Abstract
We identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein’s method of complex interpolation, we prove the conjectured inequalities when the target space is \(L^2\), and show that this recovers in the limit the celebrated Tomas-Stein theorem.
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1 Introduction
For any \(0<s<1\) and \(r>0\), we consider the function
where, using the standard notation \(\sigma _{n-1} = \frac{2\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2})}\) for the \((n-1)\)-dimensional measure of the unit sphere \({\mathbb {S}}^{n-1}\subset {\mathbb {R}}^n\), we have let
Define measures on \({\mathbb {R}}^n\) by letting
From (1.1) it immediately follows that \(A^{(s)}_r \in L^1({\mathbb {R}}^n)\). Moreover, from [9, Lemma 15.3]) one has for every \(s\in (0,1)\) and \(r>0\)
We next define the operator
It follows from (1.4) and Young’s convolution theorem that
and that for any \(f \in L^p({\mathbb {R}}^n)\), we have
which shows that \({\mathscr {A}}^{(s)}_r\) is a contraction in \(L^p({\mathbb {R}}^n)\) for every \(0<s<1\) and \(r>0\).
For any \(f\in {\mathscr {S}}({\mathbb {R}}^n)\), we denote by \({{\hat{f}}}(\xi ) = {\mathscr {F}}(f)(\xi ) = \int _{{\mathbb {R}}^n} e^{-2\pi i \langle \xi ,x\rangle } f(x) \textrm{d}x\) its Fourier transform, and ask the following
Question: Let \(\frac{1}{2}\le s<1\). If \(1\le p < \frac{2n}{n+2s -1}\), is it true that for \(1\le q \le \frac{n+1-2s}{n+1} p'\) and for every \(f\in {\mathscr {S}}({\mathbb {R}}^n)\), one has for some \(C^{(s)}(n,p)>0\)
We note that the restriction \(s\ge \frac{1}{2}\) serves to guarantee that \(\frac{2n}{n+2s -1}\le 2\). Therefore, the hypothesis \(f\in L^p({\mathbb {R}}^n)\) implies that f be in the Hausdorff-Young range [1, 2]. As a consequence, \({{\hat{f}}}\) is a function in \(L^{p'}({\mathbb {R}}^n)\).
Our interest in the above conjecture stems from the following observations. Assume that (1.7) does hold for any s such that \(\frac{1}{2} \le s < 1\). Since \(\frac{2n}{n+2s -1} \searrow \frac{2n}{n +1}\) as \(s\nearrow 1\), if we take \(1\le p < \frac{2n}{n+1}\) and \(q\le \frac{n-1}{n+1} p'\), then it is immediate to verify that for any \(s\in [\frac{1}{2},1)\)
(for the second of these inequalities simply note that \(q\le \frac{n-1}{n+1} p' < \frac{n-1+ 2(1-s)}{n+1} p' =\frac{n+1-2s}{n+1} p'\)), therefore (1.7) holds. But we now have the following fact implicitly contained in the seminal work [14] of M. Riesz (for a proof see [9, Prop. 15.4]).
Proposition 1.1
For every function \(f\in {\mathscr {S}}({\mathbb {R}}^n)\) and \(x\in {\mathbb {R}}^n\), one has
where
is the spherical average of f over the sphere \(S(x,r) = \{y\in {\mathbb {R}}^n\mid |y-x|=r\}\).
Therefore, passing to the limit in (1.7) and using Proposition 1.1, if \(C^{(s)}(n,p)\) converges to a number \(C(n,p)>0\) as \(s\rightarrow 1\), we would infer that for \(1\le p < \frac{2n}{n+1}\) and \(q\le \frac{n-1}{n+1} p'\) the following limiting inequality holds
As it is well known, this is the famous restriction conjecture of C. Fefferman and E. Stein for the Fourier transform, see [4, 6,7,8, 15] and [19].
![figure a](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs12220-023-01477-6/MediaObjects/12220_2023_1477_Figa_HTML.jpg)
Eli Stein lecturing on the restriction problem, UChicago, 1985
One obvious advantage of (1.7) over (1.8) is that the support of the measures (1.3) is \({\mathbb {R}}^n\setminus B(0,r)\), instead of the lower-dimensional manifold \({\mathbb {S}}^{n-1}\subset {\mathbb {R}}^n\). Notice that if we let \(d\sigma \) denote the surface measure concentrated on the sphere \({\mathbb {S}}^{n-1}\), then Proposition 1.1 can be equivalently stated as follows:
Concerning the measures (1.3), we recall that in his above quoted paper, M. Riesz developed the theory of the nonlocal operators \((-\Delta )^s\) and their inverses \(I_{2s}\), the operators of fractional integration which play a pervasive role in analysis, see also [16, Ch. 5]. Among other things, he solved by inversion the Dirichlet problem
and proved that for every \(x\in B_r\), the unique solution to (1.10) is provided by
see formula (3) on p. 17 in [14], but also (1.6.11’) and (1.6.2) on pages 122 and 112 in [11]. It is clear from (1.11) that \(u(0) = \int _{{\mathbb {R}}^n} f(y) A^{(s)}_r(y) \textrm{d}y = {\mathscr {A}}^{(s)}_r f(0)\). The role of the operators \({\mathscr {A}}^{(s)}_r\) is further elucidated by the following result, see [9, Prop. 15.6].
Proposition 1.2
(The Blaschke-Privalov fractional Laplacian) Let \(0<s<1\) and suppose that \(f\in {\mathscr {L}}_s({\mathbb {R}}^n)\) be in \(C^{2s+\varepsilon }\) in a neighbourhood of \(x\in {\mathbb {R}}^n\), for some \(0<\varepsilon < 1\). One has
where \(\gamma (n,s) = \frac{s 2^{2s} \Gamma (\frac{n}{2} + s)}{\pi ^{\frac{n}{2}} \Gamma (1-s)}\).
Here, for \(0<s<1\), we have denoted by \({\mathscr {L}}_s({\mathbb {R}}^n)\) the space of measurable functions \(f:{\mathbb {R}}^n\rightarrow {{\overline{{\mathbb {R}}}}}\) for which the norm
Returning to the inequality (1.7), we mention that, similarly to the restriction problem (1.8), it cannot possibly hold for every exponent \(p\in [1,2]\) in the Hausdorff-Young range. To understand the constraint \(1\le p < \frac{2n}{n+2s -1}\), denote by \(T:L^p({\mathbb {R}}^n)\rightarrow L^q({\mathbb {R}}^n,d\mu ^{(s)}_1)\) the “restriction” operator in (1.7). Then its adjoint \(T^\star :L^{q'}({\mathbb {R}}^n,d\mu ^{(s)}_1)\rightarrow L^{p'}({\mathbb {R}}^n)\) is easily seen to be given by
so that
Assuming (1.7), we would, thus, have by duality
where in the last equality, we have used (1.4). Therefore, the validity of (1.7) for some p implies that \(\widehat{A^{(s)}_1}\in L^{p'}({\mathbb {R}}^n)\). Now, in Section 3, we prove the following.
Theorem 1.3
Let \(s\in (0,1)\) and \(n\ge 2\). Then the Fourier transform of the kernel defined by (1.1) is given by
Using Theorem 1.3 in (3.9) of Corollary 3.2, we obtain the following important decay at infinity
which shows that
We conclude that the inequality (1.7) cannot possibly hold for \(p\ge \frac{2n}{n+2s -1}\).
Before proceeding, we pause to comment on an aspect of Theorem 1.3. As the reader will see its proof is somewhat more involved than its well-known counterpart in the case \(s=1\). This is due to the nonlocal nature of the measure \(d\mu ^{(s)}_1\), compared to the surface measure \(d\sigma \) of the unit sphere \({\mathbb {S}}^{n-1}\). To explain this comment, we stress that
is just a rescaled spherically symmetric eigenfunction of the (local) differential operator \(\Delta \) in \({\mathbb {R}}^n\). To see this, simply observe that for every \(\lambda >0\), the function
is a spherically symmetric solution of the Helmholtz equation \(\Delta f_\lambda = - \lambda f_\lambda \) in \({\mathbb {R}}^n\), which shows that (1.17) solves such PDE with \(\lambda = 4\pi ^2\). However, the equations (1.10), (1.11) above, and Proposition 1.2 in particular, underscore that the Fourier transform of the measure \(d\mu ^{(s)}_r\) is instead connected to the pseudodifferential operator \((-\Delta )^s\), and computations with such nonlocal operator are usually more involved. In this regard, we recall the following quote from p. 51 in [11]:...“In the theory of M. Riesz kernels, the role of the Laplace operator, which has a local character, is taken...by a non-local integral operator...This circumstance often substantially complicates the theory...”. In Lemma 3.3, we show that for \(n\ge 2\), one has for every \(\xi \in {\mathbb {R}}^n\)
This result (which can be seen as a comforting a posteriori confirmation of the correctness of the computations leading to Theorem 1.3) is of course not surprising in view of (1.9) above. It is worthwhile noting at this moment that (1.1) are clearly reminiscent of the classical Bochner-Riesz kernels
whose Fourier transform is given by
If we compare this formula with (1.17), it is clear that \({{\hat{K}}}_z\rightarrow \frac{1}{2} \ \widehat{d\sigma }\) as \(z\rightarrow -1\). While the computation of the Fourier transform of \(A^{(s)}_1\) is more involved than (1.19), there are advantages in working with (1.1) instead of (1.18). One of them is that, as we have mentioned, \(\widehat{A^{(s)}_1}\) is directly connected to the nonlocal operator \((-\Delta )^s\), while this is not the case for (1.19).
We have seen that (1.7) is only possible when p satisfies (1.16). But, given a p in such range what is the optimal range of q’s? To answer this question, we use the well-known argument of Knapp, except that because of the presence of the measure \(d\mu _1^{(s)}\) in (1.7), we need to work a bit more. In Proposition 2.1 below, we show that, given p within the range (1.16), a necessary condition for (1.7) to hold is that
Notice that when the target space is \(L^2\), then in view of (1.20), the conjecture asks whether it is true that (1.7) holds with \(q=2\) for any \(f\in {\mathscr {S}}({\mathbb {R}}^n)\) and \(2 \le \frac{n+1-2s}{n+1} p'\). This is equivalent to asking that \(\frac{1}{p'} \le \frac{n+1-2s}{2(n+1)} = \frac{1}{2} - \frac{s}{n+1}\), or equivalently \(\frac{1}{p} \ge \frac{1}{2} + \frac{s}{n+1} = \frac{n+1+2s}{2(n+1)}\), and therefore, for any
In the next result, we prove this conjecture.
Theorem 1.4
For a given \(s\in (0,1)\) let \(p = \frac{2(n+1)}{n+1+2s}\). Then there exists a constant \(C^{(s)}(n)>0\) such that for every \(f\in {\mathscr {S}}({\mathbb {R}}^n)\) one has
In order to establish Theorem 1.4, we exploit Plancherel as in [20] and reduce matters to proving that the operator \(R^{(s)} f = \widehat{A^{(s)}_1} \star f\) maps \(L^p({\mathbb {R}}^n)\rightarrow L^{p'}({\mathbb {R}}^n)\). This is ultimately achieved using Stein’s theorem of complex interpolation by embedding \(R^{(s)}\) into an analytic family of operators \(\{T_z\}_{z\in S}\) in the strip \(S = \{z\in {\mathbb {C}}\mid - \frac{n-1}{2}\le \Re z\le 1\}\). Specifically, we show that
with appropriate bounds on the operator norms. Since the constant \(C^{(s)}(n)\) in (1.22) is bounded uniformly in \(s\in (0,1)\), see (4.21) below, passing to the limit as \(s\rightarrow 1\) we recover the Tomas-Stein restriction theorem, see [20].
Remark 1.5
One should observe that the threshold exponent \(p = \frac{2(n+1)}{n+1+2s}\) in (1.21) is strictly less than 2 for any \(0<s<1\). Therefore, the limitation \(\frac{1}{2} \le s <1\) is not necessary in such case.
The plan of the paper is as follows. In Sect. 2, we adapt the well-known argument of Knapp to prove that, if for a given p in the range (1.16), the restriction inequality (1.7) does hold, then we must have \(1\le q \le \frac{n+1-2s}{n+1} p'\). Sect. 3 is devoted to proving Theorem 1.3, from which we obtain Corollary 3.2. The representation formula (3.8) contained in it will be quite important in the remainder of the paper. Finally in Sect. 4, we prove the nonlocal restriction Theorem 1.4. As a corollary of this result, we obtain the celebrated Tomas-Stein theorem. The paper closes with an appendix in Sect. 1 in which we gather some well-known facts and collect some results needed in the rest of the paper.
2 Necessary Condition for Restriction
In (1.16) above we have seen that the inequality (1.7) can possibly hold only when \(1\le p < \frac{2n}{n+2s -1}\). In this section, we adapt the well-known argument of Knapp to prove that, if for a given p in such range (1.7) does hold, then we must have \(1\le q \le \frac{n+1-2s}{n+1} p'\).
Proposition 2.1
A necessary condition for (1.7) to hold for \(1\le p < \frac{2n}{n+2s -1}\) is that \(1\le q \le \frac{n+1-2s}{n+1} p'\).
Proof
For \(\varepsilon >0\) small consider the parallelepiped \(K_\varepsilon = K'_\varepsilon \times [1-\varepsilon ,1]\) with sides parallel to the coordinate axis whose projection onto \({\mathbb {R}}^{n-1}\times \{0\}\) is the \((n-1)\)-dimensional cube \(K'_\varepsilon \) circumscribing the intersection of the hyperplane \(x_n = 1-\varepsilon \) with the unit sphere \({\mathbb {S}}^{n-1}\). If \(\theta _\varepsilon \) is the angle of aperture of the right-circular cone obtained by projecting to the origin the points of the \((n-2)\)-dimensional sphere obtained intersecting \({\mathbb {S}}^{n-1}\cap \{x_n=1-\varepsilon \}\), from elementary trigonometry, we have \(\cos \theta _\varepsilon = 1-\varepsilon \), \(\sin \theta _\varepsilon = R(\varepsilon ) = \sqrt{\varepsilon (2-\varepsilon )}\), and therefore, \(K'_\varepsilon = [-R(\varepsilon ),R(\varepsilon )]^{n-1}\).
Denoting now with \(B'(0,r) = \{x'\in {\mathbb {R}}^{n-1}\mid |x'|<r\}\), consider now the right-circular cylinders \(C_\varepsilon = B'(0,R(\varepsilon ))\times [1-\varepsilon ,1]\) and \(C^\star _\varepsilon = B'(0,\sqrt{n-1}\ R(\varepsilon ))\times [1-\varepsilon ,1]\). A moment’s thought reveals that
We note that
As in Knapp’s argument, if \({\textbf{1}}_E\) is the indicator function of a set \(E\subset {\mathbb {R}}^n\), we now consider the function \(f_\varepsilon = {\mathscr {F}}^{-1}({\textbf{1}}_{K_\varepsilon })\), so that \({{\hat{f}}}_\varepsilon = {\textbf{1}}_{K_\varepsilon }\). As it is well known
and therefore, \(f_\varepsilon \in L^p({\mathbb {R}}^n)\) for any \(p>1\) and moreover,
Next, we want to understand the asymptotic behaviour as \(\varepsilon \rightarrow 0^+\) of the quantity
In view of the inclusions (2.1), it suffices to understand the asymptotic behaviour of the right-hand side of (2.3) when the integral is performed on the set \(C_\varepsilon \cap ({\mathbb {R}}^n\setminus B(0,1))\). With this objective in mind, we obtain from Cavalieri’s principle
We now want to show that as \(\varepsilon \rightarrow 0^+\), we have
To see this we write \(G(\varepsilon ) = \int _{1-\varepsilon }^1 F(\varepsilon ,t) \textrm{d}t\), where
The chain rule gives
since \(F(\varepsilon ,1-\varepsilon ) = 0\). A simple computation gives
therefore,
Since \(R(\varepsilon )^{n-2} = (\varepsilon (2-\varepsilon ))^{\frac{n-2}{2}} \cong \varepsilon ^{\frac{n}{2} -1}\), and \(R'(\varepsilon ) \cong \varepsilon ^{-\frac{1}{2}}\), we infer that \(G(\varepsilon ) \cong \varepsilon ^{\frac{n+1}{2}-s}\), which gives the desired conclusion (2.4). In conclusion, we have shown that
Combining (2.2) with (2.5), we finally infer that a necessary condition for (1.7) to hold is
\(\square \)
3 The Fourier Transform of the Kernel \(A^{(s)}_1\)
In this section, we prove Theorem 1.3. Using Lemmas 5.3 and 5.4 in the Appendix, we establish a result which provides a key step in the proof of Theorem 1.3.
Lemma 3.1
For every \(0<s<1\), \(r>0\) and \(\xi \in {\mathbb {R}}^n\setminus \{0\}\), one has
Proof
To prove Lemma 3.1, we use Lemma 5.3 in which we take
Since \(s<1\), the condition \(\Re \beta >0\) is guaranteed. Also, \(\alpha +2\beta <\frac{7}{2}\) is equivalent to \(\frac{n}{2} + 2s > - \frac{1}{2}\), which is trivially satisfied. Furthermore, we have
We, thus, have
since from (5.12), we have \(_1F_2(0; \frac{n}{2},1-s; - (\frac{2\pi r |\xi |}{2})^2) \equiv 1\). We next want to write in a more convenient form the second hypergeometric function in the right-hand side of (5.14) below. With the above choices (3.1), we now have
and also
This gives
If we now apply Lemma 5.4 with
we obtain
Substituting in the above, and putting everything together, we find
which finally gives the desired conclusion. \(\square \)
Proof of Theorem 1.3
We begin by observing that the left-hand side of (1.14) coincides with the right-hand side when \(\xi = 0\). For this, note that on one hand (1.4) gives
On the other, we apply Lemma 5.1 with
In such case, we have \(\mu < \frac{1}{2}\) and also \(\mu + \nu = s - \frac{n}{2} + \frac{n}{2} + s - 1 = 2s - 1 > -1\), since \(s>0\). Since \(\frac{\nu -\mu +1}{2} = \frac{n}{2}\), we thus obtain
This shows that when \(\xi = 0\) the right-hand side of (1.14) becomes
and therefore (1.14) does hold in \(\xi = 0\).
Let now \(\xi \not = 0\) and recall the well-known formula of Bochner for the Fourier transform of a spherically symmetric function \(f(x) = f^\star (|x|)\),
see [3, Theor. 40 on p. 69]. Applying (3.4) to (1.1), after a simple change of variable, we find
At this point, we substitute Lemma 3.1 in (3.5), obtaining
where in the last equality we have used (3.3). Using (1.2) in (3.6), we finally obtain for every \(0<s<1\) and \(r>0\)
This completes the proof of Theorem 1.3.\(\square \)
For subsequent purposes, it will be important to have the following alternative representation of \(\widehat{A^{(s)}_r}\).
Corollary 3.2
Let \(0<s<1\). For every \(\xi \in {\mathbb {R}}^n\), we have
The identity (3.8) implies, in particular, the existence of a universal \(C(n,s)>0\), such that as \(|\xi |\rightarrow \infty \)
Proof
The proof of (3.8) follows from (3.7) by applying the recursive formula (5.3) with \(\nu = \frac{n}{2} + s\) and integrating by parts. One has in fact
Details are left to the interested reader, who should notice that, since \(s>0\), we now have \(s - \frac{n}{2}-1+ \frac{n}{2} + s = 2s-1>-1\), and therefore, the oscillatory integral \(\int _{0}^\infty t^{s - \frac{n}{2}-1} J_{\frac{n}{2} + s}(t) \textrm{d}t\) is convergent near \(t=0\) (and can in fact be explicitly computed via Lemma 5.1). To prove (3.9) it is enough to observe that when \(|\xi |\) is sufficiently large, then by (5.10) we have for some universal constant \(C>0\) and all \(t \ge 2\pi |\xi |\),
We, thus, find for the first term in the right-hand side of (3.8)
Since the second term can obviously be estimated in the same way, we are finished.
\(\square \)
The next result provides the limiting value of \(\widehat{A^{(s)}_1}(\xi )\) as \(s\rightarrow 1\). It represents the counterpart on the Fourier transform side of Proposition 1.1.
Lemma 3.3
Let \(n\ge 2\). Then for every \(\xi \in {\mathbb {R}}^n\), one has
Proof
Notice that for any \(\xi \not = 0\) and \(t>2\pi |\xi |\), we have from (5.9)
for some constant \(C(n,\xi )>0\) independent of \(s\in (0,1)\). By Lebesgue dominated convergence and (5.4), we, thus, have
This observation and (3.8) give
In view of (1.17) above, we have reached the desired conclusion when \(\xi \not = 0\). When instead \(\xi = 0\) for the left-hand side of (1.14) we obtain from (1.4)
Convergence to the same limit of the right-hand side follows from (3.3).\(\square \)
4 Proof of Theorem 1.4
In this section, we prove Theorem 1.4. The proof will be based on two central ideas: 1) To exploit the \(L^2\) nature of the inequality (1.22) via the Plancherel theorem. This reduces considerations to proving that the nonlocal Tomas-Stein operator \(R^{(s)}\) in (4.1) below maps \(L^p\) into \(L^{p'}\); 2) To accomplish this step, we embed \(R^{(s)}\) into an analytic family of operators \(T_z\). For the latter, we show that
with appropriate bounds on the operator norms.
Proof of Theorem 1.4
Similarly to [20] we write for \(f\in {\mathscr {S}}({\mathbb {R}}^n)\)
where we have defined
so that \(\widehat{R^{(s)} f}(\xi ) = {{\hat{f}}}(\xi ) A^{(s)}_1(\xi )\). By Plancherel, we, thus, obtain
where in the last inequality, we have used Hölder. The proof will be finished if we can show that for every \(0<s<1\), there exists \(M_s = M_s(n)>0\) such that for \(f\in {\mathscr {S}}({\mathbb {R}}^n)\), one has
for \(p = \frac{2(n+1)}{n+1+2s}\). We want to accomplish (4.2) by interpolating between the two endpoints \(L^1\rightarrow L^\infty \) and \(L^2\rightarrow L^2\). This means we have to choose \(\theta \in [0,1]\) such that
which gives
and, therefore,
For \(z\in {\mathbb {C}}\) such that \(\Re z\le 1\) we define a linear operator \(T_z:{\mathscr {S}}({\mathbb {R}}^n)\rightarrow {\mathbb {C}}\) by letting
where for \(\Re z \le 1\), we have let
with \(c(n,z) = \frac{2}{\sigma _{n-1}\Gamma (z)\Gamma (1-z)}\). Notice that \(c(n,z) = c(n,1-z)\) and that for \(y\in {\mathbb {R}}\), we have
From Plancherel theorem, (4.4), (4.6), and (5.1) we conclude for some universal \(C(n)>0\) and for every \(y\in {\mathbb {R}}\)
with \(M_1(y) = C(n)\ e^{\pi |y|}\).
We now introduce the kernels
Notice that, according to (3.8) in Corollary 3.2, when \(z=1-s\) we have \(K_{1-s} = \widehat{A^{(s)}_1}\), and therefore, (4.4) gives
where in the last equality, we have used (4.1). Also notice that, since by analytic continuation, (3.8) continues to be valid for any \(z\in {\mathbb {C}}\) in the strip \(0<\Re z < 1\), for any such z, we have from (4.4)
Since by (4.8), the kernel \(K_z\) defines an analytic function of z for \(-\frac{n+1}{2}< \Re z<1\) (see Lemma 5.1), we can use (4.10) to analytically extend the operator \(T_z\) to the whole strip \(S = \{z\in {\mathbb {C}}\mid -\frac{n-1}{2}<\Re z<1\}\). If we let \(z = x + i y\), then we define
Note that \(S_z\) is now defined on the strip \(\Sigma = \{z\in {\mathbb {C}}\mid 0<\Re z < 1\}\), and that (4.7) now reads
Since \(S_{i y} = T_{-\frac{n-1}{2} +i y}\), we next analyse the behaviour of \(T_z\) on the line \(L_0 = \{z\in {\mathbb {C}}\mid \Re z = - \frac{n-1}{2}\}\). Notice that (4.3) gives
Note that the equation \(-\frac{n-1}{2} + \frac{n+1}{2} x = (1-s)\) gives
see (4.3). This shows that \(S_\theta = T_{1-s}\). In view of (4.7), we conclude that, if we can show that \(T_{-\frac{n-1}{2} + i y} :L^1({\mathbb {R}}^n) \rightarrow L^\infty ({\mathbb {R}}^n)\), with appropriate bounds on the operator norms, then by Stein’s theorem of complex interpolation for an analytic family of operators, see [17] or [18, Theor. 4.1, p. 205], it will follow that \(T_{1-s} : L^p({\mathbb {R}}^n) \rightarrow L^{p'}({\mathbb {R}}^n)\), as desired.
In order to show that \(T_{-\frac{n-1}{2} + i y} :L^1({\mathbb {R}}^n) \rightarrow L^\infty ({\mathbb {R}}^n)\), we will prove that \(K_{-\frac{n-1}{2} + i y}\in L^\infty ({\mathbb {R}}^n)\) and that moreover, for some universal constant \(C>0\) depending only on n, one has
From (4.8), we obtain
Note that from (5.10), there exists a universal \(R = R(n)>0\) such that when \(|\xi | \ge R\), one has
On the other hand, for \(|\xi |\le R\), we have from (5.8)
We now have for \(n\ge 2\)
where in the last equality, we have used (5.1). This gives
Inserting this information in (4.15), and combining the resulting estimate with (4.14), we conclude that there exists \(C(n)>0\) such that for every \(\xi \in {\mathbb {R}}^n\) and any \(y\in {\mathbb {R}}\), one has
Next, we show that for some \(C = C(n)>0\) one has for every \(\gamma \in {\mathbb {R}}\)
To see this, we apply Legendre duplication formula, see e.g. (1.2.3) in [12], to write
Using the estimate \(|\Gamma (z)|\le |\Gamma (\Re z)|\), this gives
where in the last inequality, we have used (4.16). This proves (4.18). Next, we show that for every \(\xi \in {\mathbb {R}}^n\) and every \(y\in {\mathbb {R}}\)
To prove (4.19) observe that, in view of (5.9), there exists \(R>0\) depending on n such that for \(|\xi |\ge R\), we have for \(t\in [2\pi |\xi |,\infty )\)
Since
with obvious meaning of the notation, we infer
Keeping in mind that for \(z = x+iy\) we have
and that integrating by parts, we obtain
we conclude that (4.19) holds when \(|\xi |\ge R\). If instead \(|\xi |\le R\), then we write
and then use Lemma 5.1 and (5.8) to estimate
Using again Legendre duplication formula, similarly to the proof of (4.18) we recognise
If we now use (4.17), (4.18) and (4.19) in (4.13), we conclude that (4.12) does hold. It follows that for every \(f\in {\mathscr {S}}({\mathbb {R}}^n)\) and any \(y\in {\mathbb {R}}\) one has
with \(M_0(y) = C e^{\frac{3\pi }{2}|y|}\). By [18, Theor. 4.1 on p. 205] we infer that there exists \(M_s = M_{\theta (s)}>0\) such that (4.2) holds. This proves Theorem 1.4.\(\square \)
From p. 209 in [18] we see that the constant \(M_s\) is given by
From (4.3) we have
We thus find
Since \(\cos (\frac{2\pi }{n+1}) \not = \pm 1\) for any \(n\ge 2\), by Lebesgue dominated convergence we find
Since the function \(s\rightarrow \frac{2(n+1)}{n+1+2s}\) is decreasing on (0, 1), with range \((\frac{2(n+1)}{n+3},2)\), if now \(1\le p \le \frac{2(n+1)}{n+3}\), then for any \(s\in (0,1)\) we also have \(1\le p \le \frac{2(n+1)}{n+1+2s}\). From the proof of Theorem 1.4 we infer for every \(f\in {\mathscr {S}}({\mathbb {R}}^n)\) one has
with \(M_s\) give by (4.21) above. Passing to the limit for \(s\rightarrow 1\) in (4.23), and using Proposition 1.1 and (4.22), we conclude the celebrated Tomas-Stein theorem for the sphere
References
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Acknowledgements
I thank Carlos Kenig and Agnid Banerjee for their gracious preliminary reading of the manuscript and their feedback. I also want to express my appreciation to the anonymous referee for so carefully reading the entire manuscript and for bringing to my attention some annoying typos and glitches.
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The author has been supported in part by a Progetto SID (Investimento Strategico di Dipartimento): “Aspects of nonlocal operators via fine properties of heat kernels”, University of Padova, 2022. He has also been partially supported by a Visiting Professorship at the Arizona State University.
5. Appendix: Some Classical Material
5. Appendix: Some Classical Material
In this section, we collect some results needed in this paper. We recall the gamma function
Using the well-known formula \(\Gamma (z+1) = z \Gamma (z)\), this function can be extended as a meromorphic function to the whole \({\mathbb {C}}\) having simple poles in \(z = -k, k=0,1,\ldots \), with residues \({\text {Res}}(\Gamma ;-k) = \frac{(-1)^k}{k!}\). The reciprocal gamma function \(1/\Gamma (z)\) is an entire function with zeros at the negative integers. The following special values of such function will be useful subsequently, see e.g. formulas 6.1.29 and 6.1.31 on p. 256 in [1],
Since this note is about the Fourier transform, Bessel functions (introduced and developed in the seminal work [2]) play a pervasive role in it. We recall the series expansion of the Bessel function \(J_\nu (z)\) for \(|\arg z|<\pi \),
see [21]. From (5.2), it is easy to obtain the following well-known formulas
and
For every \(\nu \in {\mathbb {C}}\) such that \(\Re \nu >-\dfrac{1}{2}\), the function \(J_\nu \) admits the following Poisson representation
where z ranges in the complex plane cut along the negative real axis \((-\infty ,0]\). When \(\nu \in {\mathbb {N}}_0\) then we can take \(z\in {\mathbb {C}}\). From (5.2) one easily recognises that
Again from (5.6) one sees that when \(z\in {\mathbb {R}}\) and \(z>0\), then for \(\Re \nu >-\dfrac{1}{2}\) one has
This estimate is not useful for large \(z>0\) since \(J_\nu (z)\) decays at infinity with an oscillatory behaviour. When \(\Re \nu >-\dfrac{1}{2}\) the following result due to Hankel holds, see for instance [18, Lemma 3.11], or also (5.11.6) on p. 122 in [12]. One has for \(0<\delta <\pi \)
In particular,
The following beautiful formula can be found in 6.561.14 on p. 684 of [10], or also (19) on p. 49 in vol.2 of [5].
Lemma 5.1
Let \(a>0\), \(-\Re \nu -1<\Re \mu <\frac{1}{2}\). Then
We next recall the definition of the hypergeometric functions. The Pochammer’s symbols are defined by
Notice that since the gamma function has a pole in \(z=0\), we have
Definition 5.2
Let \(p, q\in {\mathbb {N}}\cup \{0\}\) be such that \(p\le q+1\), and let \(\alpha _1,\ldots ,\alpha _p\) and \(\beta _1,\ldots ,\beta _q\) be given parameters such that \(-\beta _j\not \in {\mathbb {N}}_0\) for \(j=1,\ldots ,q\). Given a number \(z\in {\mathbb {C}}\), the power series
is called the generalised hypergeometric function. When \(p = 2\) and \(q=1\), then the function \(_2 F_1(\alpha _1,\alpha _2;\beta _1;z)\) is the Gauss’ hypergeometric function, and it is usually denoted by \(F(\alpha _1,\alpha _2;\beta _1;z)\).
Using the ratio test one easily verifies that the radius of convergence of the above hypergeometric series is \(\infty \) when \(p\le q\), whereas it equals 1 when \(p = q+1\). Thus for instance it is 1 for Gauss’ hypergeometric function \(F(\alpha _1,\alpha _2;\beta _1;z)\). For later reference we record the following facts that follow easily from Definition 5.2:
and (see also p. 275 in [12])
The following result plays a key role in this work. The interested reader can find it in formula 2.12.4.16 on p. 178 in [13].
Lemma 5.3
Let \(c>0\), \(\Re \beta >0\), \(\Re (\alpha +2\beta )<\frac{7}{2}\). Then
In order to fully exploit Lemma 5.3, we next derive a useful representation formula of a certain hypergeometric function \(_1 F_2\) in terms of an integral involving a Bessel function. We stress that, although for convenience we have used the same letters, the parameters \(\alpha , \nu \) in the statement of the next result are not the same as those in Lemma 5.3.
Lemma 5.4
Let \(a, c>0\) and \(\Re (\alpha +\nu )>0\). Then
Proof
Performing the change of variable \(\tau = ct\) and using (5.2) we find
The change of variable \(z = \left( \frac{\tau }{2}\right) ^2\), for which \(\frac{dz}{z} = 2 \frac{d\tau }{\tau }\), gives
where the details of the last equality are left to the reader. \(\square \)
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Garofalo, N. Some Inequalities for the Fourier Transform and Their Limiting Behaviour. J Geom Anal 34, 55 (2024). https://doi.org/10.1007/s12220-023-01477-6
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DOI: https://doi.org/10.1007/s12220-023-01477-6