Explicit improvements for $\mathrm{L}^p$-estimates related to elliptic systems

Tim Böhnlein; Moritz Egert

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arXiv · 2302.09039

Explicit improvements for $\mathrm{L}^p$-estimates related to elliptic systems

Tim Böhnlein·Moritz Egert

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Abstract

We give a simple argument to obtain $\mathrm{L}^p$-boundedness for heat semigroups associated to uniformly strongly elliptic systems on $\mathbb{R}^d$ by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give $p$ explicitly in terms of ellipticity. It is optimal at the endpoint $p=\infty$. We also obtain $\mathrm{L}^p$-estimates for the gradient of the semigroup, where $p>2$ depends on ellipticity but not on dimension.

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Tim Böhnlein, Moritz Egert. 2023-02-17. Explicit improvements for $\mathrm{L}^p$-estimates related to elliptic systems. https://arxiv.org/abs/2302.09039

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