Assume \(1\le p < n\) and let \(p^*\) be the Sobolev conjugate of \(p\). There exists a constant \(C\) depending only on \(n\) and \(p\) such that

\begin{equation*} \lVert u\rVert_{L^{p^*}(\mathbb{R}^n)}\le C \lVert Du\rVert_{L^p(\mathbb{R}^n)} \end{equation*}

for all \(u\in C^1_c(\mathbb{R}^n)\). [1, 5.6 Theorem 1]

Proof idea Link to heading

Start with \(p=1\). Since \(u\) has compact support estimate \(u(x)\) by a product of integrals applying the fundamental theorem, such that

\begin{equation*} \lvert u(x)\rvert \le \prod_{i=1}^{n} \int_{-\infty}^{\infty} Du(\ldots ,y_i,\ldots ) dy_i. \end{equation*}

Integrate over every argument and apply Hölder’s general inequality after every integration step. This leads to the desired inequality with \(C=1\).

The case \(p>1\) is a consequence of the former one. Set \(v(x)=\lvert u(x)\rvert^{\gamma}\) and choose a suitable \(\gamma\). Apply the above case. The constant is then \(C=\frac{p(n-1)}{n-p}\). Note that \(p1\).

Remarks
  • We find the correct \(p^*\) considering \(u_{\lambda}(x)=u(\lambda x)\). Then the inequality gets the scaling factor \(\lambda^{1-\frac{n}{p}-\frac{n}{p^*}}\). It only vanishes if \(p^*\) is the Sobolev conjugate .
  • The statement cannot be true for \(u\) without a compact support. The inequality is not true for \(u\equiv 1\).

See also Link to heading

References Link to heading

  1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.