Sharpness of convolution bounds for measures

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.