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Search: MSC category 47D06 ( One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] )

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1. CJM Online first

Graczyk, Piotr; Kemp, Todd; Loeb, Jean-Jacques
Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions
We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.

Keywords:logarithmic Sobolev inequalities

2. CJM 2007 (vol 59 pp. 1207)

Bu, Shangquan; Le, Christian
$H^p$-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces
We consider maximal regularity in the $H^p$ sense for the Cauchy problem $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed operator on a Banach space $X$ and $f$ is an $X$-valued function defined on $\R$. We prove that if $X$ is an AUMD Banach space, then $A$ satisfies $H^p$-maximal regularity if and only if $A$ is Rademacher sectorial of type $<\frac{\pi}{2}$. Moreover we find an operator $A$ with $H^p$-maximal regularity that does not have the classical $L^p$-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces $H^p(\R;X)$, in the case when $X$ is an AUMD Banach space.

Keywords:$L^p$-maximal regularity, $H^p$-maximal regularity, Rademacher boundedness
Categories:42B30, 47D06

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