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Transactions of the American Mathematical Society

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

 
 

 

Hypercontractive semigroups and Sobolev's inequality


Author: George F. Feissner
Journal: Trans. Amer. Math. Soc. 210 (1975), 51-62
MSC: Primary 46E30; Secondary 47D05, 81.46
DOI: https://doi.org/10.1090/S0002-9947-1975-0511867-0
MathSciNet review: 0511867
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Abstract: If $ H \geqslant 0$ is the generator of a hypercontractive semigroup (HCSG), it is known that $ {(H + 1)^{ - 1/2}}$ is a bounded operator from $ {L^p}$ to $ {L^p},1 \leqslant p \leqslant \infty $. We prove that $ {(H + 1)^{ - 1/2}}$ is bounded from $ {L^2}$ to the Orlicz space $ {L^2}{\text{ I}}{{\text{n}}^ + }L$, basing the proof on the uniform semiboundedness of the operator $ H + V$, for suitable $ V$. We also prove by an interpolation argument, that $ {(H + 1)^{ - 1/2}}$ is bounded from $ {L^p}$ to $ {L^p}{\text{ I}}{{\text{n}}^ + }L,2 \leqslant p < \infty $. Another interpolation argument shows that $ {(H + 1)^{ - 1/2}}$ is bounded from $ {L^p}{({\text{I}}{{\text{n}}^ + }L)^m}$ to $ {L^p}{({\text{I}}{{\text{n}}^ + }L)^{m + 1}}$ and $ m$ a positive integer. Finally, we identify the topological duals of the spaces mentioned above.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0511867-0
Keywords: Hypercontractive semigroups, Orlicz spaces, interpolation, Sobolev inequalities, uniform semiboundedness, Gaussian measure
Article copyright: © Copyright 1975 American Mathematical Society

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