Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

TL;DR

This paper establishes a bi-sublinear embedding for semigroups generated by complex-coefficient elliptic operators in divergence form within Orlicz spaces, extending previous power-function results to more general Young functions.

Contribution

It generalizes a known embedding result from power functions to broader Orlicz spaces using a novel Bellman function construction.

Findings

Proves bi-sublinear embedding in Orlicz spaces for complex elliptic operators.

Extends previous results from power functions to general Young functions.

Introduces a generalized Bellman function for this purpose.

Abstract

We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This generalizes a result by Carbonaro and Dragi\v{c}evi\'{c} from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.