Regularization Of $m$-subharmonic Functions And H\"Older Continuity

TL;DR

This paper develops a method using sup-convolution to approximate bounded m-subharmonic functions on compact Kähler manifolds and proves Hölder continuity of solutions to complex Hessian equations with L^p data, generalizing previous results.

Contribution

It introduces a new approximation technique for m-subharmonic functions and establishes Hölder continuity for solutions to complex Hessian equations with L^p right-hand sides.

Findings

Sup-convolution provides effective upper approximations of m-subharmonic functions.

Solutions to σ_m equations are Hölder continuous when RHS is in L^p, p > n/m.

Results extend to more general complex Hessian equations.

Abstract

We use sup-convolution to find upper approximations of a bounded $m$-subharmonic function on a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. As an application, we show the H\"older continuity of solutions to $\sigma_m$ equation when the right hand side is in $L^p$, $p>\frac{n}{m}$. All these results generalize to more general complex Hessian equations.