Weyl quantization of Lebesgue spaces

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)