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

can not be true for arbitrary q. Consider ${\displaystyle u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})}$, infinitely differentiable function with compact support. Introduce ${\displaystyle u_{\lambda }(x):=u(\lambda x)}$. We have that:

The inequality (*) for ${\displaystyle u_{\lambda }}$ results in the following inequality for ${\displaystyle u}$

If ${\displaystyle 1-{\frac {n}{p}}+{\frac {n}{q}}\neq 0,}$ then by letting ${\displaystyle \lambda }$ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for

which is the Sobolev conjugate.