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 \(f\mapsto \mu*f\) associated with fractal measures \(\mu\in \mathcal P_{\alpha,\beta}(\mathbb R^d)\), namely, compactly supported Borel probability measures satisfying the \(\alpha\)-Frostman condition \[ \mu(B(x,\rho)) \lesssim \rho^\alpha, \qquad \forall (x,\rho)\in \mathbb R^d\times (0,1), \] and the \(\beta/2\)-Fourier decay condition \[ |\widehat{\mu}(\xi)| \lesssim |\xi|^{-\beta/2}, \qquad \forall \xi\in\mathbb R^d. \] 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 \((\alpha\ge\beta)\) and non-geometric \((\beta>\alpha)\) regimes, a single measure in \(\mathcal P_{\alpha,\beta}(\mathbb R^d)\) for which the corresponding threshold exponent is sharp.