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Transactions of the American Mathematical Society

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Sobolev interpolation inequalities with weights


Authors: Cristian E. Gutiérrez and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 323 (1991), 263-281
MSC: Primary 46E99; Secondary 35B45, 46M35
MathSciNet review: 994166
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Abstract: We study weighted local Sobolev interpolation inequalities of the form

\begin{displaymath}\begin{gathered}\frac{1} {{{w_2}(B)}}{\int\limits_B {\vert u(... ...\vert u(x){\vert^p}v(x)dx} } \right), \hfill \\ \end{gathered} \end{displaymath}

, where $ 1 < p < \infty,h > 1, B$ is a ball in $ {{\mathbf{R}}^n}$, and $ v$ ,$ {w_1}$, and $ {w_2}$ are weight functions. The case $ p = 2$ is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case $ u$ has compact support in $ B$, and we derive global Landau inequalities $ {\left\Vert {\nabla u} \right\Vert _{L_w^q}} \leq c\left\Vert {\nabla u} \righ... ...eft\Vert {{\nabla ^2}u} \right\Vert _{L_v^p}^a,0 < a < 1,1 < p \leq q < \infty $, when $ u$ has compact support.

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DOI: https://doi.org/10.1090/S0002-9947-1991-0994166-1
Article copyright: © Copyright 1991 American Mathematical Society