Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on crucial trilinear smoothing inequality developed herein, and bounds an anisotropic twisted paraproduct. also leads to corresponding maximal function quantitative nonlinear Roth-type theorem concerning patterns in Euclidean plane.

We propose results on interior Morrey, BMO and Hölder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L. 1. Main Results It is well known that, when dealing with elliptic systems, in contrast to the case of a single second-order elliptic equation, the solely essential boundedness of the...

In this paper, we introduce the p-adic Hardy type operator and obtain its sharp bound on the p-adic Lebesgue product spaces. Meanwhile, an analogous result is computed for the p-adic Lebesgue product spaces with power weights. In addition, we characterize a sufficient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on the p-adic Lebesgue product spa...

Let V be a finite dimensional real vector space. Let V alsm(V ) be the space of translation invariant smooth valuations on convex compact subsets of V . Let Dens(V ) be the space of Lebesgue measures on V . The goal of the article is to construct and study an isomorphism FV : V alsm(V )−̃→V alsm(V ∗)⊗Dens(V ) such that FV commutes with the natural action of the full linear group on both spaces, ...