Boundedness of Metaplectic Operators Within \(L^p\) Spaces, Applications to Pseudodifferential Calculus, and Time–Frequency Representations

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