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arXiv · 2607.02312

The structure of solution spaces for fractional-order operators, with gradient estimates

Gerd Grubb
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Abstract

The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-Δ)^{a}$ ($0 1/2$, e.g. estimating $d^{1-a+s}\nabla (u/d^a)$ in terms of norms of $f$ and $u$, both in $H_q^t$-spaces and $C_*^t$-spaces. This is entirely new in the case of Bessel-potential spaces; it extends previous results by Fall and Jarohs in Hölder spaces. A new tool is introduced: $\dot H^{s+t}_q(\barΩ)\subset d^s\dot H^{t}_q(\barΩ)$ holds for $s,t\ge 0$ with $s+t<τ$.

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Gerd Grubb. 2026-08-03. The structure of solution spaces for fractional-order operators, with gradient estimates. https://arxiv.org/abs/2607.02312

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