Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo–Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopoulos's theorem in [54]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.