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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sobolev inequalities for weight spaces and supercontractivity


Author: Jay Rosen
Journal: Trans. Amer. Math. Soc. 222 (1976), 367-376
MSC: Primary 46E35
MathSciNet review: 0425601
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Abstract: For $ \phi \in {C^2}({{\mathbf{R}}^n})$ with $ \phi (x) = a\vert x{\vert^{1 + s}}$ for $ \vert x\vert \geqslant {x_0},a,s > 0$, define the measure $ d\mu = \exp ( - 2\phi ){d^n}x$ on $ {{\mathbf{R}}^n}$. We show that for any $ k \in {{\mathbf{Z}}^ + }$

\begin{displaymath}\begin{array}{*{20}{c}} {\int {\vert f{\vert^2}\vert\lg(\vert... ...mu )}){\vert^{2sk/(s + 1)}}} } \right\}} \hfill \\ \end{array} \end{displaymath}

As a consequence we prove $ {e^{ - t{\nabla ^\ast} \cdot \nabla }}:{L_q}({{\mathbf{R}}^n},d\mu ) \to {L_p}({{\mathbf{R}}^n},d\mu ),p,q \ne 1,\infty $, is bounded for all $ t > 0$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0425601-7
PII: S 0002-9947(1976)0425601-7
Article copyright: © Copyright 1976 American Mathematical Society