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Weyl quantization of Lebesgue spaces
Author(s) -
Boggiatto Paolo,
De Donno Giuseppe,
Oliaro Alessandro
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610829
Subject(s) - mathematics , compact space , bounded function , lp space , pure mathematics , standard probability space , quantization (signal processing) , type (biology) , banach space , mathematical analysis , ecology , algorithm , biology
We study boundedness and compactness properties for the Weyl quantization with symbols in L q (ℝ 2 d ) acting on L p (ℝ d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L * q (ℝ 2 d ) and L ♯ q (R 2 d ), respectively smaller and larger than the L q (ℝ 2 d ),and showing that the Weyl correspondence is bounded on L * q (R 2 d ) (and yields compact operators), whereas it is not on L ♯ q (R 2 d ). We conclude with a remark on weak‐type L p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)