We investigate the quantum analogue of the classical Sobolev inequalities in
the phase space. These inequalities can be seen as a many body uncertainty
principle, and also lead to new bounds on the Schatten norms of the Weyl
quantization in terms of its symbol. As an intermediate tool, we define a
semiclassical analogue of the convolution together with the corresponding
Young’s and Hardy-Littlewood-Sobolev’s inequalities, and introduce quantum
Besov spaces. Explicit estimates are obtained on the optimal constants.
