Hypercontractive semigroups and Sobolev’s inequality

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.