Hardy inequalities and nonlocal capacity
Tomasz Grzywny, Julia Lenczewska
TLDR
This paper introduces nonlocal capacities, proves Hardy inequalities, and applies them to Sobolev embeddings and capacity estimation.
Key contributions
- Introduce and study capacities for nonlocal Sobolev spaces.
- Focus on capacities for zero-order nonlocal operators.
- Prove novel Hardy-type inequalities for these nonlocal spaces.
- Apply inequalities to derive Sobolev embeddings and estimate ball capacities.
Why it matters
This work provides fundamental tools for understanding nonlocal Sobolev spaces, particularly those with zero-order operators. The new Hardy inequalities and capacity estimates are crucial for further research in nonlocal analysis and its applications in PDEs and mathematical physics.
Original Abstract
In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.
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