Boundedness of Metaplectic Operators Within Lᵖ Spaces, Applications to Pseudodifferential Calculus, and Time–Frequency Representations

Hausdorff-Young's inequality establishes the boundedness of the Fourier transform from Lp to Lq spaces for 1 = p = 2 and q = p ', where p ' denotes the Lebesgue-conjugate exponent of p. This paper extends this classical result by characterizing the L p - Lq boundedness of 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 L p spaces. To achieve this, we leverage a parametrization of the symplectic group by Dopico and Johnson. We use our findings to provide boundedness results within L p spaces for pseudodifferential operators with symbols in Lebesgue spaces, and quantized by means of metaplectic operators. These quantizations consists of shift-invertible metaplecticWigner distributions, which are essential to measure local phase-space concentration of signals. Using the factorization by Dopico and Johnson, we infer a decomposition law for metaplectic operators on L2(R2d) 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).