Geodesic
Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions
Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1384-1410

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DOI

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic ( $\text{LSH}$ ) 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 $\text{LSH}$ functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.
DOI : 10.4153/CJM-2015-015-8
Mots-clés : 47D06, logarithmic Sobolev inequalities 
Graczyk, Piotr; Kemp, Todd; Loeb, Jean-Jacques. Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions. Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1384-1410. doi: 10.4153/CJM-2015-015-8
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     author = {Graczyk, Piotr and Kemp, Todd and Loeb, Jean-Jacques},
     title = {Strong {Logarithmic} {Sobolev} {Inequalities} for {Log-Subharmonic} {Functions}},
     journal = {Canadian journal of mathematics},
     pages = {1384--1410},
     year = {2015},
     volume = {67},
     number = {6},
     doi = {10.4153/CJM-2015-015-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-015-8/}
}
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