An example in the theory of hypercontractive semigroups

Korzeniowski, Andrzej; Stroock, Daniel W.

doi:10.1090/S0002-9939-1985-0781062-0

An example in the theory of hypercontractive semigroups
HTML articles powered by AMS MathViewer

by Andrzej Korzeniowski and Daniel W. Stroock

Proc. Amer. Math. Soc. 94 (1985), 87-90

DOI: https://doi.org/10.1090/S0002-9939-1985-0781062-0

PDF | Request permission

Abstract:

Let $L = x({d^2}/d{x^2}) + (1 - x)(d/dx)$ on ${C_c}((0,\infty ))$ be the Laguerre operator. It is shown that for $t > 0$, and $1 < p < q < \infty ,\;{e^{tl}}:{L^p}({e^{ - x}}dx) \to {L^q}({e^{ - x}}dx)$ has norm 1 if and only if ${e^{ - t}} \leqslant (p - 1)/(q - 1)$ and the corresponding logarithmic Sobolev constant is not equal to $2/\lambda$, where $\lambda$ is the smallest nonzero eigenvalue of $L$.

References

Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688

Probability theory and Euclidean field theory

O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Sturm-Liouville operators, J. Functional Analysis 39 (1980), no. 1, 42–56. MR 593787, DOI 10.1016/0022-1236(80)90018-X
O. S. Rothaus, Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities, J. Functional Analysis 42 (1981), no. 1, 102–109. MR 620581, DOI 10.1016/0022-1236(81)90049-5
O. S. Rothaus, Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities, J. Functional Analysis 42 (1981), no. 1, 102–109. MR 620581, DOI 10.1016/0022-1236(81)90049-5
D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. MR 755154, DOI 10.1007/978-1-4613-8514-1
Fred B. Weissler, Two-point inequalities, the Hermite semigroup, and the Gauss-Weierstrass semigroup, J. Functional Analysis 32 (1979), no. 1, 102–121. MR 533222, DOI 10.1016/0022-1236(79)90080-6
Fred B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Functional Analysis 37 (1980), no. 2, 218–234. MR 578933, DOI 10.1016/0022-1236(80)90042-7
Carl E. Mueller and Fred B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the $n$-sphere, J. Functional Analysis 48 (1982), no. 2, 252–283. MR 674060, DOI 10.1016/0022-1236(82)90069-6