2026/05/10 by Sanghyuk Lee, Sungchul Lee
#math.CA
In this paper, we determine the optimal universal \(Lp\)-\(Lq\) type sets for convolution operators \(f↦ μ*f\) associated with fractal measures μ∈ \mathcal Pα,β(\mathbb Rd), which denotes the class of compactly supported Borel probability measures satisfying the \(α\)-Frostman condition μ(B(x,ρ)) \lesssim ρα, x∈\mathbb Rd, 0<ρ<1, and the \(β/2\)-Fourier decay condition |\widehatμ(ξ)| \lesssim |ξ|-β/2, ξ∈\mathbb Rd. More precisely, we characterize the largest \(Lp\)-\(Lq\) region that is forced solely by the Frostman and Fourier decay assumptions throughout the full admissible range of \((α,β)\), with distinct optimal regions in the geometric and nongeometric regimes. We prove optimality in the worst-case sense over \(\mathcal Pα,β(\mathbb Rd)\) by constructing, for each admissible pair \((α,β)\), a single extremal measure whose support has the smallest Hausdorff dimension allowed by the hypotheses. Moreover, variants of the same constructions also yield a single-measure sharpness theorem for the \(L2\) Fourier restriction theorem of Mockenhaupt--Mitsis--Bak--Seeger: in every dimension and in both the geometric and nongeometric regimes, we construct a measure in \(\mathcal Pα,β(\mathbb Rd)\) for which the Mockenhaupt--Mitsis--Bak--Seeger threshold exponent is sharp.