Lorentz estimates for multilinear convolution operators

Abstract

We study O'Neil-type inequalities in Lorentz spaces for two extremal geometric configurations of translation-invariant multilinear convolution operators. For the one-parameter model, we complement the existing weak-kernel theory with estimates for kernels in the Lorentz space $L^{r,s}$, obtaining a stronger target Lorentz space when $s$ is finite. For the full-dimensional model, we prove a Lorentz refinement of Oberlin's multilinear Young inequality in the interior of its admissible range. For nonnegative full-dimensional kernels, we extend H\"ormander's necessary condition to the multilinear setting in Lorentz spaces and obtain an additional restriction on the secondary indices in the critical case. We also prove the failure of the full-dimensional weak $L^1$ endpoint, use additive combinatorics to construct one-parameter restricted weak-type counterexamples for $p<1$, and obtain complementary positive results for kernels of special structure.