  We investigate the quantum analogue of the classical Sobolev inequalities in
the phase space, with the quantum Sobolev norms defined in terms of Schatten
norms of commutators. These inequalities provide an uncertainty principle for
the Wigner-Yanase skew information, and also lead to new bounds on the Schatten
norms of the Weyl quantization in terms of its symbol. As an intermediate tool,
we obtain the analogue of Hardy-Littlewood-Sobolev's inequalities for a
semiclassical analogue of the convolution, and introduce quantum Besov spaces.
Explicit estimates are obtained on the optimal constants.
