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.

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