Hypercontractive semigroups and Sobolev’s inequality
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- by George F. Feissner PDF
- Trans. Amer. Math. Soc. 210 (1975), 51-62 Request permission
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.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- 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