Weyl quantization of Lebesgue spaces

✍ Scribed by Paolo Boggiatto; Giuseppe De Donno; Alessandro Oliaro

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 131 KB
• Volume: 282
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200610829

✦ Synopsis

Abstract

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)