Sharpness of convolution bounds for measures

Abstract: In this paper, we determine the sharp ((p,q)) range for (L^p)--(L^q) bounds of convolution operators (f\mapsto μ*f) associated with fractal measures (μ\in \mathcal P_{α,β}(\mathbb R^d)), namely, compactly supported Borel probability measures satisfying the (α)-Frostman condition [ μ(B(x,ρ)) \lesssim ρ^α, \qquad \forall (x,ρ)\in \mathbb R^d\times (0,1), ] and the (β/2)-Fourier decay condition [ |\widehatμ(ξ)| \lesssim |ξ|^{-β/2}, \qquad \forall ξ\in\mathbb R^d. ] Sharpness is established by constructing measures satisfying these conditions together with a suitable lower regularity condition. Modifications of the same constructions also refine previous sharpness results for the (L^2) restriction estimate of Mockenhaupt--Mitsis--Bak--Seeger by producing, in every dimension and in both the geometric ((α\geβ)) and non-geometric ((β>α)) regimes, a single measure in (\mathcal P_{α,β}(\mathbb R^d)) for which the corresponding threshold exponent is sharp.