On semigroup maximal operators associated with divergence-form operators with complex coefficients

TL;DR

This paper proves the boundedness of a maximal operator associated with divergence-form operators with complex coefficients on arbitrary domains, extending classical ergodic theorems to non-positive, non-contractively generated semigroups.

Contribution

It establishes $L^p$ boundedness of the maximal operator for divergence-form operators with complex coefficients under $p$-ellipticity, even when the semigroup is not contractive or positive.

Findings

Maximal operator ${ mf M}^A$ is bounded in $L^p( ext{domain})$ for $p$-elliptic operators.

Range of $L^p$ boundedness improves under certain Sobolev embedding conditions.

Analysis of two-parameter maximal operators extends the scope of maximal function theory.

Abstract

Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $\Omega\subseteq\mathbb{R}^{d}$. We prove that the maximal operator ${\mathscr M}^{A} f=\sup_{t>0}|\exp(-tL_{A})f|$ is bounded in $L^{p}(\Omega)$, whenever $A$ is $p$-elliptic in the sense of [10]. The relevance of this result is that, in general, the semigroup generated by $-L_{A}$ is neither contractive in $L^{\infty}$ nor positive, therefore neither the Hopf--Dunford--Schwartz maximal ergodic theorem [15, Chap.~VIII] nor Akcoglu's maximal ergodic theorem [1] can be used. We also show that if $d\geq 3$ and the domain of the sesquilinear form associated with $L_{A}$ embeds into $L^{2^{*}}(\Omega)$ with $2^{*}=2d/(d-2)$, then the range of $L^{p}$-boundedness of ${\mathscr M}^{A}$ improves into the interval $(rd/((r-1)d+2),rd/(d-2))$, where $r\geq 2$ is such that $A$ is $r$-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator $\sup_{s,t>0}|T^{A_{1}}_{s}T^{A_{2}}_{t}f|$.