We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón–Zygmund type estimates below the natural exponent are also obtained for $\mathcal{A}$ -superharmonic functions in the whole space ℝ^{$n$}. This answers a question raised in our earlier work (On Calderón–Zygmund theory for$p$- and $\mathcal{A}$ -superharmonic functions, to appear in$Calc. Var. Partial Differential Equations$, DOI10.1007/s00526-011-0478-8) and thus greatly improves the result there.

Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory.

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Necessary and sufficient conditions are given in order for a Borel measure on the Euclidean sphere to have an affine image that is isotropic. A sharp reverse affine isoperimetric inequality for Borel measures on the sphere is presented. This leads to sharp reverse affine isoperimetric inequalities for convex bodies.

We show that expanding Kahler-Ricci solitons which have positive holomorphic bisectional curvature and are C^2-asymptotic to a conical Kahler manifold at infinity must be the U(n)-rotationally symmetric expand solitons constructed by Cao.