Boundedness of metaplectic operators within $L^p$ spaces, applications to pseudodifferential calculus, and time-frequency representations

Abstract: Housdorff-Young's inequality establishes the boundedness of the Fourier transform from $L^p$ to $L^q$ spaces for $1\leq p\leq2$ and $q=p'$, where $p'$ denotes the Lebesgue-conjugate exponent of $p$. This paper extends this classical result by characterizing the $L^p-L^q$ 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 $L^p$ 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 $L^p$ 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 $L^2(\mathbb{R}^{2d})$ in terms of shift-invertible metaplectic operators, establish the density of shift-invertible symplectic matrices in $Sp(2d,\mathbb{R})$, and prove that the lack of shift-invertibility prevents metaplectic Wigner distributions to define the so-called modulation spaces $M^p(\mathbb{R}^d)$.