Reproducing Kernel for the Herglotz Functions in $\mathbb {R}^n$ and Solutions of the Helmholtz Equation

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.

1 Introduction and Preliminaries

Consider the Fourier extension operator

$$\begin{aligned} W\phi (x):=\widehat{\phi d\sigma (x)}=(2\pi )^{(1-n)/2}\int _{\mathbb {S}^{n-1} }e^{ix\cdot \xi }\phi (\xi )d\sigma (\xi ), \end{aligned}$$

(1)

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

$$\begin{aligned} \Delta u+u=0. \end{aligned}$$

(2)

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

$$\begin{aligned} \limsup _{R\rightarrow \infty }\frac{1}{R}\int _{\left| x\right|<R}{\left| u(x)\right| }^{2}\,dx<\infty . \end{aligned}$$

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

$$\begin{aligned} \left\| \widehat{f}_{\mid _{\mathbb {S}^{n-1}}}\right\| _{L^{q} (\mathbb {S}^{n-1})}\le C\left\| f\right\| _{L^{p}(\mathbb {R}^{n} )},\quad f\in S(\mathbb {R}^{n}). \end{aligned}$$

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

$$\begin{aligned} \left\| \widehat{f}_{\mid _{\mathbb {S}^{n-1}}}\right\| _{L^{2} (\mathbb {S}^{n-1})}\le C_{p,n}\left\| f\right\| _{L^{p}(\mathbb {R} ^{n})}. \end{aligned}$$

Or equivalent, if $f\in L^{2}(\mathbb {S}^{n-1})$

$$\begin{aligned} \left\| Wf\right\| _{L^{q}(\mathbb {R}^{n})}\le C_{q,n}\left\| f\right\| _{L^{2}(\mathbb {S}^{n-1})} \end{aligned}$$

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

$$\begin{aligned} {\left\| u\right\| _{\mathcal {H}^{2}}^{2}}=\int _{\left| x\right| >1}\left( {\left| u(x)\right| }^{2}+{\left| \frac{\partial u}{\partial r}(x)\right| }^{2}+{\left| \frac{\partial u}{\partial \theta }(x)\right| }^{2}\right) \frac{dx}{{\left| x\right| }^{3}}<\infty . \end{aligned}$$

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

$$\begin{aligned} {\left\| u\right\| _{\mathcal {H}^{p,2}}^{2}}=\int _{0}^{\infty }\left( \int _{0}^{2\pi }\left( {\left| u(r\theta )\right| }^{2}+{\left| \frac{\partial u}{\partial \theta }(x)\right| }^{2}\right) {d\theta }\right) ^{p/2}\frac{rdr}{(1+r^{2})^{3/2}}. \end{aligned}$$

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

$$\begin{aligned} {\left\| u\right\| _{\mathcal {H}^{2}}}=\left( {\left\| u\right\| }_{L^{2}\Big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\Big )}^{2}+{\left\| \frac{\partial u}{\partial r}\right\| }_{L^{2}\Big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\Big )}^{2}+{\left\| |\nabla _{S}u|\right\| }_{L^{2}\Big (\mathbb {R}^{n},\frac{dx}{{\langle x\rangle }^{3}}\Big )}^{2}\right) ^{1/2}, \end{aligned}$$

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

$$\begin{aligned} T U(x)=C_n\int _{\mathbb {R}^n}|x||y|\frac{J_{n/2+1}(\vert x-y\vert )}{(\vert x-y\vert )^{n/2+1}}\left( (x-\mathbf {P}_yx)\cdot U(y)\right) (y-\mathbf {P}_xy)\frac{dy}{{\langle y\rangle }^3}, \end{aligned}$$

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 $\mathbb {R}$:

$$\begin{aligned} M_\mu f(x)=\sup _{I:x\in I}\frac{1}{\mu (I)}\int _I \left| f(y)\right| d\,\mu (y), \end{aligned}$$

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

$$\begin{aligned} \left( \frac{1}{\mu (I)}\int _I w(r) d\,\mu (r)\right) \left( \frac{1}{\mu (I)}\int _I w(r)^{1-p'} d\,\mu (r)\right) ^{p-1}\le C, \end{aligned}$$

for $1<p<\infty$ and

$$\begin{aligned} M_\mu w(r)\le Cw(r)\; \mathrm {a.e.} \end{aligned}$$

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 $\nu$

$$\begin{aligned} J_{\nu }(z)=\left( \frac{z}{2}\right) ^{\nu }\sum _{k=0}^{\infty }\frac{(-1)^{k}}{k!\,\Gamma (k+v+1)}\left( \frac{z}{2}\right) ^{2k}. \end{aligned}$$

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

$$\begin{aligned} \left( \frac{J_\nu (t)}{t^\nu }\right) '=-\frac{J_{\nu +1}(t)}{t^{\nu }}. \end{aligned}$$

(3)

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

$$\begin{aligned} J_\nu (r)=\sqrt{\frac{2}{\pi r}}\cos (r-\frac{\nu \pi }{2}-\frac{\pi }{4})+O(r^{-3/2}) \end{aligned}$$

(4)

as $r\rightarrow \infty$. In particular,

$$\begin{aligned} J_\nu (r)=O(r^{-1/2})\quad if \quad r\rightarrow \infty . \end{aligned}$$

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

$$\begin{aligned} \frac{1}{C}\nu ^{\frac{1}{3}-\frac{p}{3}}\sum _{j=0}^{K-1}2^{j(1-\frac{p}{4})} \le \int _{\frac{\nu }{a}}^{2\nu }|J_{\nu }(r)|^p dr\le C\nu ^{\frac{1}{3}-\frac{p}{3}}\sum _{j=0}^{K-1}2^{j(1-\frac{p}{4})}, \end{aligned}$$

(5)

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

$$\begin{aligned} \int _{0}^{\infty }J_{\nu (m)}^{2}(r)\frac{dr}{r^{2}}=\frac{1}{\pi }\frac{1}{\nu (m)^{2}-1/4}, \end{aligned}$$

(6)

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

$$\begin{aligned} d_m=\left\{ \begin{array}{ll} 1 &{} \quad \mathrm {if}\; m=0\\ \frac{(2m+n-2)(m+n-3)!}{m!(n-2)!} &{} \quad \mathrm {if}\; m\ge 1. \end{array} \right. \end{aligned}$$

Theorem 2

(Spherical Harmonic Addition Theorem) Let $\{Y_m^j\}$, $j=1,\ldots ,d_m$ be an orthonormal basis for $\mathcal {Y}_m$. Then

$$\begin{aligned} Z_m(\xi ,\eta )=\frac{d_m}{\sigma _n}P_m(\xi \cdot \eta ), \end{aligned}$$

(7)

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

$$\begin{aligned} \mathcal {J}_0(n;\left| x-y \right| )=\sum _{m=0}^\infty d_m\mathcal {J}_m(n;r)\mathcal {J}_m(n;s)P_m(\xi \cdot \theta ), \end{aligned}$$

(8)

where

$$\begin{aligned} \mathcal {J}_m(n;r)=\Gamma \left( \frac{n}{2}\right) \left( \frac{r}{2}\right) ^{\frac{2-n}{2}}J_{\nu (m)}(r). \end{aligned}$$

(9)

We have that

$$\begin{aligned} \nabla =\frac{\partial }{\partial r}\xi +\frac{1}{r}\nabla _{S}, \end{aligned}$$

that is,

$$\begin{aligned} \nabla _{S}u=r\left( \nabla u-\frac{\partial u}{\partial r}\xi \right) . \end{aligned}$$

(10)

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

$$\begin{aligned} \int _{\mathbb {S}^{n-1}}{|\nabla _{S}Y_{k}(\xi )|}^{2}\,d\sigma (\xi )=k(k+n-2)\int _{\mathbb {S}^{n-1}}{|Y_{k}(\xi )|}^{2}\,d\sigma (\xi ), \end{aligned}$$

(11)

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

$$\begin{aligned} c_{p}\big \Vert (-\Delta _{S})^{1/2}u\big \Vert _{L^{p}(\mathbb {S}^{n-1})}\le \big \Vert |\nabla _{S}u|\big \Vert _{L^{p}(\mathbb {S}^{n-1})}\le C_{p}\big \Vert (-\Delta _{S} )^{1/2}u\big \Vert _{L^{p}(\mathbb {S}^{n-1})} \end{aligned}$$

(12)

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,

$$\begin{aligned} u(r\xi )=(2\pi )^{\frac{1}{2}}r^{-(n-2)/2}\sum _{m=0}^{\infty } \sum _{j=1}^{d_{m}}i^{m}A_{mj}(r)Y_{m}^{j}(\xi ), \end{aligned}$$

with

$$\begin{aligned} (2\pi )^\frac{1}{2}r^{-(n-2)/2}i^mA_{mj}(r)= \int _{\mathbb {S}^{n-1}}u(r\eta )\overline{Y_{m}^j(\eta )}d\sigma (\eta ). \end{aligned}$$

Thus, we can apply term by term the Helmholtz operator in polar coordinates

$$\begin{aligned} \frac{{\partial }^2}{\partial r^2}+\frac{n-1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\Delta _{S}+1 \end{aligned}$$

to the representation of u. We obtain

$$\begin{aligned} \sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}}\left( A_{mj}^{\prime \prime }(r)+\frac{1}{r}A_{mj}^\prime (r) +\left( 1-\frac{{\nu (m)}^2}{r^2}\right) A_{mj}(r)\right) Y_{m}^j(\xi )=0, \end{aligned}$$

and using the orthogonality of the spherical harmonics we have that

$$\begin{aligned} A_{mj}^{\prime \prime }(r)+\frac{1}{r}A_{mj}^\prime (r) +\left( 1-\frac{{\nu (m)}^2}{r^2}\right) A_{mj}(r)=0, \end{aligned}$$

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,

$$\begin{aligned} A_{mj}(r)=a_{mj}J_{\nu (m)}(r)+b_{mj}N_{\nu (m)}(r), \end{aligned}$$

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

$$\begin{aligned} \mathcal {K}(x,y)&=\sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}} \frac{F_{m}^{j}(x)\overline{F_{m}^{j}(y)}}{\beta _{m}^{2}}\\&=2\pi (rs)^{-(n-2)/2}\sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}}\frac{J_{\nu (m)}(r)J_{\nu (m)}(s)}{\beta _{m}^{2}}Y_{m}^{j}(\xi ) \overline{Y_{m}^{j} (\theta )}, \end{aligned}$$

where $x=r\xi$, $y=s\theta$. Using directly the Addition Theorem 2 we have

$$\begin{aligned} \mathcal {K}(x,y)&=2\pi (rs)^{-(n-2)/2}\sum _{m=0}^{\infty }\frac{J_{\nu (m)} (r)J_{\nu (m)}(s)}{\beta _{m}^{2}}Z_m(\xi ,\theta )\end{aligned}$$

(19)

$$\begin{aligned}&=2\pi (rs)^{-(n-2)/2}\sum _{m=0}^{\infty } \frac{J_{\nu (m)}(r)J_{\nu (m)}(s)}{\beta _{m}^{2}} \frac{d_{m}}{\sigma _{n}}P_{m}(\xi \cdot \theta ). \end{aligned}$$

(20)

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

$$\begin{aligned} \mathcal {P}u=\sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}}\big \langle u, \beta _{m}^{-1}F_{m}^{j}\big \rangle _{\mathcal {H}^{2}}\beta _{m}^{-1}F_{m}^{j}, \end{aligned}$$

with convergence in $\mathcal {W}^{2}$ and also pointwise.

For $x\in \mathbb {R}^{n}$ fixed, we have

(21)

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

$$\begin{aligned} \mathcal {M}\left( \sum _{m,j}a_{mj}Y_{m}^{j}(\xi )\right)&=\sum _{m,j} \mu _ma_{mj}Y_{m}^{j}(\xi ) \end{aligned}$$

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

$$\begin{aligned} \mathcal {M}\left( \sum _{m,j}a_{mj}F_{m}^{j}(\xi )\right)&=\sum _{m,j} \beta ^2_ma_{mj}F_{m}^{j}(\xi ). \end{aligned}$$

Moreover, the kernel function of the composition $\mathcal {M}\circ \mathcal {P}$ is

$$\begin{aligned} \widetilde{\mathcal {K}}(x,y)=(2\pi \left| x-y \right| )^{-(n-2)/2} J_{\frac{n-2}{2}}(\left| x-y \right| ), \end{aligned}$$

(22)

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

$$\begin{aligned} \widetilde{\mathcal {K}}(x,y)&:=\mathcal {M}\mathcal {K}(x,y)\\&=\frac{2\pi }{\sigma _n}\sum _{m=0}^\infty d_m r^{-(n-2)/2}J_{m+\frac{n-2}{2}}(r)s^{-(n-2)/2}J_{m+\frac{n-2}{2}} (s)P_m(\xi \cdot \theta )\\&=(2\pi \left| x-y \right| )^{-(n-2)/2}J_{\frac{n-2}{2}}(\left| x-y \right| ), \end{aligned}$$

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

$$\begin{aligned} |x^{k}m^{(k)}(x)|\le A\quad for\quad k=0,\ldots ,a. \end{aligned}$$

If $m_{j}=m(j)$ then $\{m_{j}\}\in \mathcal {M}_{p}(\mathbb {S}^{n-1})$ for

$$\begin{aligned} \left| \frac{1}{p}-\frac{1}{2}\right| <\frac{a}{n-1},\,p\ne 1,\,\infty , \end{aligned}$$

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 $m\ge 2$,

$$\begin{aligned} \beta _m^2=2+R(m) \end{aligned}$$

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,

$$\begin{aligned} \vert x^kR^{(k)}(x)\vert \le A, \end{aligned}$$

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

$$\begin{aligned} \gamma _m=\frac{1}{1+\frac{R(m)}{2}}. \end{aligned}$$

For m large and $L\in \mathbb {N}$ fixed, there exists a sequence $\{r_m\}$ such that

$$\begin{aligned} \gamma _m=1-\frac{R(m)}{2}+\frac{R(m)^2}{2^2}-\cdots +\frac{R(m)^{L-1}}{2^{L-1}}+r(m), \end{aligned}$$

$|r(m)|\sim O(\frac{1}{m^{2L}})$. 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,

$$\begin{aligned} \sum \limits _{m=2^j}^{2^{j+1}}\left| \Delta ^Nr_m\right|&=\sum \limits _{m=2^j}^{2^{j+1}}\left| \sum _{i=0}^N(-1)^{i} \left( {\begin{array}{c}N\\ i\end{array}}\right) r_{m-i}\right| \\&\le \frac{C_{N,L}}{2^{2jL}}\\ \end{aligned}$$

for all $j\ge 2$. Therefore,

$$\begin{aligned} 2^{j(N-1)}\sum \limits _{m=2^j}^{2^{j+1}}\left| \Delta ^Nr_m\right| \le C_{L}2^{j(N-2L)}=O(1) \end{aligned}$$

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

$$\begin{aligned} \left\| u\right\| _{\mathcal {H}^2}\sim \left( \int _{\mathbb {R}^n} \left( {\left| u(x)\right| }^2+{\left| \nabla _S u(x) \right| }^2 \right) \frac{dx}{{\langle x \rangle }^3}\right) ^{1/2} \end{aligned}$$

for $u\in \mathcal {H}^2$.

Hence we may replace $\mathcal {H}^2$ by the Hilbert space $\mathcal {H'}^2$ with the norm

$$\begin{aligned} \left\| u\right\| _{\mathcal {H'}^2}=\left( \int _{\mathbb {R}^n} \left( {\left| u(x)\right| }^2+{\left| \nabla _S u(x) \right| }^2 \right) \frac{dx}{{\langle x \rangle }^3}\right) ^{1/2}, \end{aligned}$$

(23)

to define the kernel

$$\begin{aligned} \mathcal {K'}(x,y)&=\sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}} \frac{F_{m}^{j}(x)\overline{F_{m}^{j}(y)}}{\gamma _{m}^{2}}, \end{aligned}$$

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

$$\begin{aligned} \mathcal {P'}u(x)&=\int _{\mathbb {R}^{n}} \left( \mathcal {K'}(x,y)u(y)+(-\Delta _{S_{\theta } })^{1/2}{\mathcal {K'}}(x,y)(-\Delta _{S_{\theta } })^{1/2}u(y)\right) \frac{dy}{ {\langle y \rangle }^{3}}. \end{aligned}$$

(24)

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:

$$\begin{aligned} \left| \widetilde{\mathcal {K}}(x,y)\right|&\le \frac{C}{(1+|x-y|)^{\frac{n-1}{2}}},\end{aligned}$$

(25)

$$\begin{aligned} \left| \frac{\partial }{\partial s}\widetilde{\mathcal {K}}(x,y)\right|&\le \frac{C}{(1+|x-y|)^{\frac{n-1}{2}}},\end{aligned}$$

(26)

$$\begin{aligned} \left| \nabla _{{S}_\theta }\widetilde{\mathcal {K}}(x,y)\right|&\le \frac{C|x|\,|y|}{(1+|x-y|)^{\frac{n+1}{2}}}. \end{aligned}$$

(27)

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

$$\begin{aligned} |D_\nu \widetilde{\mathcal {K}}(x,s\theta )|&= s|\nabla _y\widetilde{\mathcal {K}}(x,y)\cdot \nu |\\&=C|y|\left| \frac{J_{\frac{n}{2}}(|x-y|)}{|x-y|^{n/2}}(x-y)\cdot \nu \right| \\&=C|y|\left| \frac{J_{\frac{n}{2}}(|x-y|)}{|x-y|^{\frac{n}{2}}}x\cdot \nu \right| \\&\le C|y|\left| \frac{J_{\frac{n}{2}}(|x-y|)}{|x-y|^{\frac{n}{2}}}\right| |x|. \end{aligned}$$

Thus in particular we obtain (27). $\square$

Proposition 1

Let

$$\begin{aligned} \alpha _n=\left\{ \begin{array}{ll} 1 &{} \quad \mathrm {if}\; n=2,3,4,5\\ \frac{2(n-3)}{n-1} &{} \quad \mathrm {if}\; n>5. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \left( \int _{\mathbb {R}^n}{\left| \widetilde{\mathcal {K}}(x,y)\right| }^p\frac{dy}{ {\langle y \rangle }^{3}}\right) ^{1/p}&\le C\left( \int _{\mathbb {R}^n}\frac{1}{(1+|x-y|)^{\frac{n-1}{2}p}}\frac{dy}{{\langle y\rangle }^3}\right) ^{1/p}\\&\le C(1+|x|)^{\frac{n-1}{2}}\left( \int _{\mathbb {R}^n}\frac{1}{(1+|y|)^{\frac{n-1}{2}p}}\frac{dy}{{\langle y\rangle }^3}\right) ^{1/p}\\&\le C(x)<\infty . \end{aligned}$$

Similarly, $\frac{\partial }{\partial s}\widetilde{\mathcal {K}}(x,.)\in L^p\Big (\frac{dy}{{\langle y\rangle }^3}\Big )$.

Finally,

$$\begin{aligned} \left( \int _{\mathbb {R}^n}{\left| \nabla _{S_{\theta }}\widetilde{\mathcal {K}}(x,y)\right| }^p\frac{dy}{ {\langle y \rangle }^{3}}\right) ^{1/p}&\le C|x|\left( \int _{\mathbb {R}^n}\frac{|y|^p}{(1+|x-y|)^{\frac{n+1}{2}p}}\frac{dy}{{\langle y\rangle }^3}\right) ^{1/p}\\&\le C|x|(1+|x|)^{\frac{n+1}{2}}\left( \int _{\mathbb {R}^n}\frac{|y|^p}{(1+|y|)^{\frac{n+1}{2}p}}\frac{dy}{{\langle y\rangle }^3}\right) ^{1/p}\\&\le C|x|(1+|x|)^{\frac{n+1}{2}}\left( \int _{\mathbb {R}^n}\frac{1}{(1+|y|)^{\frac{n-1}{2}p}}\frac{dy}{{\langle y\rangle }^3}\right) ^{1/p}\\&\le C(x)<\infty . \end{aligned}$$

$\square$

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)

$$\begin{aligned} F_m^j\in L^p({\langle x\rangle }^{-3}{dx})&\Longleftrightarrow \int _0^\infty {\left| \frac{J_{\nu (m)}(r)}{r^{(n-2)/2}}\right| }^p\frac{r^{n-1}dr}{(1+r^2)^{3/2}}<\infty \\&\Longleftrightarrow \int _0^\infty r^{(n-1)-\frac{p}{2}(n-1)+3}dr<\infty \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {R}^n}\big \vert u(x)Y_k^l(\xi )\varphi (\vert x\vert )\big \vert \frac{dx}{{\langle x \rangle }^{3}}\le C, \end{aligned}$$

which implies that

$$\begin{aligned} \int _0^\infty \left| \frac{J_{\nu (k)}(r)}{r^{(n-2)/2}}\varphi (r) \right| \frac{r^{n-1}}{{\langle r \rangle }^{3}}dr \le C. \end{aligned}$$

Consequently, by duality

$$\begin{aligned} \int _0^\infty {\left| \frac{J_{\nu (k)}(r)}{r^{(n-2)/2}}\right| }^p\frac{r^{n-1}dr}{(1+r^2)^{3/2}}<\infty , \end{aligned}$$

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

$$\begin{aligned} \Phi (x,y)=\frac{i}{4}(2\pi \left| x-y \right| )^{-(n-2)/2}H_{\frac{n-2}{2}}^1(\left| x-y \right| ). \end{aligned}$$

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 $\rho >R$,

$$\begin{aligned} v(x)=\rho ^{n-1}\int _{\mathbb {S}^{n-1}}\left( \frac{\partial v}{\partial s}(\rho \omega )\Phi (x,\rho \omega )-\frac{\partial \Phi }{\partial s}(x,\rho \omega )v(\rho \omega )\right) \,d\sigma (\omega ). \end{aligned}$$

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 $\mathcal {B}_R$,

$$\begin{aligned} v(x)=C_R\int _{2R\le \vert y \vert \le 3R}\left( \frac{\partial v}{\partial s}(y)\Phi (x,y)-\frac{\partial \Phi }{\partial s}(x,y)v(y)\right) \frac{dy}{{\langle y \rangle }^3}. \end{aligned}$$

(28)

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

$$\begin{aligned} \left| {\frac{\partial ^\alpha u}{\partial x^\alpha }(x)}\right| \le C_{K,\alpha }\Vert u\Vert _{\mathcal {H}^p}, \quad u\in \mathcal {W}^p,\;x\in K. \end{aligned}$$

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

$$\begin{aligned} R_N^\delta f(\xi )=\sum _{k=0}^N\sum _{j=1}^{d_k}\left( 1-\frac{k}{N+1}\right) ^\delta a_{kj}Y_k^j(\xi ). \end{aligned}$$

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 $f\in L^{p}(\mathbb {S}^{n-1})$,

$$\begin{aligned} R_{N}^{\delta }f\rightarrow f\quad \,in\,L^{p}(\mathbb {S}^{n-1}), \end{aligned}$$

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

$$\begin{aligned} {\left\| R_{N}^{\delta }f\right\| }_{L^{p}(\mathbb {S}^{n-1})}\le C_{p,\delta }{\left\| f\right\| }_{L^{p}(\mathbb {S}^{n-1})} \end{aligned}$$

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

$$\begin{aligned} u(r\xi )=\sum _{m=0}^{\infty }\sum _{j=1}^{d_{m}}a_{mj}F_m^j(r\xi ), \end{aligned}$$

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

$$\begin{aligned} \Lambda _N^p(r)=&\int _{\mathbb {S}^{n-1}}\left( {\big \vert (R_N^\delta u-u)(r\xi )\big \vert }^p+{\left| \frac{\partial }{\partial r}(R_N^\delta u-u)(r\xi )\right| }^p\right. \\&+\, \left. {\big \vert (-\Delta _{S})^{1/2}(R_N^\delta u-u)(r\xi )\big \vert }^p \right) d\sigma (\xi ). \end{aligned}$$

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

$$\begin{aligned} \lim _{N\rightarrow \infty } \Lambda _N^p(r)=0. \end{aligned}$$

Also, using the uniform boundedness of the Riesz means (Theorem 9) we obtain

$$\begin{aligned} \Lambda _N^p(r)\le C\int _{\mathbb {S}^{n-1}}\left( {\left| u(r\xi )\right| }^p+{\left| \frac{\partial }{\partial r}u(r\xi )\right| }^p+{\left| (-\Delta _{S})^{1/2}u(r\xi )\right| }^p\right) d\sigma (\xi ), \end{aligned}$$

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

$$\begin{aligned} 0=\int _0^\infty \lim _{N\rightarrow \infty } \Lambda _N^p(r)r^{n-1}\frac{dr}{(1+r^2)^{3/2}} =\lim _{N\rightarrow \infty }\int _0^\infty \Lambda _N^p(r)r^{n-1}\frac{dr}{(1+r^2)^{3/2}}. \end{aligned}$$

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,

$$\begin{aligned} \left| \mathcal {P}u_n(x)-\mathcal {P}u(x)\right|&\le \int _{\mathbb {R}^n}\left| \mathcal {K}(x,y)\right| \left| (u_n-u)(y)\right| \frac{dy}{{\langle y\rangle }^3}\\&+\int _{\mathbb {R}^n}\left| \frac{\partial \mathcal {K}}{\partial s}(x,y)\right| \left| \frac{\partial }{\partial s}(u_n-u)(y)\right| \frac{dy}{{\langle y\rangle }^3}\\&+\int _{\mathbb {R}^n}\left| \nabla _{S_\theta }\mathcal {K}(x,y)\right| \left| \nabla _{S_\theta }(u_n-u)(y)\right| \frac{dy}{{\langle y\rangle }^3}. \end{aligned}$$

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

$$\begin{aligned} \left| u_n(x)-\mathcal {P}u(x)\right| =\left| \mathcal {P}u_n(x)- \mathcal {P}u(x)\right| \le C(x){\left\| u_n-u\right\| }^p_{\mathcal {H}^p}\longrightarrow 0. \end{aligned}$$

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

$$\begin{aligned} \left\| f\right\| ^{p}_{L^{p,2}}:=\int _{0}^{\infty }\left( \int _{\mathbb {S}^{n-1}} {\left| f(r\xi )\right| }^{2} d\sigma (\xi )\right) ^{\frac{p}{2}}d\mu (r)<\infty , \end{aligned}$$

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

$$\begin{aligned} \left\| u\right\| ^{p}_{\mathcal {H}^{p,2}}&:=\int _{0}^{\infty }\left( \int _{\mathbb {S}^{n-1}} ( {\left| u(r\xi )\right| }^{2}+{\left| \nabla _{S} u(r\xi ) \right| }^{2} )d\sigma (\xi )\right) ^{\frac{p}{2}} d\mu (r)\\&\sim \int _{0}^{\infty }\left( \int _{\mathbb {S}^{n-1}} ( {\left| u(r\xi )\right| }^{2}+{\vert (-\Delta _{S})^{1/2}u(r\xi )\vert }^{2} )d\sigma (\xi )\right) ^{\frac{p}{2}}d\mu (r), \end{aligned}$$

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

$$\begin{aligned} Tu(r\xi )=(-\Delta _{S_{\xi }})^{1/2}\int _{\mathbb {R}^{n}} (-\Delta _{S_{\theta } })^{1/2}{\mathcal {K'}}(x,y)u(y)\frac{dy}{ {\langle y \rangle }^{3}}. \end{aligned}$$

(29)

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

$$\begin{aligned}&\left| \int _{\mathbb {R}^{n}}(-\Delta _{S_{\theta }})^{1/2} {\mathcal {K'}}(x,y)u(y)\frac{dy}{ {\langle y \rangle }^{3}}\right| \\&\le \left\| (-\Delta _{S_{\theta }})^{1/2}{\mathcal {K'}} (x,\cdot )\right\| _{L^{p',2}\big (\mathbb {R}^{n},\frac{dy}{\langle y \rangle ^{3}}\big )}\left\| u\right\| _{L^{p,2}\big (\mathbb {R}^{n},\frac{dy}{\langle y \rangle ^{3}}\big )}\\&\le C\left\| \nabla _{S}{\mathcal {K'}}(x,\cdot )\right\| _{L^{p',2}\big (\mathbb {R}^{n},\frac{dy}{\langle y \rangle ^{3}}\big )}\left\| u\right\| _{L^{p,2}\big (\mathbb {R}^{n},\frac{dy}{\langle y\rangle ^{3}}\big )}\\&<\infty . \end{aligned}$$

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

$$\begin{aligned} n^4\int _0^{\infty }|J_{n}(r)|^2 w(r)d\tilde{\mu }(r)\int _0^{\infty }|J_{n}(r)|^2 w^{-1}(r)d\tilde{\mu }(r)\le C, \end{aligned}$$

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

$$\begin{aligned} m^4\int _0^{\infty }|J_{\nu (m)}(r)|^2r^aw(r)d\tilde{\mu }(r)\int _0^{\infty }|J_{\nu (m)}(r)|^2r^{-a}w^{-1}(r)d\tilde{\mu }(r)\le C, \end{aligned}$$

(30)

where C is independent of m.

Proof

Let $I^1$ and $I^2$ be the integrals given by

$$\begin{aligned} I^1=\int _0^{\infty }|J_{\nu (m)}(r)|^2r^aw(r)d\tilde{\mu }(r) \end{aligned}$$

and

$$\begin{aligned} I^2=\int _0^{\infty }|J_{\nu (m)}(r)|^2r^{-a}w^{-1}(r)d\tilde{\mu }(r), \end{aligned}$$

respectively.

We split these integrals as

$$\begin{aligned} I^1=\int _0^1+\int _1^{\nu (m)\text {sech}\alpha _0}+\int _{\nu (m) \text {sech}\alpha _0}^{2\nu (m)}+ \int _{2\nu (m)}^\infty =\sum _{i=1}^4 I^1_i \end{aligned}$$

and

$$\begin{aligned} I^2=\int _0^1+\int _1^{\nu (m)\text {sech}\alpha _0} +\int _{\nu (m)\text {sech}\alpha _0}^{2\nu (m)}+ \int _{2\nu (m)}^\infty =\sum _{j=1}^4 I^2_j. \end{aligned}$$

We proceed as in the proof of Lemma 7. We will prove that

$$\begin{aligned} m^4I^1_iI^2_j\le C;\quad i,j\in \{1,2,3,4\}. \end{aligned}$$

Suppose $m\ge 1$, then by Hölder’s inequality and the estimates of Bessel functions $\left( D1\right)$–$\left( D4\right)$ we have

$$\begin{aligned} I^1_1&\le \tilde{\mu }([0,1])^{1/\beta }\left( \int _0^1|J_{\nu (m)}(r)|^{2\beta ^{'}} r^{a\beta ^{'}}d\tilde{\mu }(r)\right) ^{1/\beta ^{'}} \left( \frac{1}{\tilde{\mu }([0,1])}\int _0^1 w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }\\&\le \frac{C}{(2^mm!)^2}\left( \int _0^1r^{(n-2)\beta ^{'} +a\beta ^{'}}d\tilde{\mu }(r)\right) ^{1/\beta ^{'}} \left( \frac{1}{\tilde{\mu }([0,1])}\int _0^1 w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }\\&\le \frac{C}{(2^mm!)^2}\left( \frac{1}{\tilde{\mu }([0,1])} \int _0^1 w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }, \end{aligned}$$

$$\begin{aligned} I^1_4&\le \frac{C}{\nu (m)^{1/\beta }}\left( \int _{2\nu (m)}^\infty r^{-\beta ^{'}+a\beta ^{'}}d\tilde{\mu }(r)\right) ^{1/\beta ^{'}} \left( \frac{1}{\tilde{\mu }([2\nu (m),\infty ])}\int _{2\nu (m)}^\infty w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }\\&\le \frac{C\nu (m)^a}{m^2}\left( \frac{1}{\tilde{\mu }([2\nu (m),\infty ])} \int _{2\nu (m)}^\infty w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta } \end{aligned}$$

and

$$\begin{aligned} I^1_2&\le C\left( \int _1^{\nu (m)c}e^{-2\nu (m)\beta ^{'}\phi (r)}\,dr\right) ^{1/\beta ^{'}} \left( \frac{1}{\tilde{\mu }([1,\nu (m)c])}\int _1^{\nu (m)c} w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }\\&\le \frac{C}{e^{2m\beta _0}}\left( \frac{1}{\tilde{\mu }([1,\nu (m)c])}\int _1^{\nu (m)c} w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }, \end{aligned}$$

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

$$\begin{aligned} I^1_3&\le \frac{C\nu (m)^a}{m^2}\left( \frac{1}{\tilde{\mu }([\nu (m)c,2\nu (m)])}\int _{\nu (m)c}^{2\nu (m)} w^{\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }. \end{aligned}$$

Similarly, we have that

$$\begin{aligned} I^2_1&\le \frac{C}{(2^mm!)^2}\left( \frac{1}{\tilde{\mu }([0,1])}\int _0^1 w^{-\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta },\\ I^2_2&\le \frac{C\nu (m)^{-a}}{e^{2m\beta _0}}\left( \frac{1}{\tilde{\mu }([1,2\nu (m)c])} \int _1^{2\nu (m)c}w^{-\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta },\\ I^2_3&\le \frac{C\nu (m)^{-a}}{m^2}\left( \frac{1}{\tilde{\mu }([\nu (m)c,2\nu (m)])} \int _{\nu (m)c}^{2\nu (m)}w^{-\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }. \end{aligned}$$

Furthermore, using that $a>\frac{1}{\beta }-2$ it follows that

$$\begin{aligned} I^2_4&\le \frac{C\nu (m)^{-a}}{m^2}\left( \frac{1}{\tilde{\mu }([2\nu (m),\infty ])} \int _{2\nu (m)}^\infty w^{-\beta }(r)d\tilde{\mu }(r)\right) ^{1/\beta }, \end{aligned}$$

Consequently, since $w^{\beta }\in A_2(d\tilde{\mu }(r))$,

$$\begin{aligned} m^4I^1_iI^2_j\le C;\quad i,j\in \{1,2,3,4\}. \end{aligned}$$

$\square$

Proposition 3

Let $\beta _n\in (1,\infty )$ such that

$$\begin{aligned} \beta '_n=\left\{ \begin{array}{ll} \infty &{} \quad \mathrm {if}\; n=2,3\\ 2+\frac{4}{n-3} &{} \quad \mathrm {if}\; n>3. \end{array} \right. \end{aligned}$$

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,

$$\begin{aligned} u(r\xi )=\sum _{m,j}u_{mj}(r)Y_m^j(\xi ), \end{aligned}$$

and using the Fourier expansion of the kernel ${\mathcal {K'}}$ we have

$$\begin{aligned} Tu(r\xi )=\sum _{m,j} T_{mj} u_{mj}(r)Y_m^j(\xi ), \end{aligned}$$

(31)

where

$$\begin{aligned} T_{mj}f_{mj}(r)=C m(m+n-2)J_{\nu (m)}(r)r^{-(n-2)/2}\int _0^\infty J_{\nu (m)}(s)s^{-(n-2)/2}f_{mj}(s)\,d\mu (s). \end{aligned}$$

(32)

Showing that T is bounded on $L^{p,2}$ is equivalent to prove the vector-valued inequality,

$$\begin{aligned} \left( \int _0^{\infty }\left( \sum _{m,j}{|T_{mj}u_{mj}(r)|}^2\right) ^{\frac{p}{2}}d\mu (r)\right) ^{\frac{1}{p}}\le C\left( \int _0^{\infty }\left( \sum _{m,j}{|u_{mj}(r)|}^2\right) ^{\frac{p}{2}}d\mu (r)\right) ^{\frac{1}{p}}, \end{aligned}$$

(33)

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

$$\begin{aligned} \left( \int _0^{\infty }\left( \sum _{m,j}{|T_{mj}u_{mj}(s)|}^2\right) ^{\frac{p}{2}}d\mu (s)\right) ^{\frac{2}{p}}= \int _0^{\infty }\sum _{m,j}{|T_{mj}u_{mj}(s)|}^2h(s)d\mu (s). \end{aligned}$$

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

$$\begin{aligned}&\left( \int _0^{\infty }\left( \sum _{m,j}{|T_{mj}u_{mj}(s)|}^2\right) ^{\frac{p}{2}}d\mu (s)\right) ^{\frac{2}{p}}\\&\quad =\int _0^{\infty }\sum _{m,j}{|T_{mj}u_{mj}(s)|}^2s^{-(n-2)/r}g(s)d\mu (s)\\&\quad =\int _0^{\infty }\sum _{m,j}{|T_{mj}u_{mj}(s)|}^2s^{(n-2)(1-1/r)}g(s)d\tilde{\mu }(s)\\&\quad \le \sum _{m,j}\int _0^{\infty }{|T_{mj}u_{mj}(s)|}^2s^{2(n-2)/p} {(M_{\tilde{\mu }}[g^{\gamma }](s))}^{\frac{1}{\gamma }}d\tilde{\mu }(s)\\&\quad \le C\sum _{m,j}m^4\int _0^{\infty }{|J_{\nu (m)}(s)s^{-(n-2)/2}|}^2s^{2(n-2)/p}w(s)d\tilde{\mu }(s)\\&\qquad \times \int _0^{\infty }{|J_{\nu (m)}(s)s^{-(n-2)/2}|^2}s^{2(n-2)/q}w^{-1}(s)d\tilde{\mu }(s) \nonumber \\&\int _0^{\infty }{|u_{mj}(s)|}^2s^{2(n-2)/p}w(s)d\tilde{\mu }(s)\\&\quad \le C\sum _{m,j}\int _0^{\infty }{|u_{mj}(s)|}^2s^{2(n-2)/p}w(s)d\tilde{\mu }(s)\\&\qquad \times m^4\int _0^{\infty }{|J_{\nu (m)}(s)|}^2s^{(n-2)(\frac{2}{p}-1)}w(s)d\tilde{\mu }(s) \nonumber \\&\qquad \int _0^{\infty }{|J_{\nu (m)}(s)|^2}s^{-(n-2)(\frac{2}{p}-1)}w^{-1}(s)d\tilde{\mu }(s), \end{aligned}$$

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

$$\begin{aligned}&\left( \int _0^{\infty }\left( \sum _{m,j}{|T_{mj}u_{mj}(s)|}^2\right) ^{\frac{p}{2}}d\mu (s)\right) ^{\frac{2}{p}}\\&\quad \le C\int _0^{\infty }\sum _{m,j}{|u_{mj}(s)|}^2s^{2(n-2)/p}{(M_{\tilde{\mu }}[g^{\gamma }](s))}^{\frac{1}{\gamma }}d\tilde{\mu }(s)\\&\quad \le C\left( \int _0^{\infty }\left( \sum _{m,j}{|u_{mj}(s)|}^2s^{2(n-2)/p}\right) ^{\frac{p}{2}}d\tilde{\mu }(s)\right) ^{\frac{2}{p}}\left( \int _0^{\infty }{(M_{\tilde{\mu }}[g^{\gamma }](s))}^{\frac{r}{\gamma }}d\tilde{\mu }(s)\right) ^{\frac{1}{r}}\\&\quad \le C\left( \int _0^{\infty }\left( \sum _{m,j}{|u_{mj}(s)|}^2\right) ^{\frac{p}{2}}d\mu (s)\right) ^{\frac{2}{p}}. \end{aligned}$$

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

$$\begin{aligned} u_{mj}(r\xi )=r^\alpha {|J_{\nu (m)}(r)|}^{p'-1}sgn(J_{\nu (m)}(r)) \chi _{[\nu (m),2\nu (m)]}Y_m^j(\xi ) \end{aligned}$$

(34)

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

$$\begin{aligned} {\Vert u_{mj}\Vert }_{p,2}={\left( \int _{\nu (m)}^{2\nu (m)}{|J_{\nu (m)}(r)|}^{p'}r^{-(n-2)p'/2}d \mu (r)\right) }^{1/p} \end{aligned}$$

and

$$\begin{aligned} \Vert T_{mj}u_{mj}\Vert _{p,2} \ge Cm(m+n-2){\left( \int _{\nu (m)}^{2\nu (m)} {|J_{\nu (m)}(r)|}^{p} r^{-(n-2)p/2}d\mu (r)\right) }^{1/p}\times {\Vert u_{mj}\Vert }_{p,2}^p. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\Vert T_{mj}u_{mj}\Vert _{p,2}}{{\Vert u_{mj}\Vert }_{p,2}}\ge C{\left( \int _{\nu (m)}^{2\nu (m)} {|J_{\nu (m)} (r)|}^{p}dr\right) }^{1/p}{\left( \int _{\nu (m)}^{2\nu (m)} {|J_{\nu (m)}(r)|}^{p'}dr\right) }^{1/p'}, \end{aligned}$$

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

$$\begin{aligned} {\mathcal {K'}}(x,y),\;(-\Delta _{S_\xi })^{1/2} {\mathcal {K'}}(x,y),\;(-\Delta _{S_\xi })^{1/2}(-\Delta _{S_\theta })^{1/2} {\mathcal {K'}}(x,y), \end{aligned}$$

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

$$\begin{aligned} {\mathcal {K'}}(x,y)=\mathcal {M}_1(-\Delta _{S_\xi })^{1/2} (-\Delta _{S_\theta })^{1/2}{\mathcal {K'}}(x,y) \end{aligned}$$

and

$$\begin{aligned} (-\Delta _{S_\xi })^{1/2}{\mathcal {K'}}(x,y)= \mathcal {M}_2(-\Delta _{S_\xi })^{1/2}(-\Delta _{S_\theta })^{1/2}{\mathcal {K'}}(x,y), \end{aligned}$$

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

$$\begin{aligned} \mathcal {L}=(-\Delta _{S_\xi })^{1/2}\circ \mathcal {P'}\circ (-\Delta _{S_\xi })^{-1/2} \end{aligned}$$

(35)

is continuous in $L^{p,2}$.

But

$$\begin{aligned} \mathcal {L}u(x)=\int _{\mathbb {R}^{n}} \mathcal {K'}(x,y)u(y)\frac{dy}{ {\langle y \rangle }^{3}}+Tu(x), \end{aligned}$$

hence, in the notation of Proposition 3,

$$\begin{aligned} \mathcal {L}u(x)=\sum _{m,j} \left( \frac{1}{m(m+n-2)}+1\right) T_{mj} u_{mj}(r)Y_m^j(\xi ) \end{aligned}$$

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

$$\begin{aligned} \mathcal {T} u(x)=\nabla _{S_\xi }\int _{\mathbb {R}^n}\nabla _{S_\theta } \widetilde{\mathcal {K}}(x,y)\cdot u(y)\frac{dy}{{\langle y\rangle }^3},\;x=r\xi ,\;y=s\theta , \end{aligned}$$

(36)

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

$$\begin{aligned} \mathcal {T}_1 u(x)=C_n\int _{\mathbb {R}^n}|x||y|F_{n/2}(\vert x-y\vert )\left( \mathbf {A}(u,y)- \mathbf {A}(u,y)\cdot \frac{x}{|x|}\frac{x}{|x|}\right) \frac{dy}{{\langle y\rangle }^3}, \end{aligned}$$

(37)

$$\begin{aligned} \mathcal {T}_2 u(x)=C_n\int _{\mathbb {R}^n}|x||y|F_{n/2+1}(\vert x-y\vert )(x-\mathbf {P}_yx)\cdot \nabla _{S_\theta }u(y) (y-\mathbf {P}_xy)\frac{dy}{{\langle y\rangle }^3}, \end{aligned}$$

(38)

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

$$\begin{aligned} \left| A'_k\right| =\int _{0}^{\frac{\pi }{4}}\int _{\varphi _2}^{\varphi _1} \int _{\alpha (k)}^{\alpha (k)+l}r^2\sin {\varphi }dr\;d\varphi \;d\theta \ge Ck^2(\cos {\varphi _2}-\cos {\varphi _1}). \end{aligned}$$

Hence, to complete the proof of the lemma, it suffices to show that there exists $c>0$ such that

$$\begin{aligned} \cos {\phi _2}-\cos {\phi _1}\ge c. \end{aligned}$$

(39)

Denoting $\alpha (k)$ just by $\alpha$, we observe that $P_2=(c_2^{-1}t_2,t_2)$ with

$$\begin{aligned} \frac{t_2}{\alpha }=\frac{m+\sqrt{m^2+(\alpha ^2-m^2) (c_2^{-2}+1)}}{(c_2^{-2}+1)\alpha }. \end{aligned}$$

Let $\lambda >0$ and $\alpha >\lambda m$. Then $1-\frac{1}{\lambda ^2}<1-\frac{m^2}{\alpha ^2}$, and

$$\begin{aligned} \frac{t_2}{\alpha } \ge \frac{\sqrt{c_2^{-2}+1}\sqrt{\alpha ^2-m^2}}{(c_2^{-2}+1)\alpha } \ge \frac{1}{\sqrt{c_2^{-2}+1}}\sqrt{1-\frac{1}{\lambda ^2}}. \end{aligned}$$

(40)

Similarly, we have that $P_1=(c_1^{-1}t_1,t_1)$ and

$$\begin{aligned} \frac{t_1}{\alpha +l}=\frac{m+\sqrt{m^2+[(\alpha +l)^2-m^2](c_1^{-2}+1)}}{(c_1^{-2}+1)(\alpha +l)}. \end{aligned}$$

Since $\alpha >\lambda m$ then $\frac{m}{\alpha +l}<\frac{1}{\lambda }$, hence

$$\begin{aligned} \frac{t_1}{\alpha +l}\le \frac{1}{(c_1^{-2}+1)\lambda } +\frac{\sqrt{\frac{1}{\lambda ^2}+(c_1^{-2}+1)}}{c_1^{-2}+1}. \end{aligned}$$

(41)

By (40) and (41), we have that

$$\begin{aligned} \frac{t_2}{\alpha }-\frac{t_1}{\alpha +l}\ge \frac{1}{\sqrt{c_2^{-2}+1}}\sqrt{1-\frac{1}{\lambda ^2}} -\frac{1}{(c_1^{-2}+1)\lambda }-\frac{\sqrt{\frac{1}{\lambda ^2} +(c_1^{-2}+1)}}{c_1^{-2}+1}. \end{aligned}$$

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

$$\begin{aligned} \cos {\varphi _2}-\cos {\varphi _1}&=\frac{t_2-m}{\alpha }-\frac{t_1-m}{\alpha +\lambda }\\&=\left( \frac{t_2}{\alpha }-\frac{t_1}{\alpha +\lambda }\right) +h, \end{aligned}$$

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

$$\begin{aligned} (\mathbf {P}_xy)_3=(\mathbf {P}_xm\mathbf {e_3})_3+(\mathbf {P}_xy')_3<Cm+\epsilon . \end{aligned}$$

Therefore,

$$\begin{aligned} (y-\mathbf {P}_xy)_3\ge (m-\epsilon )-(Cm+\epsilon )=(1-C)m-2\epsilon \ge Cm\ge C|y| \end{aligned}$$

(42)

for all $\epsilon$ sufficiently small, m sufficiently large and choosing $C<1$.

On the other hand, we have that

$$\begin{aligned} (x-\mathbf {P}_yx)\cdot u_m(y)=x_1-\frac{y\cdot x}{|y|}\frac{y}{|y|}\cdot \mathbf {e_1}, \end{aligned}$$

estimating above the right hand side we have

$$\begin{aligned} \left| \frac{y\cdot x}{|y|}\frac{y}{|y|}\cdot \mathbf {e_1}\right| \le \frac{\epsilon }{m^2}\left( |x_1y_1|+|x_2y_2|+|x_3y_3|\right) \le C|x_1|\epsilon /m. \end{aligned}$$

Hence,

$$\begin{aligned} (x-\mathbf {P}_yx)\cdot u_m(y)=x_1-O(|x_1|\epsilon /m)>C|x_1|>C\left| x\right| . \end{aligned}$$

(43)

Let $x\in A_k$. For (42), (43) and (4), we deduce that

$$\begin{aligned} \left| \mathcal {T}_2 u_m(x)\right| \ge C\int _{B_m}|x|m\frac{1}{k^3}|x|m\frac{dy}{{\langle y\rangle }^3}\ge \frac{C|x|^2}{k^3m}. \end{aligned}$$

By Lemma 9,

$$\begin{aligned} {\left\| \mathcal {T}_2 u_m\right\| }_{L^p(\bigcup _{k\ge Cm}R\cap A_k)}^p&= \int _{\bigcup _{k\ge Cm}R\cap A_k}{\left| \mathcal {T}_2 u_m(x)\right| }^p\frac{dx}{{\langle x\rangle }^3} {\ge } C\sum _{k\ge Cm}\int _{A_k}\left( \frac{k^2}{k^3m}\right) ^p\frac{dx}{{\langle x\rangle }^3}\\&\ge C\sum _{k\ge Cm}\left( \frac{1}{km}\right) ^p\frac{1}{k^3}\left| R\cap A_k\right| \ {\ge }\frac{C}{m^p}\sum _{k\ge Cm}\frac{1}{k^{p+1}}\ge \frac{C}{m^{2p}}, \end{aligned}$$

and so

$$\begin{aligned} {\left\| \mathcal {T}_2 u_m\right\| }_{L^p(\bigcup _{k\ge Cm}R\cap A_k)}\ge \frac{C}{m^{2}}. \end{aligned}$$

(44)

Furthermore,

$$\begin{aligned} \left| \mathcal {T}_1 u_m(x)\right| \le C\int _{B_m}|x|\frac{m}{k^2}\frac{dy}{{\langle y\rangle }^3}\le \frac{C|x|}{m^2k^2}. \end{aligned}$$

Then,

$$\begin{aligned} {\left\| \mathcal {T}_1 u_m\right\| }_{L^p(\bigcup _{k\ge Cm}R\cap A_k)}^p&= \int _{\bigcup _{k\ge Cm}R\cap A_k}{\left| \mathcal {T}_1 u_m(x)\right| }^p\frac{dx}{{\langle x\rangle }^3} \\&\le C\int _{\bigcup _{k\ge Cm}R\cap A_k}\left( \frac{|x|}{m^2k^2}\right) ^p\frac{dx}{{\langle x\rangle }^3}\\&\le C\sum _{k\ge Cm}\int _{R\cap A_k}\left( \frac{k}{m^2k^2}\right) ^p\frac{dx}{{\langle x\rangle }^3} \\&\le \frac{C}{m^{2p}}\sum _{k\ge Cm}\int _{R\cap A_k}\frac{1}{k^{p+3}}\left| R\cap A_k\right| \\&\le \frac{C}{m^{2p}}\sum _{k\ge Cm}\frac{1}{k^{p+1}}\le \frac{C}{m^{3p}}. \end{aligned}$$

Consequently,

$$\begin{aligned} {\left\| \mathcal {T}_1 u_m\right\| }_{L^p(\bigcup _{k\ge Cm}R\cap A_k)}\le \frac{C}{m^{3}}. \end{aligned}$$

(45)

Finally, by (44) and (45)

$$\begin{aligned} {\left\| \mathcal {T} u_m\right\| }_p&={\left\| (\mathcal {T}_2u_m-(-\mathcal {T}_1u_m)\right\| }_p\\&\ge {\left\| \mathcal {T}_2u_m\right\| }_{L^p(\bigcup _{k\ge Cm}R\cap A_k)}-{\left\| \mathcal {T}_1 u_m\right\| }_{L^p(\bigcup _{k\ge Cm}R\cap A_k)} \\&\ge C\left( \frac{1}{m^2}-\frac{1}{m^3}\right) \ge \frac{C}{m^2}, \end{aligned}$$

then, since ${\left\| u_m\right\| }_p\sim m^{-3/p}$,

$$\begin{aligned} \frac{{\left\| \mathcal {T} u_m\right\| }_p}{{\left\| u_m\right\| }_p}\ge Cm^{3/p-2}. \end{aligned}$$

(46)

Hence $\mathcal {T}$ is not bounded if $p\in (1,3/2)$. $\square$