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arXiv · 1710.09453

On the interpolation space $(L^p(\Omega), W^{1,p}(\Omega))_{s,p}$ in non-smooth domains

Irene Drelichman·Ricardo G. Durán
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Abstract

We show that, for certain non-smooth bounded domains $\Omega\subset\mathbb{R}^n$, the real interpolation space $(L^p(\Omega), W^{1,p}(\Omega))_{s,p}$ is the subspace $\widetilde W^{s,p}(\Omega) \subset L^p(\Omega)$ induced by the restricted fractional seminorm $$ |f|_{\widetilde W^{s,p}(\Omega)} = \Big( \int_\Omega \int_{|x-y|<\frac{d(x)}2} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} \, dy \,dx \Big)^\frac{1}{p}. $$ In particular, the above result includes simply connected uniform domains in the plane, for which a characterization of the interpolation space was previously unknown.

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Irene Drelichman, Ricardo G. Durán. 2017-10-25. On the interpolation space $(L^p(\Omega), W^{1,p}(\Omega))_{s,p}$ in non-smooth domains. https://arxiv.org/abs/1710.09453

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