}\).
Theorem 6
Let \(N=[\frac{n-1}{2}]\) and \({\{\mu _{k}\}}_{k\ge 0}\) be a sequence of complex numbers such that
-
\(\left( A_{0}\right) \) \(|\mu _{k}|\le C,\)
-
\(\left( A_{N}\right) \) \(\sup _{j\ge 0}2^{j(N-1)}\sum _{k=2^{j}}^{2^{j+1} }|\Delta ^{N}\mu _{k}|\le C.\)
Then \(\{\mu _{k}\}\in \mathcal {M}_{p}(\mathbb {S}^{n-1})\) for \(1<p<\infty \). Here \(\Delta \) denotes the forward difference operator given by \(\Delta \mu _k=\mu _{k+1}-\mu _k\).
Theorem 7
For \(1<p<\infty \), the operators \(\mathcal {M}\) and \(\mathcal {M}^{-1}\) are continuous on \(L^{p}(\mathbb {S}^{n-1})\). That is, the sequences \(\left\{ \beta _{m}^{2}\right\} \) and \(\left\{ \beta _{m}^{-2}\right\} \) define bounded multipliers on \(L^{p}(\mathbb {S}^{n-1})\).
Proof
By (6) and (18) we obtain that for all \(m\ge 2\),
with \(R(m)=\frac{P(m)}{Q(m)}\) for some polynomials P y Q of degree 4 and 6, respectively. Thus, to prove the continuity of \(\mathcal {M}\) it is enough to show that the sequence \(\{R(m)\}\) defines a bounded multiplier on \(L^p(\mathbb {S}^{n-1})\). We have that \(R^{(k)}(x)\sim \frac{1}{{\vert x\vert }^{k+2}}\) for \(\left| x\right| \) large. Hence,
for all \(k\in \mathbb {N}\cup \{0\}\). Then by Theorem 5, the above inequality implies that \(\{R(m)\}\) defines a bounded multiplier on \(L^p(\mathbb {S}^{n-1})\) for \(1<p<\infty \). To prove the continuity of \(\mathcal {M}^{-1}\), it suffices to prove the continuity of the multiplier defined by the sequence \(\{\gamma _m\}\) given by
For m large and \(L\in \mathbb {N}\) fixed, there exists a sequence \(\{r_m\}\) such that
\(|r(m)|\sim O(\frac{1}{m^{2L}})\). Using Strichartz’s Theorem we see that each \(\{R(m)^k\}\) defines a bounded multiplier in \(L^p(\mathbb {S}^{n-1})\) for \(1<p<\infty \) and \(k=0,1,\ldots ,L-1\). Thus, to end the proof we will show that if we choose L large enough, \(\{r(m)\}\) defines a bounded multiplier in \(L^p(\mathbb {S}^{n-1})\) . Let \(N=\big [\frac{n-1}{2}\big ]\), then for m large,
for all \(j\ge 2\). Therefore,
if we choose any \(L>N/2\). By Theorem 6, we conclude that \(\{r_m\}\) defines a bounded multiplier on \(L^p(\mathbb {S}^{n-1})\). \(\square \)
Remark 2
By (18), we have that
for \(u\in \mathcal {H}^2\).
Hence we may replace \(\mathcal {H}^2\) by the Hilbert space \(\mathcal {H'}^2\) with the norm
to define the kernel
where \(\gamma _{m}=\left\| F_m\right\| _{\mathcal {H'}^2}\sim \sqrt{2}+O\left( \frac{1}{m^2}\right) \). In this case, the orthogonal projection \(\mathcal {P'}\) on \(\mathcal {H'}^2\) is given by
3 Structure and Properties of \(\mathcal {W}^{p}\)
Now we give estimates of the kernel \(\widetilde{\mathcal {K}}(x,y).\)
Lemma 6
Consider \(\widetilde{\mathcal {K}}(x,y)=\frac{J_{\frac{n-2}{2}}(\left| x-y \right| )}{(2\pi \left| x-y \right| )^{(n-2)/2}}\), \(y=s\theta \) in the polar form. Then we have the following pointwise estimates:
Proof
The inequality (25) follows from (4) and the fact that the function \(J_{\frac{n-2}{2}}(r)\) has a zero of order \((n-2)/2\) at \(r=0\). Similarly, we can obtain (26).
To prove (27) we estimate any directional derivative \(D_\nu \) of \(\widetilde{\mathcal {K}}\) in the direction of a unit vector \(\nu \) tangent to \(\mathbb {S}^{n-1}\). Using (3), we have that
Thus in particular we obtain (27). \(\square \)
Proposition 1
Let
If \(p>\alpha _n\) then \(\widetilde{\mathcal {K}}(x,.)\), \(\frac{\partial }{\partial s}\widetilde{\mathcal {K}}(x,.)\) and \(\nabla _{S_{\theta }}\widetilde{\mathcal {K}}(x,.)\) belong to \(L^p(\frac{dy}{{\langle y\rangle }^3})\) for each \(x\in \mathbb {R}^n\).
Proof
In fact, using the estimates given in the Lemma 6 and Peetre’s inequality \((1+|x-y))^{-1}\le C(1+|x|)/(1+|y|)\), we have
Similarly, \(\frac{\partial }{\partial s}\widetilde{\mathcal {K}}(x,.)\in L^p\Big (\frac{dy}{{\langle y\rangle }^3}\Big )\).
Finally,
\(\square \)
Proposition 2
If \(p>\alpha _n\) then \(F_m^j\in \mathcal {W}^{p}\) for any m, j. Moreover, \(\mathcal {W}^{p}\ne \{\mathbf {0}\}\) if and only if \(p>\alpha _n\).
Proof
We know that \(F_m^j\) is an entire solution of the Helmholtz equation and if \(p>\alpha _n\), \(F_m^j\in L^p\Big (\frac{dx}{{\langle x\rangle }^3}\Big )\). In fact, by (4)
whenever \(p>\alpha _n\). Thus, \(F_m^j\in \mathcal {W}^{p}\).
Now suppose that \(\mathcal {W}^{p}\ne \{\mathbf {0}\}\). Let \(u\in \mathcal {W}^{p}\), \(u\ne 0\). Then \(u=\sum _{m,j}a_{mj}F_{mj}\) with some \(a_{mj}\ne 0\). We have that \(u(r\xi )Y_k^l(\xi )\in L^p\Big (\frac{dx}{{\langle x\rangle }^3}\Big )\). If \(\varphi \) is a radial function such that \(\varphi (\vert x\vert )\in L^{p'}\Big (\frac{dx}{{\langle x\rangle }^3}\Big )\) and \({\left\| \varphi \right\| }_{L^{p'}(\frac{r^{n-1}}{{\langle r\rangle }^3})}\le 1\), then by Hölder’s inequality
which implies that
Consequently, by duality
and this implies that \(p>\alpha _n\). \(\square \)
Theorem 8
For \(1<p<\infty \), \(\mathcal {W}^{p}\) is a Banach space.
Proof
Let v any entire solution of the Helmholtz equation and let \(\Phi (x,y)\) be the fundamental solution of the Helmholtz equation in \(\mathbb {R}^n\) [1, p. 42], given as
Let \(x\in \mathcal {B}_R\) fixed with \(R>1\). Using a Green’s identity for the functions v and \(\Phi (x,\cdot )\) we have (see [7, p. 68–69]) for \(\rho >R\),
Next, integrating both sides above with respect to \(\frac{d\rho }{(1+\rho ^2)^{3/2}}\) on the interval [2R, 3R], we have the integral representation of v for points of \(\mathcal {B}_R\),
Now we prove that \(\mathcal {W}^p\) is closed in \(\mathcal {H}^p\). Differentiating under the integral in (28) and using Hölder’s inequality we have that on any compact set K, any partial derivative
Let \(\{u_n\}\) be a sequence in \(\mathcal {W}^p\) converging to \(u\in \mathcal {H}^p\). Taking a subsequence if necessary, assume that the convergence is also almost everywhere. The relation (28) implies that \(\{u_n\}\) (and all their derivatives) is a Cauchy sequence uniformly in compact subsets of \(\mathbb {R}^n\), converging to a limit \(\tilde{u}\), that satisfies the Helmholtz equation. Then \(u=\tilde{u}\) and \(u\in \mathcal {W}^p\). \(\square \)
Remark 3
Using the integral representation (28) we can see that the evaluation functional \(\mathcal {W}^{p}\longrightarrow \mathbb {C}\), \(v\longmapsto v(x)\) is continuous for every \(x\in \mathbb {R}^n\).
Given \(f(\xi )=\sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}}a_{mj}Y_m^j(\xi )\in L^{p}(\mathbb {S}^{n-1})\), the Riesz means \(R_N^\delta \) of f of order \(\delta \) is defined by
We will need the following theorem (see [6]) about the convergence of Riesz means to study the density of the linear span of \(\{F_m^j\}\) in \(\mathcal {W}^p\).
Theorem 9
Let \(1\le p\le \infty \). If \(\delta >(n-2)/2\), then for \(f\in L^{p}(\mathbb {S}^{n-1})\),
moreover, the Riesz means are uniformly bounded on \(L^{p}(\mathbb {S}^{n-1})\), that is, there exists a uniform constant \(C_{p,\delta }\) such that
for all N.
Theorem 10
Let \(p>\alpha _n\) and \(\mathcal {W} ^{p}_{0} \) the linear span of \(\{F_{m}^{j}\}_{m,j}\). Then \(\mathcal {W} ^{p}_{0}\) is dense in \(\mathcal {W}^{p}\).
Proof
Given \(u\in \mathcal {W}^p\), the proof of the surjectivity in Theorem 4 shows that there exists \(a_{mj}\in \mathbb {C}\) such that
where the convergence is absolute and uniform in compact subsets of \(\mathbb {R}^{n}\). Let r fixed and \(\delta >(n-2)/2\), and we consider the Riesz means \(R_N^\delta \) of u of order \(\delta \). By Proposition 2, \(R_N^\delta u\in \mathcal {W}^p\) for \(p>\alpha _n\).
Let \(\Lambda _N^p(r)\) the integral given by
By the Theorem 9 we have that \(R^\delta _Nu\longrightarrow u\) and \(\frac{\partial }{\partial r} R^\delta _Nu\longrightarrow \frac{\partial u}{\partial r}\) in \(L^p(\mathbb {S}^{n-1})\) as \(N\rightarrow \infty \). Since \((-\Delta _{S})^{1/2}(R_N^\delta u)=R_N^\delta ((-\Delta _{S})^{1/2}u)\) we deduce that \((-\Delta _{S})^{1/2}R_N^\delta u\) converges to \((-\Delta _{S})^{1/2}u\) in \(L^p(\mathbb {S}^{n-1})\). Hence
Also, using the uniform boundedness of the Riesz means (Theorem 9) we obtain
that is, \(\Lambda _N^p(r)\le Cg(r)\) with \(g\in L^1\big (\mathbb {R}^{+},\frac{r^{n-1}dr}{(1+r^2)^{3/2}}\big )\). Then applying the Lebesgue’s Dominated Convergence Theorem we have
Therefore, \(R_N^\delta u\) converges to u in \(\mathcal {H}^p\). So, we conclude that the linear span of \(\{F_{m}^{j}\}_{m,j}\) is dense in \(\mathcal {W}^p\). \(\square \)
Remark 4
By Theorems 7 and 10, we have that \(\mathcal {M}\) and \(\mathcal {M}^{-1}\) are continuous in \(\mathcal {W}^p\) for any \(p>\alpha _n\).
Now we will prove a reproducing property of the orthogonal projection \(\mathcal {P}\) for the space \(\mathcal {W}^p\).
Theorem 11
Let \(\alpha _n<p<\alpha '_n\). Given \(u\in \mathcal {H}^{p}\), then \(u\in \mathcal {W}^{p}\) if and only if \(\mathcal {P}u=u\).
Proof
Let \(u\in \mathcal {W}^p\) and \(\alpha _n<p<\alpha '_n\). By Theorem 10, there exists a sequence \(\{u_n\}\subseteq \mathcal {W}^p_0\subseteq \mathcal {W}^2\) such that \(u_n\rightarrow u\) in \(\mathcal {H}^p\) for \(p>\alpha _n\). Also, since \(\mathcal {P}\) is continuous in \(\mathcal {W}^2\), then \(\mathcal {P}u_n=u_n\). On the other hand, by Remark 3, we have that \(u_n(x)\rightarrow u(x)\) for every \(x\in \mathbb {R}^n\). So, to end the proof it is enough to see that \(\mathcal {P}u_n(x)\rightarrow \mathcal {P}u(x)\) for all \(x\in \mathbb {R}^n\). In effect,
Since by Proposition 1, \(\widetilde{\mathcal {K}}(x,.)\), \(\frac{\partial \widetilde{\mathcal {K}}}{\partial s}(x,.)\) and \(|\nabla _{S_\theta }\widetilde{\mathcal {K}}(x,.)|\in L^{p'}\Big (\frac{dy}{{\langle y\rangle }^3}\Big )\), applying the Hölder’s inequality we have that
Since we also have that \(u_n(x)\longrightarrow u(x)\) we conclude that \(Pu(x)=u(x)\).
To prove the converse, let \(u\in \mathcal {H}^p\) and suppose \(u=Pu\), then
since \(\mathcal {K}(.,y)\) satisfies the Helmholtz equation in \(\mathbb {R}^n\) for each \(y\in \mathbb {R}^n\). Therefore, \(u\in \mathcal {W}^p\). \(\square \)
4 Continuity of \(\mathcal {P'}\) in Mixed-Normed Spaces
In this section we prove a positive result about the continuity of \(\mathcal {P}\) on mixed-normed spaces, generalizing the results in [4] for \(n>2\).
Definition 2
Let \(1\le p<\infty \), the mixed-normed space \(L^{p}(\mathbb {R}^{+};d\mu )(L^{2}(\mathbb {S}^{n-1}))\) consisting of all the measurable functions \(f(r\xi )\) such that
where \(d\mu (r):=r^{n-1}/(1+r^{2})^{3/2}dr\).
From now on we will write \(L^{p}(\mathbb {R}^{+};d\mu )(L^{2}(\mathbb {S} ^{n-1}))\) as \(L^{p,2}\).
Definition 3
For \(1\le p<\infty \), we denote by \(\mathcal {H}^{p,2}\) the closure of \(C_{c}^{\infty }(\mathbb {R}^{n})\) with respect to the mixed-norm
and denote by \(\mathcal {W}^{p,2}\) the space of all functions \(u\in \mathcal {H}^{p,2}\) satisfying the Helmholtz \(\Delta u+u=0\) in \(\mathbb {R}^{n}\).
To study the continuity of \(\mathcal {P'}\) in \(\mathcal {H}^{p,2}\), we introduce the operator T defined by
T is well defined when \(p<\alpha '_n\). In fact, for \(u\in L^{p,2}\), by Hölder’s inequality, Theorem 3 and Proposition 1, we have
Lemma 7
Let w(r) be a non-negative function such that \(w^{\beta }\in A_2(d\tilde{\mu }(r))\) for some \(\beta >2\). Then
where C independent of n.
The proof of this lemma can be found in [4] and we have the following version.
Lemma 8
Let w(r) be a non-negative function and suppose there exists \(\beta >2\) such that \(w^{\beta }\in A_2(d\tilde{\mu }(r))\) and \(-a=(n-2)(1-\frac{2}{p})<2-\frac{1}{\beta }\). Then
where C is independent of m.
Proof
Let \(I^1\) and \(I^2\) be the integrals given by
and
respectively.
We split these integrals as
and
We proceed as in the proof of Lemma 7. We will prove that
Suppose \(m\ge 1\), then by Hölder’s inequality and the estimates of Bessel functions \(\left( D1\right) \)–\(\left( D4\right) \) we have
and
where \(c=\text {sech}\alpha _0\) for some \(\alpha _0>0\), \(\phi (r)=\alpha (r)-\tanh \alpha (r)\), \(\beta _0=\phi (\nu (m)c)=\alpha _0-\tanh \alpha _0>0\) and the function \(\alpha (r)\) is defined by the equation \(\nu (m)\sinh \alpha (r)=r\).
In addition, by Lemma 1 we see that
Similarly, we have that
Furthermore, using that \(a>\frac{1}{\beta }-2\) it follows that
Consequently, since \(w^{\beta }\in A_2(d\tilde{\mu }(r))\),
\(\square \)
Proposition 3
Let \(\beta _n\in (1,\infty )\) such that
If \(p\in (\beta _n,\beta '_n)\cap (4/3,4)\) then T is a bounded operator on \(L^{p,2}\). Moreover, if \(p\notin (4/3,4)\) then T cannot be extended to a bounded operator on \(L^{p,2}\).
Proof
First we note that \(p\in (\beta _n,\beta '_n)\subset (\alpha _n,\alpha '_n)\) and then p satisfies \((n-1)(1-\frac{2}{p})<2\).
It suffices to prove the proposition for \(( \beta _n,\beta _n ')\) and \(p \ge 2\), since T is self adjoint with respect to the duality \((f,g)\rightarrow \int _{\mathbb {R}^n}fg\frac{dx}{\langle x\rangle ^3}\) of \(L^{p,2}\) and \(L^{p',2}\).
Next, expanding u in spherical harmonics, that is,
and using the Fourier expansion of the kernel \({\mathcal {K'}}\) we have
where
Showing that T is bounded on \(L^{p,2}\) is equivalent to prove the vector-valued inequality,
with C independent of m.
Let r be the dual exponent of p / 2. By duality, there exists \(h\in L^r(d\mu )\) with \({\Vert h\Vert }_{L^r(d\mu )}=1\) such that
Let \(g(s)=s^{\frac{n-2}{r}}h(s)\) and \(\tilde{\mu }\) the measure given by \(d\tilde{\mu }(r)=\frac{rdr}{(1+r^2)^{3/2}}\). Notice that since \(p<4\) we have that \(r>2\), so we can choose \(\gamma \) such that \(2<\gamma \le r\), then \(g^{\gamma }\in L_{loc}^1(d\tilde{\mu })\), \(g^{\gamma }\le M_{\tilde{\mu }}(g^\gamma )\;\mathrm {a.e.}\) and
where \(w(s)={(M_{\tilde{\mu }}[g^{\gamma }](s))}^{\frac{1}{\gamma }}\). Furthermore, since \((n-1)\big (1-\frac{2}{p}\big )<2\), we have that \((n-2)\big (\frac{2}{p}-1\big )-\frac{1}{r}>-2\). Then we can choose \(\gamma \) close enough to r so that for some \(2<\beta <\gamma \) we have \((n-2)\big (\frac{2}{p}-1\big )-\frac{1}{\beta }>-2\). We know (see [8, Theorem 7.7(1)]) that \({M_{\tilde{\mu }}(g^{\gamma })}^{\frac{\beta }{\gamma }}\in A_1(\tilde{\mu })\). Then since \(M_{\tilde{\mu }}\) is bounded on \(L^s(\tilde{\mu })\) for \(s>1\), by Lemma 8 and Hölder’s inequality, we have
Now we prove that T is not continuous on \(L^{p,2}\) for \(p\notin (4/3,4)\).
Let \(u(r\xi )=\sum _{m,j}u_{mj}(r)Y_m^j(\xi )\), where
with \(\alpha =-\frac{(n-2)}{2}\frac{1}{p-1}\) (see in [4], the sequence \(\{f_n\}\) in the proof of Theorem 4). Writing \(Tu(r\xi )=\sum _{m,j} T_{mj} u_{mj}(r)Y_m^j(\xi )\) as in (31), we have that
and
Therefore,
and using the Lemma 1 we see that this last expression is not bounded if \(p\notin (4/3,4)\). \(\square \)
Now, we are ready to demonstrate the main theorem of this section.
Theorem 12
If \(p\in (\beta _n,\beta '_n)\cap (4/3,4)\) then \(\mathcal {P'}\) can be extended to a bounded operator on \(\mathcal {H}^{p,2}\). Moreover, if \(p\notin (4/3,4)\) then \(\mathcal {P'}\) cannot be extended to a bounded operator on \(\mathcal {H}^{p,2}\). In particular, for \(n=\)2, 3, 4, 5, \(\mathcal {P'}\) is continuous on \(\mathcal {H}^{p,2}\) if and only if \(p\in (4/3,4)\).
Proof
Let \(p\in (\beta _n,\beta '_n)\cap (4/3,4).\) To prove the \(L^{p,2}\) boundedness of \(\mathcal {P'}\), it suffices to prove that the operators \(T_1\), \(T_2\), \(T_3\) with kernels
are bounded on \(L^{p,2}\). By Proposition 3, we know that \(T_3\) is continuous on \(L^{p,2}\). To prove the continuity of \(T_1\) and \(T_2\) notice that
and
where \(\mathcal {M}_1\) and \(\mathcal {M}_2\) are the multipliers in \(\mathbb {S}^{n-1}\) corresponding to the sequences \(\frac{1}{m(m+n-2)}\) and \(\frac{1}{\sqrt{m(m+n-2)}}\) respectively. Then proceeding as in Theorem 3 we see that the required vector valued inequalities for \(T_1\) and \(T_2\) are less demanding than (33).
Now we show that \(\mathcal {P'}\) is not continuous in \(\mathcal {H}^{p,2}\) for \(p\notin (4/3,4)\).
If \(\mathcal {P'}\) is continuous in \(\mathcal {H}^{p,2}\) then since \((-\Delta _{S_\xi })^{-1/2}:\mathcal {H}^{p,2}\rightarrow \mathcal {H}^{p,2}\) is bounded (due to the fact that \((-\Delta _{S_\xi })^{-1/2}\) is bounded in \(L^2(\mathbb {S}^{n-1})\)), we have that
is continuous in \(L^{p,2}\).
But
hence, in the notation of Proposition 3,
and it follows proceeding as in Proposition 3, that \(\mathcal {L}\) is not bounded in \(L^{p,2}\) for \(p\notin (4/3,4)\). \(\square \)
Now we will obtain a negative result relative to the continuity of projection \(\mathcal {P}\). Notice that by Remark 4 the operators \(\mathcal {P}\) and \(\widetilde{\mathcal {P}}\) have the same continuity properties on \(\mathcal {H}^p\). This motivates the study of the continuity of the integral operator \(\mathcal {T}\) given by
since the most singular part of \(\widetilde{\mathcal {P}}\) is precisely \(\mathcal {T}(\nabla _{S_\theta }u)\).
Using (10), we can split the operator in the sum \(\mathcal {T}=\mathcal {T}_1+\mathcal {T}_2\), where
where \(F_{\alpha }(t)=\frac{J_{\alpha }(t)}{t^{\alpha }}\), \(\mathbf {A}(u,y)=u(y)-u(y)\cdot \frac{y}{|y|}\frac{y}{|y|}\) and \(\mathbf {P}_a b=\frac{a\cdot b}{\vert a \vert }\frac{a}{\vert a \vert }\) is the orthogonal projection of b in the direction of a.
We will assume that \(n=3\) and we will prove that \(\mathcal {T}\) cannot be extended in general to a bounded operator on \(L^p({\langle x\rangle }^{-3}dx)\). Let \(m\in \mathbb {N}\) and \(B_m\) be the unit ball of center (0, 0, m) and fixed radius \(\epsilon <1\). Define \(u_m=\chi _{B_m}\mathbf {e_1}\) .
We consider the region R of the upper half-space between two cones \(c_1^2 \big (x_1^2+x_2^2\big )\le x_3^2\le c_2^2\big (x_1^2+x_2^2\big )\) and such that \(|x_1|>|x_2|\). Now, for fixed \(\lambda >0\) and \(k>\lambda m\), let \(A_k\) be the annulus between the spheres centered in (0, 0, m) and radii \(\alpha (k)\) and \(\alpha (k)+l\), with \(\alpha (k)=2\pi k+C\) and where C and \(l>0\) are chosen so that \(\cos {\big (t-(\frac{n}{2}+1)\frac{\pi }{2}-\frac{\pi }{4}\big )}\ge 1/2\) for \(t\in [\alpha (k),\alpha (k)+l].\)
Lemma 9
There exists positive constant \(\lambda \) such that, if \(k>\lambda m\), then \(\left| R\cap A_k\right| \sim k^2\) uniformly for large m.
Proof
Clearly \(\left| R\cap A_k\right| =O(k^2)\). Now consider spherical coordinates \(\lbrace (r,\theta ,\varphi ):r>0,\theta \in [0,2\pi ],\varphi \in [0,\pi ] \rbrace \) centered at the point (cartesian) (0, 0, m). Notice that as a subset of \(\mathbb {R}^2\), every vertical section \(R\cap A_k\cap \lbrace (r,\theta _0,\varphi ):r>0,\varphi \in [0,\pi ]\rbrace \) is independent of \(\theta _0\in [0,\pi /4]\) . This subset of \(\mathbb {R}^2\) contains the region in \(S_k\) described as follows.
Let \(P_1\) be the intersection of \((\alpha (k)+l)\mathbb {S}^{1}\) and the line \(s=c_1^{-1}t\) in the plane (s, t) and \(P_2\) the intersection of \(\alpha (k)\mathbb {S}^{1}\) and the line \(s=c_2^{-1}t\) in the plane (s, t) both with \(t>m\).
Then define \(S_k\) as the intersection of the annulus \(\alpha _k<\left| x-(0,m)\right| <\alpha _k+l\) and the region in the first quadrant between the line \(l_1\) through (0, m) and \(P_1\) and the line \(l_2\) passing through (0, m) and \(P_2\). Let \(\varphi _i\) be such that \(\tan {(\pi /2-\varphi _i)}\) is the slope of the line \(l_i\) for \(i=1,2\).
It follows that if \(A'_k\subset R\cap A_k\) in spherical coordinates centered on (0, 0, m) is given by the inequalities \(\alpha (k)\le r\le \alpha (k)+l\), \(0\le \theta \le \frac{\pi }{4}\), \(\varphi _2\le \varphi \le \varphi _1\), then we have
Hence, to complete the proof of the lemma, it suffices to show that there exists \(c>0\) such that
Denoting \(\alpha (k)\) just by \(\alpha \), we observe that \(P_2=(c_2^{-1}t_2,t_2)\) with
Let \(\lambda >0\) and \(\alpha >\lambda m\). Then \(1-\frac{1}{\lambda ^2}<1-\frac{m^2}{\alpha ^2}\), and
Similarly, we have that \(P_1=(c_1^{-1}t_1,t_1)\) and
Since \(\alpha >\lambda m\) then \(\frac{m}{\alpha +l}<\frac{1}{\lambda }\), hence
By (40) and (41), we have that
Since the limit of the right side is positive as \(\lambda \rightarrow \infty \), we conclude that choosing \(\lambda \) large enough \(t_2/{\alpha }-t_1/(\alpha +\lambda )\ge \epsilon \), for some \(\epsilon >0\).
Finally for such \(\lambda \), if \(\alpha >\lambda m\) we have that
where \(\left| h\right| \sim O\big (\frac{1}{m}\big )\). Therefore, since \(t_2/{\alpha }-t_1/(\alpha +\lambda )\ge c\) then (39) holds for large m and the proof is complete. \(\square \)
Theorem 13
\(\mathcal {T}\) cannot be extended to a bounded operator on \(L^p({\langle x\rangle }^{-3}dx)\) for \(p\in (1,3/2)\).
Proof
Let \(y\in B_m\), then we can write to \(y=m\mathbf {e_3}+y'\) with \(|y'|<\epsilon \), so that
Therefore,
for all \(\epsilon \) sufficiently small, m sufficiently large and choosing \(C<1\).
On the other hand, we have that
estimating above the right hand side we have
Hence,
Let \(x\in A_k\). For (42), (43) and (4), we deduce that
By Lemma 9,
and so
Furthermore,
Then,
Consequently,
then, since \({\left\| u_m\right\| }_p\sim m^{-3/p}\),
Hence \(\mathcal {T}\) is not bounded if \(p\in (1,3/2)\). \(\square \)
References
Agmon, S.: A Representation Theorem for Solutions of the Helmholtz Equation and Resolvent Estimates for the Laplacian. Analysis. et cetera, pp. 39–76. Academic Press, Boston (1990)
Alvarez, J., Folch-Gabayet, M., Esteva, S.P.: Banach spaces of solutions of the Helmholtz equation in the plane. J. Fourier Anal. Appl. 7(1), 49–62 (2001)
Bakry, D.: Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. In: Séminaire de Probabilités, XXI, Lecture Notes in Math., vol. 1247, pp. 137–172. Springer, Berlin (1987)
Barceló, J.A., Bennet, J., Ruiz, A.: Mapping properties of a projection related to the Helmholtz equation. J. Fourier Anal. Appl. 9(6), 541–562 (2003)
Barceló Valcárcel, J.A.: Funciones de banda limitada. Ph.D. thesis, Universidad Autónoma de Madrid (1988)
Bonami, A., Clerc, J.L.: Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques. Trans. Am. Math. Soc. 183, 223–263 (1973)
Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Krieger Publishing Company, Malabar (1992)
Duoandikoetxea, J.: Fourier Analysis. Studies in Mathematics, vol. Graduate. Americal Mathematical Society, Providence (2001)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic Press, California (2000)
Hartman, P., Wilcox, C.: On solutions of the Helmholz equation in exterior domains. Math. Z. 75, 228–255 (1960/1961)
Helgason, S.: Topics in Harmonic Analysis on Homogeneous Spaces. Birkhäuser, Boston (1981)
Müller, C.: Analysis of Spherical Symmetries in Euclidean Spaces. Springer, New York (1998)
Strichartz, R.S.: Multipliers for spherical harmonic expansions. Trans. Am. Math. Soc. 167, 115–124 (1972)
Acknowledgments
S. Pérez-Esteva was partially supported by the Mexican Grant PAPIIT-UNAM IN102915. S. Valenzuela-Díaz was sponsored by the SEP-CONACYT Project No. 129280 (México).
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Communicated by Luis Vega.
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Pérez-Esteva, S., Valenzuela-Díaz, S. Reproducing Kernel for the Herglotz Functions in \(\mathbb {R}^n\) and Solutions of the Helmholtz Equation. J Fourier Anal Appl 23, 834–862 (2017). https://doi.org/10.1007/s00041-016-9492-8
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DOI: https://doi.org/10.1007/s00041-016-9492-8