Hypercontractive semigroups and Sobolev’s inequality

Abstract

ABSTRACT. If H > 0 is the generator of a hypercontractive semigroup (HCSG), it is known that (H + 1)~'''2 is a bounded operator from Lp to IP, 1 < p < °°. We prove that (H + 1)~'2 is bounded from L to the Orlicz space L ln"*"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 + l)-" is bounded from LP to iPXn^L, 2 < p < °°. Another interpolation argument shows that (H + 1)~'/2 is bounded from LP(ln+L)m to ¿P(ln+L)m + 1, 2 < p < °° and m a positive integer. Finally, we identify the topological duals of the spaces mentioned above.

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

“…In a somewhat different direction, Feissner's thesis [17] under Gross, takes up the embedding implied by (1.1), namely W 1 2 R n , dγ n ⊂ L 2 (LogL) R n , dγ n ,…”

Discrete logarithmic Sobolev inequalities in Banach spaces

“…As Ledoux himself points out, his theorem was only new in the endpoint case $$p=1$$. For $$p\in (1,\infty)$$, the corresponding inequality also follows by a concatenation of Meyer's Riesz transform inequalities in Gauss space [34] (see also [38]) and a delicate result of Bakry and Meyer [1] on the boundedness of operators of the form $$\mathcal {L}^{-\ualpha}$$ in Orlicz spaces, where $$\mathcal {L}$$ is the generator of a hypercontractive semigroup (see also [18]).…”

The clamped plate in Gauss space

“…It is not immediately clear that the above form is well defined, i.e., that the integral converges for all u and v. It turns out that convergence of a(u, v) follows from, and is closely linked with, the logarithmic Sobolev inequality of Gross [20]. Related results and generalizations may be found in the work of Adams [2] and Feissner [19]; we need only the following, narrower result. Proposition 2 Let ⊆ R n be a domain and say u ∈ H 1 0 ( , γ ).…”