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

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Antonella Carbonaro , Oliver Dragičević · YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2207.11045

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 $Ω\subseteq\mathbb{R}^{d}$. We prove that the maximal operator ${\mathscr M}^{A} f=\sup_{t>0}|\exp(-tL_{A})f|$ is bounded in $L^{p}(Ω)$, 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 $L^{2^{*}}(Ω)$ 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 $\sup_{s,t>0}|T^{A_{1}}_{s}T^{A_{2}}_{t}f|$.

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preprint

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en

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http://arxiv.org/abs/2207.11045

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https://doi.org/10.48550/arxiv.2207.11045

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

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preprint

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en

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2022

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2022-07-22

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Antonella Carbonaro, Oliver Dragičević

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https://arxiv.org/abs/2207.11045

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Mathematics, Semigroup, Ergodic theory, Elliptic operator, Bounded function, Combinatorics, Omega, Domain (mathematical analysis), Operator (biology), Discrete mathematics, Pure mathematics, Mathematical analysis, Physics, Transcription factor, Chemistry, Biochemistry, Gene, Repressor, Quantum mechanics

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