Boundedness of metaplectic operators within Lᵖ spaces, applications to pseudodifferential calculus, and time-frequency representations

Housdorff-Young's inequality establishes the boundedness of the Fourier transform from Lᵖ to Lq spaces for 1 p2 and q=p', where p' denotes the Lebesgue-conjugate exponent of p. This paper extends this classical result by characterizing the Lᵖ-Lq boundedness of all metaplectic operators, which play a significant role in harmonic analysis. We demonstrate that metaplectic operators are bounded on Lebesgue spaces if and only if their symplectic projection is either free or lower block triangular. As a byproduct, we identify metaplectic operators that serve as homeomorphisms of Lᵖ spaces. To achieve this, we leverage a parametrization of the symplectic group by F. M. Dopico and C. R. Johnson involving products of complex exponentials with quadratic phase, Fourier multipliers, linear changes of variables, and partial Fourier transforms. Then, we use our findings to provide boundedness results within Lᵖ spaces for pseudodifferential operators with symbols in Lebesgue spaces, and quantized by means of metaplectic operators. These quantizations consists of shift-invertible metaplectic Wigner distributions, which play a fundamental role in measuring local phase-space concentration of signals. Using the Dopico-Johnson factorization, we infer a decomposition law for metaplectic operators on L² (R^2d) in terms of shift-invertible metaplectic operators, establish the density of shift-invertible symplectic matrices in Sp (2d, R), and prove that the lack of shift-invertibility prevents metaplectic Wigner distributions to define the so-called modulation spaces Mᵖ (Rᵈ).