Sobolev conjugate

The Sobolev conjugate of p for ${\displaystyle 1\leq p<n}$, where n is space dimensionality, is

This is an important parameter in the Sobolev inequalities.

A question arises whether u from the Sobolev space ${\displaystyle W^{1,p}(\mathbb {R} ^{n})}$ belongs to ${\displaystyle L^{q}(\mathbb {R} ^{n})}$ for some q>p. More specifically, when does ${\displaystyle \|Du\|_{L^{p}(\mathbb {R} ^{n})}}$ control ${\displaystyle \|u\|_{L^{q}(\mathbb {R} ^{n})}}$? It is easy to check that the following inequality

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