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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Sobolev inequalities for weight spaces and supercontractivity
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by Jay Rosen PDF
Trans. Amer. Math. Soc. 222 (1976), 367-376 Request permission

Abstract:

For $\phi \in {C^2}({{\mathbf {R}}^n})$ with $\phi (x) = a|x{|^{1 + s}}$ for $|x| \geqslant {x_0},a,s > 0$, define the measure $d\mu = \exp ( - 2\phi ){d^n}x$ on ${{\mathbf {R}}^n}$. We show that for any $k \in {{\mathbf {Z}}^ + }$ \[ \begin {array}{*{20}{c}} {\int {|f{|^2}|\lg (|f|){|^{2sk/(s + 1)}}d\mu } } \hfill \\ { \leqslant c\left \{ {\sum \limits _{|\alpha | = 0}^k {\left \|{D^\alpha }f\right \|_{{L_2}(d\mu )}^2 + \left \|f\right \|_{{L_2}(d\mu )}^2 \bullet |\lg (\left \|f\right \|_{{L_2}(d\mu )}){|^{2sk/(s + 1)}}} } \right \}} \hfill \\ \end {array} \] 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
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 222 (1976), 367-376
  • MSC: Primary 46E35
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0425601-7
  • MathSciNet review: 0425601