Moser-Trudinger and Beckner-Onofri’s inequalities on the CR sphere

Abstract

We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for general spectrally defined operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CR-pluriharmonic functions on the sphere and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss.

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    @book {VK, MRKEY = {1220225},
      AUTHOR = {Vilenkin, N. Ja. and Klimyk, A. U.},
      TITLE = {Representation of {L}ie groups and Special Functions. {V}ol. 2. Class I Representations, Special Functions, and Integral Transforms},
      SERIES = {Math. Appl. (Soviet Series)},
      VOLUME = {74},
      PUBLISHER = {Kluwer Academic Publishers Group},
      ADDRESS = {Dordrecht},
      YEAR = {1993},
      PAGES = {xviii+607},
      ISBN = {0-7923-1492-1},
      MRCLASS = {22-02 (33C80 33D80)},
      MRNUMBER = {1220225},
      MRREVIEWER = {P. D. F. Ion},
      ZBLNUMBER = {0809.22001},
     }
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    @book {Zhu, MRKEY = {2115155},
      AUTHOR = {Zhu, Kehe},
      TITLE = {Spaces of Holomorphic Functions in the Unit Ball},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {226},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2005},
      PAGES = {x+271},
      ISBN = {0-387-22036-4},
      MRCLASS = {46E15 (32A36 32A37 46E20 46E22)},
      MRNUMBER = {2115155},
      MRREVIEWER = {Marco M. Peloso},
      ZBLNUMBER = {1067.32005},
      }

Authors

Thomas P. Branson

Luigi Fontana

Dipartimento di Matematica e Applicazioni
Universitá degli Studi di Milano-Bicocca
20125 Milano
Italy

Carlo Morpurgo

Mathematics Department
University of Missouri
Columbia, MO 65211