Sharpness of convolution bounds for measures

In this paper, we determine the optimal universal $L^p$-$L^q$ type sets for convolution operators $f\mapsto \mu \ast f$ associated with fractal measures $\mu \in {\mathcal{P}}_{\alpha ,\beta }({\mathbb{R}}^d)$, which denotes the class of compactly supported Borel probability measures satisfying the $\alpha$-Frostman condition $$\mu (B(x,\rho ))\lesssim {\rho }^{\alpha },x\in {\mathbb{R}}^d,0<\rho <1,$$ and the $\beta \mathrm{/}2$-Fourier decay condition $$\mathrm{\mid }\widehat{\mu }(\xi )\mathrm{\mid }\lesssim \mathrm{\mid }\xi {\mathrm{\mid }}^{-\beta \mathrm{/}2},\xi \in {\mathbb{R}}^d\mathrm{.}$$ More precisely, we characterize the largest $L^p$-$L^q$ region that is forced solely by the Frostman and Fourier decay assumptions throughout the full admissible range of $(\alpha ,\beta )$, with distinct optimal regions in the geometric and nongeometric regimes. We prove optimality in the worst-case sense over ${\mathcal{P}}_{\alpha ,\beta }({\mathbb{R}}^d)$ by constructing, for each admissible pair $(\alpha ,\beta )$, 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 $L^2$ Fourier restriction theorem of Mockenhaupt–Mitsis–Bak–Seeger: in every dimension and in both the geometric and nongeometric regimes, we construct a measure in ${\mathcal{P}}_{\alpha ,\beta }({\mathbb{R}}^d)$ for which the Mockenhaupt–Mitsis–Bak–Seeger threshold exponent is sharp.