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Weighted norm estimates for Sobolev spaces


Author: Martin Schechter
Journal: Trans. Amer. Math. Soc. 304 (1987), 669-687
MSC: Primary 46E35; Secondary 26D20
DOI: https://doi.org/10.1090/S0002-9947-1987-0911089-3
MathSciNet review: 911089
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Abstract: We give sufficient conditions for estimates of the form

$\displaystyle {\int {\left\vert {u(x)} \right\vert} ^q}d\mu (x) \leqslant C\left\Vert u \right\Vert _{s,p}^1,\qquad u \in {H^{s,p}},$

to hold, where $ \mu (x)$ is a measure and $ {\left\Vert u \right\Vert _{s,p}}$ is the norm of the Sobolev space $ {H^{s,p}}$. If $ d\mu = dx$, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.

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DOI: https://doi.org/10.1090/S0002-9947-1987-0911089-3
Article copyright: © Copyright 1987 American Mathematical Society

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