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arXiv · 2407.01584

The mapping properties of fractional derivatives in weighted fractional Sobolev space

Cailing Li
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Abstract

We study the mapping behavior of the Marchaud fractional derivative with different extensions in the scale of fractional weighted Sobolev spaces. In particular we show that the $\alpha$--order Riemann--Liouville fractional derivative maps $W^{p,s}_0(\Omega)$ to $W^{p,s-\alpha}(\Omega)$, for all $0<\alpha<s<1$ and the $\alpha$--order Marchaud fractional derivative with even extension maps the fractional Sobolev space $W^{p,s}((0,\infty))$ to $W^{p,s-\alpha}(\real)$ for all $0<\alpha<s<1$ and $ps\geq1$ . The proof is based on the Calder\'{o}n--Lions interpolation theorem.

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Cailing Li. 2024-06-18. The mapping properties of fractional derivatives in weighted fractional Sobolev space. https://arxiv.org/abs/2407.01584

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