Weyl Quantization of Lebesgue Spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in $L^q(\mathbb{R}^{2d})$ acting on $L^p(\mathbb{R}^{d})$. 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 $L^q_*(\R^{2d})$ and $L^q_\sharp(\R^{2d})$, respectively smaller and larger than the $L^q(\R^{2d})$, and showing that the Weyl correspondence is bounded on $L^q_*(\R^{2d})$ (and yields compact operators), whereas it is not on $L^q_\sharp(\R^{2d})$. We conclude with a remark on weak-type $L^p$ boundedness.