Weyl quantization of Lebesgue spaces
Copyright policy )Weyl quantization of Lebesgue spaces
AbstractWe study boundedness and compactness properties for the Weyl quantization with symbols in Lq (ℝ2d ) acting on Lp (ℝd ). 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 Lp setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L*q (ℝ2d ) and L♯q (R2d ), respectively smaller and larger than the Lq (ℝ2d ),and showing that the Weyl correspondence is bounded on L*q (R2d ) (and yields compact operators), whereas it is not on L♯q (R2d ). We conclude with a remark on weak‐type Lp boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
- University of Turin Italy
Integral operators, Compactness, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Interpolation between normed linear spaces, Weyl operators, \(L^{p}\) spaces, quantization, Quantization; Weyl operators; $L^p$ spaces, Pseudodifferential operators
Integral operators, Compactness, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Interpolation between normed linear spaces, Weyl operators, \(L^{p}\) spaces, quantization, Quantization; Weyl operators; $L^p$ spaces, Pseudodifferential operators
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