For 0<λ<120<\lambda<\frac{1}{2}, let BλB_\lambda be the Bochner–Riesz multiplier on R2\mathbb{R}^2 with symbol (1−∣ξ∣2)+λ(1-|\xi|^2)_+^\lambda and pλ=43+2λp_\lambda=\frac{4}{3+2\lambda} its critical exponent. Kesler and Lacey (2018) proved the (pλ,q)(p_\lambda,q) sparse bound for 4<q<pλ′4<q<p_\lambda' and asked whether it extends to 41+6λ<q≤4\frac{4}{1+6\lambda}<q\le 4; Beltran, Roos, and Seeger (published 2025, online 2024) answered this affirmatively, proving the bound for every q>41+6λq>\frac{4}{1+6\lambda}. What survives is the sharp boundary itself. The conjectured maximal sparse-domination region is the closed trapezoid T2(λ)\mathcal{T}_2(\lambda), sharpness of which follows from known necessary conditions; after that work its interior and both critical-exponent sides are settled except at the corners P2P_2 and P3P_3, which are equivalent by duality. The open core is: (i) the endpoint bound ∥Bλ:(pλ,41+6λ)∥Sp<∞\|B_\lambda:(p_\lambda,\frac{4}{1+6\lambda})\|_{Sp}<\infty, known only at the special index λ=16\lambda=\frac{1}{6}; (ii) sparse bounds along the top side of the trapezoid for λ≠16\lambda\ne\frac{1}{6}; and (iii) the sharp reverse-Hölder threshold σ∘(λ)=3+2λ8λ\sigma_\circ(\lambda)=\frac{3+2\lambda}{8\lambda} in the endpoint weighted weak-type estimate for weights in A1∩RHσA_1\cap RH_\sigma. The identified obstruction is an endpoint case of the vector-valued Bochner–Riesz inequalities at exponents r≠2r\ne 2.

Background

The Bochner–Riesz multiplier of index λ>0\lambda>0 on the plane is the Fourier multiplier BλB_\lambda acting by Bλf^(ξ)=(1−∣ξ∣2)+λf^(ξ)\widehat{B_\lambda f}(\xi)=(1-|\xi|^2)_+^\lambda \hat f(\xi), where (t)+=max⁡(t,0)(t)_+=\max(t,0); it is the model operator for smoothing the sharp frequency cutoff at the boundary of a disc, and its LpL^p mapping properties control the convergence of Fourier integrals in two dimensions. The theorem of Carleson and Sjölin (1972), which settles the two-dimensional case of the Bochner–Riesz conjecture, states that BλB_\lambda is bounded on Lp(R2)L^p(\mathbb{R}^2) exactly for 1−1pλ<1p<1pλ1-\frac{1}{p_\lambda}<\frac{1}{p}<\frac{1}{p_\lambda}, where pλ=43+2λp_\lambda=\frac{4}{3+2\lambda} is the critical exponent. At the endpoint p=pλp=p_\lambda only a weak-type estimate holds: Christ (1987) proved such endpoint bounds in an initial range, and Seeger (1996) established ∥Bλ∥Lpλ→Lpλ,∞<∞\|B_\lambda\|_{L^{p_\lambda}\to L^{p_\lambda,\infty}}<\infty in the plane, which is the sharp Lebesgue-space endpoint result.

Sparse domination is a finer, quantified strengthening of such boundedness. A collection S\mathcal{S} of dyadic cubes is called sparse if to each Q∈SQ\in\mathcal{S} one can assign a subset EQ⊆QE_Q\subseteq Q with ∣EQ∣≥γ∣Q∣|E_Q|\ge\gamma|Q| such that the sets EQE_Q are pairwise disjoint; the associated sparse form is Λr,sS(f,g)=∑Q∈S∣Q∣⟨f⟩Q,r⟨g⟩Q,s\Lambda^{\mathcal{S}}_{r,s}(f,g)=\sum_{Q\in\mathcal{S}}|Q|\langle f\rangle_{Q,r}\langle g\rangle_{Q,s}, where ⟨f⟩Q,r=(∣Q∣−1∫Q∣f∣r)1/r\langle f\rangle_{Q,r}=\big(|Q|^{-1}\int_Q|f|^r\big)^{1/r}. An operator TT satisfies an (r,s)(r,s) sparse bound, written ∥T:(r,s)∥Sp<∞\|T:(r,s)\|_{Sp}<\infty, if ∣⟨Tf,g⟩∣≤Csup⁡SΛr,sS(f,g)|\langle Tf,g\rangle|\le C\sup_{\mathcal{S}}\Lambda^{\mathcal{S}}_{r,s}(f,g) with CC independent of ff and gg, the supremum taken over all sparse families. Sparse bounds refine unweighted boundedness and, through the sparse-to-weighted transfer of Frey and Nieraeth, yield weighted weak-type inequalities for weights in the Muckenhoupt class A1A_1 (weights ww with Mw≤CwMw\le Cw for the Hardy–Littlewood maximal operator MM) satisfying a reverse Hölder condition RHσRH_\sigma (meaning (\fintQwσ)1/σ≤C\fintQw(\fint_Q w^\sigma)^{1/\sigma}\le C\fint_Q w on all cubes QQ).

For the Bochner–Riesz problem the conjectured maximal sparse region is the closed trapezoid T2(λ)\mathcal{T}_2(\lambda) in the (1/r,1/s)(1/r,1/s)-plane with corners P1=(3+2λ4,1−2λ4)P_1=(\frac{3+2\lambda}{4},\frac{1-2\lambda}{4}), P2=(3+2λ4,1+6λ4)P_2=(\frac{3+2\lambda}{4},\frac{1+6\lambda}{4}), P3=(1+6λ4,3+2λ4)P_3=(\frac{1+6\lambda}{4},\frac{3+2\lambda}{4}) and P4=(1−2λ4,3+2λ4)P_4=(\frac{1-2\lambda}{4},\frac{3+2\lambda}{4}); sparse bounds fail outside the closure of this trapezoid, a necessary condition first observed by Benea–Bernicot–Luque and recovered by the general necessary conditions in the multi-scale sparse-domination framework of Beltran–Roos–Seeger. Sparse bounds throughout the interior of T2(λ)\mathcal{T}_2(\lambda) follow from a single-scale analysis (Benea–Bernicot–Luque; Lacey–Mena–Reguera). On the critical vertical line 1r=1pλ\frac{1}{r}=\frac{1}{p_\lambda}, Kesler and Lacey (2018) proved ∥Bλ:(pλ,q)∥Sp<∞\|B_\lambda:(p_\lambda,q)\|_{Sp}<\infty for 4<q<pλ′4<q<p_\lambda', a sparse strengthening of Seeger's theorem, and posed the question whether this extends to 41+6λ<q≤4\frac{4}{1+6\lambda}<q\le 4, that is, down to the corner P2P_2, where the endpoint exponent qopt,λ=41+6λq_{opt,\lambda}=\frac{4}{1+6\lambda} is defined by 1qopt,λ=14+3λ2\frac{1}{q_{opt,\lambda}}=\frac{1}{4}+\frac{3\lambda}{2}. Beltran, Roos and Seeger answered this affirmatively: ∥Bλ:(pλ,q)∥Sp<∞\|B_\lambda:(p_\lambda,q)\|_{Sp}<\infty for all q>41+6λq>\frac{4}{1+6\lambda}, and at the special index λ=16\lambda=\frac{1}{6}, where qopt,λ=2q_{opt,\lambda}=2, they proved fully optimal sparse bounds on the entire closed trapezoid. The unresolved part of the boundary is the subject of this problem: the corner P2P_2 for λ≠16\lambda\ne\frac{1}{6}, the top side P2P3P_2P_3 for λ≠16\lambda\ne\frac{1}{6}, and the sharp weighted endpoint classes that these bounds induce.

Problem Statement

For 0<λ<120<\lambda<\frac{1}{2}, let BλB_\lambda be the planar Bochner–Riesz multiplier, that is, the Fourier multiplier on R2\mathbb{R}^2 acting by Bλf^(ξ)=(1−∣ξ∣2)+λf^(ξ)\widehat{B_\lambda f}(\xi)=(1-|\xi|^2)_+^\lambda\hat f(\xi) with (t)+=max⁡(t,0)(t)_+=\max(t,0); let pλ=43+2λp_\lambda=\frac{4}{3+2\lambda} be its critical exponent, and let T2(λ)\mathcal{T}_2(\lambda) be the trapezoid with corners P1=(3+2λ4,1−2λ4)P_1=(\frac{3+2\lambda}{4},\frac{1-2\lambda}{4}), P2=(3+2λ4,1+6λ4)P_2=(\frac{3+2\lambda}{4},\frac{1+6\lambda}{4}), P3=(1+6λ4,3+2λ4)P_3=(\frac{1+6\lambda}{4},\frac{3+2\lambda}{4}), P4=(1−2λ4,3+2λ4)P_4=(\frac{1-2\lambda}{4},\frac{3+2\lambda}{4}) in the (1/r,1/s)(1/r,1/s)-plane. Define qopt,λq_{opt,\lambda} by 1qopt,λ=14+3λ2,i.e.qopt,λ=41+6λ.\frac{1}{q_{opt,\lambda}}=\frac{1}{4}+\frac{3\lambda}{2},\qquad\text{i.e.}\quad q_{opt,\lambda}=\frac{4}{1+6\lambda}. Write ∥T:(r,s)∥Sp<∞\|T:(r,s)\|_{Sp}<\infty for the (r,s)(r,s) sparse bound defined in the background, i.e. ∣⟨Tf,g⟩∣≤Csup⁡SΛr,sS(f,g)|\langle Tf,g\rangle|\le C\sup_{\mathcal{S}}\Lambda^{\mathcal{S}}_{r,s}(f,g) over all sparse families S\mathcal{S} of dyadic cubes, with CC independent of f,gf,g. Determine the exact boundary of the set of pairs (r,s)(r,s) for which the sparse bound ∥Bλ:(r,s)∥Sp<∞\|B_\lambda:(r,s)\|_{Sp}<\infty holds. After Beltran–Roos–Seeger the interior of T2(λ)\mathcal{T}_2(\lambda) and both critical-exponent sides are settled except at the corners P2P_2 and P3P_3, which are equivalent by duality; the remaining questions are: (1) Endpoint corner: does ∥Bλ:(pλ,qopt,λ)∥Sp<∞\|B_\lambda:(p_\lambda,q_{opt,\lambda})\|_{Sp}<\infty hold for λ∈(0,12)∖{16}\lambda\in(0,\frac{1}{2})\setminus\{\frac{1}{6}\}? (At λ=16\lambda=\frac{1}{6} the bound is known.) (2) Top side interior: does ∥Bλ:(r,s)∥Sp<∞\|B_\lambda:(r,s)\|_{Sp}<\infty hold for (1r,1s)(\frac{1}{r},\frac{1}{s}) on the open segment P2P3P_2P_3 (the top side with the corner P2P_2, and its dual P3P_3, removed) when λ≠16\lambda\ne\frac{1}{6}? (3) Weighted endpoint sharpness: the best current endpoint weighted bound is Bλ:Lpλ(w)→Lpλ,∞(w)B_\lambda:L^{p_\lambda}(w)\to L^{p_\lambda,\infty}(w) (weak-LpλL^{p_\lambda} target) for all w∈A1∩RHσw\in A_1\cap RH_\sigma with σ>3+2λ8λ\sigma>\frac{3+2\lambda}{8\lambda} (and σ≥52\sigma\ge\frac{5}{2} when λ=16\lambda=\frac{1}{6}). Is this reverse-Hölder threshold sharp — does the estimate fail for suitable weights with smaller σ\sigma — or does it extend to a larger weight class? These three components form one scientific objective: the sharp sparse-domination boundary of BλB_\lambda and its endpoint inclusions, of which (3) is the weighted corollary face via the sparse-to-weighted transfer. An affirmative resolution of (1) and (2) would confirm the closed trapezoid T2(λ)\mathcal{T}_2(\lambda) as the exact sparse-domination region for every 0<λ<120<\lambda<\frac{1}{2}; a negative answer at any boundary point, established by explicit examples, would identify the true boundary.

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