For n≥2n\ge 2, Goldberg (2016) proved that the negative-order Bochner–Riesz-type multiplier S−αS^{-\alpha}, (S−αf)  ^(ξ)=(1−∣ξ∣2)+−αf^(ξ)(S^{-\alpha}f)\hat{\;} (\xi)=(1-|\xi|^2)_+^{-\alpha}\hat f(\xi), maps L(2n+2)/(n+1+4α)(Rn)L^{(2n+2)/(n+1+4\alpha)}(\mathbb{R}^n) boundedly into L2(Rn)L^2(\mathbb{R}^n) on the subspace of data whose Fourier transform vanishes on the unit sphere, for every 12≤α<32\tfrac12\le\alpha<\tfrac32 except the single endpoint pair (α,p)=(12,2n+2n+3)(\alpha,p)=(\tfrac12,\tfrac{2n+2}{n+3}). The estimate is false without the vanishing (Fredholm) condition, is false in one dimension even with it, and the case n≥2n\ge 2 with vanishing is open: it was posed by the author as Problem 4 of the 2017 problem collection ‘Some Problems in Harmonic Analysis’, phrased to ask for a counterexample if one exists. The problem is to decide, for n≥2n\ge2, whether ∥S−1/2f∥L2(Rn)≤Cn∥f∥L(2n+2)/(n+3)(Rn)\|S^{-1/2}f\|_{L^2(\mathbb{R}^n)}\le C_n\|f\|_{L^{(2n+2)/(n+3)}(\mathbb{R}^n)} for every ff whose Fourier transform vanishes on the unit sphere in the L2L^2-trace sense (the Fredholm condition); the public posing uses the two-sided weight ∣1−∣ξ∣2∣−1/2|1-|\xi|^2|^{-1/2}, whose estimate implies the one-sided one, while a counterexample whose inner-region integral diverges refutes both formulations. Either answer closes the last endpoint of the Lp→L2L^p\to L^2 solvability theory of the Helmholtz equation with Fredholm data and of the corresponding multiplier theory on the Fourier-vanishing subspace for the formulation it addresses; since only the two-sided-to-one-sided implication is known, the one-sided and two-sided formulations must be settled separately.

Background

For the inhomogeneous Helmholtz equation (−Δ−1)u=f(-\Delta-1)u=f in Rn\mathbb{R}^n, a formal solution is given in frequency space by u^(ξ)=f^(ξ)/(1−∣ξ∣2)\hat u(\xi)=\hat f(\xi)/(1-|\xi|^2), whose multiplier blows up on the unit sphere Sn−1={ξ:∣ξ∣=1}\mathbb{S}^{n-1}=\{\xi:|\xi|=1\}. Solvability in L2L^2 therefore requires a cancellation condition on the data: the Fredholm condition, namely that f^\hat f vanish on Sn−1\mathbb{S}^{n-1} in the L2L^2-trace sense — the restrictions of f^\hat f to the spheres {∣ξ∣=r}\{|\xi|=r\} tend to zero in L2(Sn−1)L^2(\mathbb{S}^{n-1}) as r→1r\to1 — which expresses orthogonality of the data to all unit-frequency plane waves. Agmon's classical limiting-absorption theorem, which constructs solutions as limits of resolvents with the frequency parameter approaching the real axis, gives an L2L^2 solution under a weighted L2L^2 hypothesis plus this condition, and Goldberg (2016) showed that unweighted LpL^p data suffice for a narrow range of exponents: for n≥3n\ge 3 and max⁡(1,2nn+4)≤p≤2n+2n+5\max(1,\tfrac{2n}{n+4})\le p\le\tfrac{2n+2}{n+5} (with (n,p)≠(4,1)(n,p)\ne(4,1)), every f∈Lpf\in L^p satisfying the Fredholm condition admits a unique L2L^2 solution.

The engine behind this solvability is a family of Bochner–Riesz multipliers of negative order — Fourier multipliers whose symbol is a power of 1−∣ξ∣21-|\xi|^2 cut off to the unit ball, singular on the sphere when the power is negative. For α≤0\alpha\le0 define (Sαf)  ^(ξ)=(1−∣ξ∣2)+αf^(ξ)(S^{\alpha}f)\hat{\;} (\xi)=(1-|\xi|^2)^\alpha_+\hat f(\xi), where the subscript ++ indicates that the multiplier is supported in the unit ball; for α≤−1\alpha\le-1 the operator is defined by formal positivity rather than by analytic continuation, which preserves the multiplicative structure SαSβ=Sα+βS^\alpha S^\beta=S^{\alpha+\beta} at the cost of not acting boundedly on general Schwartz functions. For negative α\alpha these multipliers are unbounded on all of LpL^p; Goldberg's Theorem 3 states that the multipliers S−αS^{-\alpha} become bounded from Lp(Rn)L^p(\mathbb{R}^n) to L2(Rn)L^2(\mathbb{R}^n) on the closed subspace {f∈Lp:f^∣Sn−1=0}\{f\in L^p:\hat f|_{\mathbb{S}^{n-1}}=0\} whenever n≥2n\ge2, 12≤α<32\tfrac12\le\alpha<\tfrac32, and 1≤p≤2n+2n+1+4α1\le p\le\tfrac{2n+2}{n+1+4\alpha} — with precisely one pair excluded: (α,p)=(12,2n+2n+3)(\alpha,p)=(\tfrac12,\tfrac{2n+2}{n+3}). The exponent 2n+2n+1+4α\tfrac{2n+2}{n+1+4\alpha} is sharp up to endpoints by Knapp-type examples (packets of f^\hat f concentrated in a δ\delta-slab tangent to the sphere), and the endpoint exponent 2n+2n+3=2(n+1)n+3\tfrac{2n+2}{n+3}=\tfrac{2(n+1)}{n+3} is exactly the endpoint exponent of the Stein–Tomas restriction theorem, which bounds the Fourier transform of an L2L^2 density supported on a curved surface in Lq(Rn)L^q(\mathbb{R}^n). The two sides of α=12\alpha=\tfrac12 behave differently: for 0<α<120<\alpha<\tfrac12 the weight ∣1−∣ξ∣2∣−α|1-|\xi|^2|^{-\alpha} is integrable in the transverse (radial) variable and no vanishing condition is needed, while for 12<α<32\tfrac12<\alpha<\tfrac32 the vanishing condition restores boundedness (Goldberg 2016). Without vanishing, the two-sided weight fails for every α≥12\alpha\ge\tfrac12 already for a Gaussian, whose Fourier transform is nonzero on the sphere; Goldberg–Lau (2024) study this two-sided behavior through transverse difference-quotient operators Tαf(ξ,η)=(f^(ξ,h(ξ)+η)−f^(ξ,h(ξ)))/∣η∣αT_\alpha f(\xi,\eta)=(\hat f(\xi,h(\xi)+\eta)-\hat f(\xi,h(\xi)))/|\eta|^\alpha (differences of f^\hat f across a surface {ξn=h(ξ′)}\{\xi_n=h(\xi')\} divided by ∣η∣α|\eta|^\alpha), show by Gaussian examples that no L2L^2 bound for TαT_\alpha can hold without vanishing when ∣α−1∣≥12|\alpha-1|\ge\tfrac12, and prove such bounds for α∈(12,32)\alpha\in(\tfrac12,\tfrac32) under a trace-smoothness hypothesis that is vacuous when f^\hat f vanishes on the surface.

At α=12\alpha=\tfrac12 the weight ∣1−∣ξ∣2∣−1/2|1-|\xi|^2|^{-1/2} is borderline non-integrable in the transverse direction, and the qualitative trace condition f^∣Sn−1=0\hat f|_{\mathbb{S}^{n-1}}=0 supplies no quantitative decay rate of f^\hat f near the sphere. The endpoint fails without the vanishing condition, and in dimension n=1n=1 it fails even with it: there S0={±1}\mathbb{S}^0=\{\pm1\} has no curvature, S−1/2S^{-1/2} is closely related to the one-dimensional fractional integral operator of order 12\tfrac12, and the correct sufficient condition is the strictly stronger requirement e±ixf∈H1(R)e^{\pm ix}f\in H^1(\mathbb{R}), where H1(R)H^1(\mathbb{R}) is the real Hardy space, a strict subspace of L1(R)L^1(\mathbb{R}) of functions with additional cancellation; explicit one-dimensional counterexamples are built from translated packets (Goldberg 2016, and the 2017 problem collection). Those constructions exploit flatness and do not extend to curved spheres, where curvature is an asset. The derivative-trace program around this problem (Cho–Guo–Lee 2015; Goldberg–Stolyarov 2020; Goldberg–Lau 2024) settled when the restrictions of ∇kf^\nabla^k\hat f to the sphere belong to L2(Sn−1)L^2(\mathbb{S}^{n-1}) or to Sobolev spaces H−s(Sn−1)H^{-s}(\mathbb{S}^{n-1}) of negative order — including all endpoints of that trace family — and settled the two-sided difference-quotient estimates for α∈(12,32)\alpha\in(\tfrac12,\tfrac32), but its constants and parameter ranges degenerate exactly at α=12\alpha=\tfrac12, p=2n+2n+3p=\tfrac{2n+2}{n+3}, leaving this endpoint untouched.

Problem Statement

Let n≥2n\ge2 and let S−1/2S^{-1/2} be the negative-order Bochner–Riesz-type multiplier acting by (S−1/2f)  ^(ξ)=(1−∣ξ∣2)+−1/2f^(ξ)(S^{-1/2}f)\hat{\;} (\xi)=(1-|\xi|^2)_+^{-1/2}\hat f(\xi); the squared L2L^2 norm of S−1/2fS^{-1/2}f is ∫∣ξ∣<1∣f^(ξ)∣2 /(1−∣ξ∣2) dξ\int_{|\xi|<1}|\hat f(\xi)|^2\,/(1-|\xi|^2)\,d\xi. The publicly posed form of the question (Problem 4 of the 2017 collection ‘Some Problems in Harmonic Analysis’) uses the two-sided weight ∣1−∣ξ∣2∣−1/2|1-|\xi|^2|^{-1/2}, which adds the outer region {∣ξ∣>1}\{|\xi|>1\} to this integral; the two-sided estimate implies the one-sided estimate, and a counterexample whose inner-region integral diverges refutes both formulations.

Decide the endpoint pair excluded from Goldberg's Theorem 3: does there exist a constant CnC_n such that

∥S−1/2f∥L2(Rn)  ≤  Cn ∥f∥L2n+2n+3(Rn)\|S^{-1/2}f\|_{L^2(\mathbb{R}^n)}\;\le\;C_n\,\|f\|_{L^{\frac{2n+2}{n+3}}(\mathbb{R}^n)}

for every f∈L2n+2n+3(Rn)f\in L^{\frac{2n+2}{n+3}}(\mathbb{R}^n), n≥2n\ge2, whose Fourier transform vanishes on the unit sphere in the L2L^2-trace sense (f^∣Sn−1=0\hat f|_{\mathbb{S}^{n-1}}=0, the Fredholm condition)? The publicly posed Problem 4 asks the related, strictly stronger two-sided question of whether ∣1−∣ξ∣2∣−1/2f^|1-|\xi|^2|^{-1/2}\hat f belongs to L2(Rn)L^2(\mathbb{R}^n) for every such ff; a proof of the two-sided estimate also settles the one-sided endpoint, whereas a proof of the one-sided estimate alone leaves Problem 4 open.

A counterexample is given by a function f∈L2n+2n+3(Rn)f\in L^{\frac{2n+2}{n+3}}(\mathbb{R}^n), for some n≥2n\ge2, with f^∣Sn−1≡0\hat f|_{\mathbb{S}^{n-1}}\equiv0 but ∫∣ξ∣<1∣f^(ξ)∣2 /(1−∣ξ∣2) dξ=∞\int_{|\xi|<1}|\hat f(\xi)|^2\,/(1-|\xi|^2)\,d\xi=\infty, which refutes both the one-sided and the two-sided formulations in that dimension; a counterexample whose weighted integral diverges only in the outer region {∣ξ∣>1}\{|\xi|>1\} refutes the two-sided formulation alone and leaves the one-sided endpoint open. The estimate fails without the vanishing condition, fails in n=1n=1 even with it (where e±ixf∈H1(R)e^{\pm ix}f\in H^1(\mathbb{R}) is the correct sufficient condition), and holds, with the vanishing condition, throughout 12≤α<32\tfrac12\le\alpha<\tfrac32, 1≤p≤2n+2n+1+4α1\le p\le\tfrac{2n+2}{n+1+4\alpha}, except at the single endpoint pair in question.

References

  1. Michael Goldberg, The Helmholtz equation with LpL^p data and Bochner–Riesz multipliers, Mathematical Research Letters 23 (2016), no. 6, 1665–1679. DOI: 10.4310/MRL.2016.v23.n6.a5; arXiv:1502.02066. https://arxiv.org/abs/1502.02066
  2. Loukas Grafakos, Diogo Oliveira e Silva, Malabika Pramanik, Andreas Seeger, Betsy Stovall, Some Problems in Harmonic Analysis (open-problem collection; Problem 4 contributed by Michael Goldberg), 2017. arXiv:1701.06637. https://arxiv.org/abs/1701.06637
  3. Michael Goldberg, Dmitriy Stolyarov, Restrictions of higher derivatives of the Fourier transform, Transactions of the American Mathematical Society, Series B 7 (2020), 46–96. DOI: 10.1090/btran/45; arXiv:1809.04159. https://arxiv.org/abs/1809.04159
  4. Michael Goldberg, Chun Ho Lau, Restriction of Fractional Derivatives of the Fourier Transform, preprint, 2024. arXiv:2410.06092. https://arxiv.org/abs/2410.06092
  5. Dmitriy M. Stolyarov, Functions whose Fourier transform vanishes on a surface, preprint, 2016. arXiv:1601.04604. https://arxiv.org/abs/1601.04604
  6. Yonggeun Cho, Zihua Guo, Sanghyuk Lee, A Sobolev estimate for the adjoint restriction operator, Mathematische Annalen 362 (2015), no. 3–4, 799–815. DOI: 10.1007/s00208-014-1130-7. https://doi.org/10.1007/s00208-014-1130-7
  7. Benjamin Foster, The inhomogeneous wave equation with LpL^p data, Proceedings of the American Mathematical Society 148 (2020), 4479–4490. DOI: 10.1090/proc/15123; arXiv:1911.01584. https://arxiv.org/abs/1911.01584
  8. Michael J. Puls, Sets of pp-restriction and pp-spectral synthesis, Demonstratio Mathematica 52 (2019), no. 1, 397–403. DOI: 10.1515/dema-2019-0033; arXiv:1904.02282. https://doi.org/10.1515/dema-2019-0033