We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp-Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in the literature including the Prékopa-Leindler and Barthe inequalities.
Bibliographical noteFunding Information:
The second author is supported by the action Supporting Postdoctoral Researchers of the operational program Education and Lifelong Learning (Action's Beneficiary: General Secretariat for Research and Technology ) and is co-financed by the European Social Fund (ESF) and the Greek State .
The last author is supported by the A. Sloan foundation , United States-Israel Binational Science Foundation ( BSF-2010288 ) and US National Science Foundation ( NSF-CAREER-1151711 ).
- Brascamp-Lieb inequality
- Correlation inequalities
- Gaussian hypercontractivity