Some inequalities for the Fourier transform and their limiting behaviour

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.


Introduction
For any 0 < s < 1 and r > 0 we consider the function (1.1) where, using the standard notation σ n−1 = 2π n 2

Γ( n
2 ) for the (n − 1)-dimensional measure of the unit sphere S n−1 ⊂ R n , we have let Define measures on R n by letting (1.3) dµ (s) r (x) = A (s) r (x)dx.
Since from (1.1) it immediately follows that A (s) r ∈ L 1 (R n ), from [9,Lemma 15.3]) one has for every s ∈ (0, 1) and r > 0 We next define the operator (1.5)A (s) r f (x) = A (s) r ⋆ f (x).It follows from (1.4) and Young's convolution theorem that and that for any f ∈ L p (R n ) we have which shows that A (s) r is a contraction in L p (R n ) for every 0 < s < 1 and r > 0.
For any f ∈ S (R n ) we denote by f (ξ) = F (f )(ξ) = R n e −2πi⟨ξ,x⟩ f (x)dx its Fourier transform, and ask the following Question: Let 1 2 ≤ s < 1.If 1 ≤ p < 2n n+2s−1 , is it true that for 1 ≤ q ≤ n+1−2s n+1 p ′ and for every f ∈ S (R n ) one has for some C (s) (n, p) > 0 (1.7) We note that the restriction s ≥ 1  2 serves to guarantee that 2n n+2s−1 ≤ 2. Therefore, the hypothesis f ∈ L p (R n ) implies that f be in the Hausdorff-Young range [1,2].As a consequence, f is a function in L p ′ (R n ).
Proposition 1.1.For every function f ∈ S (R n ) and x ∈ R n , one has where is the spherical average of f over the sphere S(x, r) = {y ∈ R n | |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 → 1, we would infer that for 1 ≤ p < 2n n+1 and q ≤ 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 [6], [7], [8], [18], [4] and [21].
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 R n \ B(0, 1), instead of the lower dimensional manifold S n−1 ⊂ R n .Notice that if we let dσ denote the surface measure concentrated on the sphere S n−1 , then Proposition 1.1 can be equivalently stated as follows (1.9) dµ Concerning the measures (1.3) we recall that in his above quoted paper M. Riesz developed the theory of the nonlocal operators (−∆) s and their inverses I 2s , the operators of fractional integration which play a pervasive role in analysis, see also [17,Ch. 5].Among other things, he solved by inversion the Dirichlet problem and proved that for every x ∈ B r the unique solution to (1.10) is provided by |y − x| n dy, 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 r f (0).The role of the operators 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 Here, for 0 < s < 1 we have denoted by L s (R n ) the space of measurable functions f : R n → 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 ∈ [1,2] in the Hausdorff-Young range.To understand the constraint 1 ≤ p < 2n n+2s−1 , denote by T : ) the "restriction" operator in (1.7).Then its adjoint T ⋆ : L q ′ (R n , dµ (s) 1 ) → L p ′ (R n ) is easily seen to be given by (1.13) 1 (ξ)dx, so that 1 .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 A (s) 1 ∈ L p ′ (R n ).Now, in Section 3 we prove the following.Theorem 1.3.Let s ∈ (0, 1) and n ≥ 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 (1.15) We conclude that the inequality (1.7) cannot possibly hold for p ≥ 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 dµ (s) 1 , compared to the surface measure dσ of the unit sphere S n−1 .To explain this comment we stress that (1.17) is just a rescaled spherically symmetric eigenfunction of the (local) differential operator ∆ in R n .
To see this, simply observe that for every λ > 0 the function is a spherically symmetric solution of the Helmholtz equation ∆f λ = −λf λ in R n , which shows that (1.17) solves such PDE with λ = 4π 2 .However, the equations (1.10), (1.11) above, and Proposition 1.2 in particular, underscore that the Fourier transform of the measure dµ r is instead connected to the pseudodifferential operator (−∆) 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 ≥ 2 one has for every 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 Kz → 1 2 dσ as z → −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, A (s) 1 is directly connected to the eigenfunctions of the nonlocal operator (−∆) s (to be further analysed in a future work), 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 dµ (s) 1 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 ∈ S (R n ) and 2 ≤ n+1−2s n+1 p ′ .This is equivalent to asking that 1 , and therefore for any In the next result, we prove this conjecture.
Theorem 1.4.For a given s ∈ (0, 1) let p = 2(n+1) n+1+2s .Then there exists a constant In order to establish Theorem 1.4 we exploit Plancherel as in [22] and reduce matters to proving that the operator ).This is ultimately achieved using Stein's theorem of complex interpolation by embedding R (s) into an analytic family of operators with appropriate bounds on the operator norms.Since the constant C (s) (n) in (1.22) is bounded uniformly in s ∈ (0, 1), see (4.21) below, passing to the limit as s → 1 we recover the Tomas-Stein restriction theorem, see [22].
Remark 1.5.One should observe that the threshold exponent p = 2(n+1)  n+1+2s in (1.21) is strictly less than 2 for any 0 < s < 1.Therefore, the limitation 1 2 ≤ s < 1 is not necessary in such case.The plan of the paper is as follows.In Section 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 ≤ q ≤ n+1−2s n+1 p ′ .Section 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 Section 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 Section 5 in which we gather some well-known facts and collect some results needed in the rest of the paper.
Acknowledgment: I thank Carlos Kenig and Agnid Banerjee for their gracious preliminary reading of the manuscript and their feedback.

Necessary condition for restriction
In (1.16) above we have seen that the inequality (1.7) can possibly hold only when 1 ≤ p < 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 ≤ q ≤ n+1−2s n+1 p ′ .
Proposition 2.1.A necessary condition for (1.7) to hold for with sides parallel to the coordinate axis whose projection onto R n−1 ×{0} is the (n−1)-dimensional cube K ′ ε circumscribing the intersection of the hyperplane x n = 1 − ε with the unit sphere S n−1 .If θ ε 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 S n−1 ∩ {x n = 1 − ε}, from elementary trigonometry we have cos and therefore f ε ∈ L p (R n ) for any p > 1 and moreover Next, we want to understand the asymptotic behaviour as ε → 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 ε ∩ (R n \ B(0, 1)).With this objective in mind, we obtain from Cavalieri's principle Cε∩(R n \B(0,1)) We now want to show that as ε → 0 + we have To see this we write The chain rule gives 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 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.
Proof.To prove Lemma 3.1 we use Lemma 5.3 in which we take , which is trivially satisfied.Furthermore, we have We thus have , since from (5.12) we have 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 t)dt.Substituting in the above, and putting everything together, we find which finally gives the desired conclusion.□ Proof of Theorem 1.3.We begin by observing that the left-hand side of (1.14) coincides with the right-hand side when ξ = 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 µ < 1 2 and also µ .
This shows that when ξ = 0 the right-hand side of (1.14) becomes 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.

□
For subsequent purposes, it will be important to have the following alternative representation of A (s) r .
Corollary 3.2.Let 0 < s < 1.For every ξ ∈ R n we have The identity (3.8) implies, in particular, the existence of a universal C(n, s) > 0, such that as |ξ| → ∞ (3.9) Proof.The proof of (3.8) follows from (3.7) by applying the recursive formula (5.3) with ν = 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 and therefore the oscillatory integral dt 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 |ξ| is sufficiently large, then by (5.10) we have for some universal constant C > 0 and all t ≥ 2π|ξ|, 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.□ The next result provides the limiting value of A Proof.Notice that for any ξ ̸ = 0 and t > 2π|ξ| we have from (5.9) for some constant C(n, ξ) > 0 independent of s ∈ (0, 1).By Lebesgue dominated convergence we thus have This observation and (3.8) give In view of (1.17) above, we have reached the desired conclusion when ξ ̸ = 0.When instead ξ = 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).□

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 [22] we write for where we have defined . 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 for p = 2(n+1) n+1+2s .We want to accomplish (4.2) by interpolating between the two endpoints L 1 → L ∞ and L 2 → L 2 .This means we have to choose θ ∈ [0, 1] such that and therefore For z ∈ C such that ℜz ≤ 1 we define a linear operator where for ℜz ≤ 1 we have let (4.5) From Plancherel theorem, (4.4), (4.6) and (5.1) we conclude for some universal C(n) > 0 and for every y ∈ R We now introduce the kernels Notice that, according to (3.8) in Corollary 3.2, when z = 1 − s we have 1 , and therefore (4.4) gives (4.9) , 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 ∈ C in the strip 0 < ℜz < 1, for any such z we have from (4.4) (4.10) Since by (4.8) the kernel K z defines an analytic function of z for − n+1 2 < ℜz < 1 (see Lemma 5.1), we can use (4.10) to analytically extend the operator T z to the whole strip 2 x+iy .Note that S z is now defined on the strip Σ = {z ∈ C | 0 < ℜz < 1}, and that (4.7) now reads Since S iy = T − n−1 2 +iy , we next analyse the behaviour of T z on the line the line Note that the equation . This shows that S θ = T 1−s .In view of (4.7) we conclude that, if we can show that ), with appropriate bounds on the operator norms, then by Stein's theorem of complex interpolation for an analytic family of operators, see [16] or [20, Theor.4.1, p. 205], it will follow that In order to show that ) and that moreover for some universal constant C > 0 depending only on n, one has (4.12) From (4.8) we obtain (4.13) Note that from (5.10) there exists a universal R = R(n) > 0 such that when |ξ| ≥ R one has On the other hand, for |ξ| ≤ R we have from (5.8) We now have for n ≥ 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 ξ ∈ R n and any y ∈ R one has (4.17) 1 Next, we show that for some C = C(n) > 0 one has for every γ ∈ R To see this we apply Legendre duplication formula, see e.g.(1.2.3) in [12], to write Using the estimate where in the last inequality we have used (4.16).This proves (4.18).Next, we show that for every ξ ∈ R n and every y ∈ R To prove (4.19) observe that, in view of (5.9) there exists R > 0 depending on n such that for |ξ| ≥ R we have for t ∈ [2π|ξ|, ∞) ), n even, (−1) n cos(t + i π 2 γ), n odd, with obvious meaning of the notation, we infer Keeping in mind that for z = x + iy we have cos z = cos x cosh y − i sin x sinh y, sin z = sin x cosh y + i cos x sinh y, and that integrating by parts we obtain −iy (t)dt, and then use Lemma 5.1 and (5.8) to estimate Using again Legendre duplication formula, similarly to the proof of (4.From p. 209 in [20] we see that the constant M s is given by n+1+2s is decreasing on (0, 1), with range ( n+3 , then for any s ∈ (0, 1) we also have 1 ≤ p ≤ 2(n+1) n+1+2s .From the proof of Theorem 1.4 we infer for every f ∈ S (R n ) one has (4.23) with M s give by (4.21) above.Passing to the limit for s → 1 in (4.23), and using Proposition 1.1 and (4.22), we conclude the celebrated Tomas-Stein theorem for the sphere
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].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 α, ν in the statement of the next result are not the same as those in Lemma 5.3.

1 . 3 . 3 .A
as s → 1.It represents the counterpart on the Fourier transform side of Proposition 1.Lemma Let n ≥ 2. Then for every ξ ∈ R n one has lim s→1