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## A Littlewood–Paley type inequality with applications to the elliptic Dirichlet problem

### Tom 90 / 2007

Annales Polonici Mathematici 90 (2007), 105-130 MSC: 35J25, 42B25. DOI: 10.4064/ap90-2-2

#### Streszczenie

Let $L$ be a strictly elliptic second order operator on a bounded domain ${\mit \Omega } \subset {{\mathbb R}}^{n}$. Let $u$ be a solution to $Lu=\mathop {\rm div}\vec {f}$ in ${\mit \Omega }$, $u=0$ on $\partial {\mit \Omega }$. Sufficient conditions on two measures, $\mu$ and $\nu$ defined on ${\mit \Omega }$, are established which imply that the $L^{q}({\mit \Omega } ,d\mu )$ norm of $| \nabla u|$ is dominated by the $L^{p}({\mit \Omega } ,dv)$ norms of $\mathop {\rm div}\vec {f}$ and $| \vec {f}|$. If we replace $| \nabla u|$ by a local Hölder norm of $u$, the conditions on $\mu$ and $\nu$ can be significantly weaker.

#### Autorzy

• Caroline SweezyDepartment of Mathematical Sciences
New Mexico State University
Box 30001 3MB
Las Cruces, NM 88003-8001, U.S.A.
e-mail

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