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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Potential space estimates for Green potentials in convex domains


Author: Stephen J. Fromm
Journal: Proc. Amer. Math. Soc. 119 (1993), 225-233
MSC: Primary 35J25; Secondary 35B65
MathSciNet review: 1156467
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Abstract: Weak type $ (1,1)$ bounds are demonstrated for the operators

$\displaystyle f \mapsto \int {{\nabla _x}{\nabla _x}G(x,y)f(y)dy} \quad {\text{and}}\quad f \mapsto \int {{\nabla _x}{\nabla _y}G(x,y)f(y)dy,} $

where $ G$ is the Green operator for the Dirichlet problem for the Poisson equation on a bounded convex domain in $ {\mathbb{R}^n}$. These results are used to investigate smoothing properties of the Green operator in potential spaces. An application is given to the restriction of the potential space to the boundary of the domain.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1156467-3
PII: S 0002-9939(1993)1156467-3
Keywords: Green potential, convex domain, potential spaces
Article copyright: © Copyright 1993 American Mathematical Society