The aim of this paper is to study the Lelong number, the integrability index and the Monge–Ampère mass at the origin of an S^1-invariant plurisubharmonic function on a balanced domain in C^n under the Schwarz symmetrization. We prove that n times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge–Ampère mass is always decreasing under the symmetrization.

A condition on a Banach function space X is given under which the Coifman–Fefferman and Fefferman–Stein inequalities on X are equivalent.

In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound ϰ(E)≤πH^1(E)is sharp.

We obtain an asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, relevant for random matrix theory. As a consequence we show that the density of states in the random normal matrix model has universal error function transition across smooth interfaces for the limiting eigenvalue density. The key ingredient in the proof of the asymptotic expansion is the construction of the orthogonal foliation flow: a smooth flow of closed curves foliating a neighborhood of the interface, where the weighted L^2-mass of the orthogonal polynomial concentrates. The defining property of the flow is that the orthogonal polynomial of a given degree with respect to the disintegrated weighted area measure on the curves is approximately stationary in the flow parameter. To compute the coefficient functions in the expansion, we develop an algorithm which collapses the orthogonality relations to conditions along the interface, which become inhomogeneous Toeplitz kernel equations.