  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
$M^p(\mathbb{R}^d)$.
