Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Logarithmic Sobolev inequalities for the heat-diffusion semigroup
HTML articles powered by AMS MathViewer

by Fred B. Weissler PDF
Trans. Amer. Math. Soc. 237 (1978), 255-269 Request permission

Abstract:

An explicit formula relating the Hermite semigroup ${e^{ - tH}}$ on R with Gauss measure and the heat-diffusion semigroup ${e^{t\Delta }}$ on R with Lebesgue measure is proved. From this formula it follows that Nelson’s hypercontractive estimates for ${e^{ - tH}}$ are equivalent to the best norm estimates for ${e^{t\Delta }}$ as a map ${L^q}(R)$ into ${L^p}(R),1 < q < p < \infty$. Furthermore,the inequality \[ \frac {d}{{dq}}\log \left \| \phi \right \|_q^q \leqslant \frac {n}{{2q}}\log \left [ {\frac {{{q^2}}}{{2\pi ne(q - 1)}} \cdot \frac {{\operatorname {Re} \langle - \Delta \phi ,{J^q}\phi \rangle }}{{\left \| \phi \right \|_q^q}}} \right ] + \log {\left \| \phi \right \|_q},\] where $1 < q < \infty ,{J^q}\phi = (\operatorname {sgn} \phi )|\phi {|^{q - 1}}$, and the norms and sesquilinear form $\langle ,\rangle$ are taken with respect to Lebesgue measure on ${R^n}$, is shown to be equivalent to the best norm estimates for ${e^{t\Delta }}$ as a map from ${L^q}({R^n})$ into ${L^p}({R^n})$. This inequality is analogous to Gross’ logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D05, 46E35
  • Retrieve articles in all journals with MSC: 47D05, 46E35
Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 237 (1978), 255-269
  • MSC: Primary 47D05; Secondary 46E35
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0479373-2
  • MathSciNet review: 479373