Assume \(1\le p < \frac{n}{k}\). Let \(U\subset \mathbb{R}^n\) be a bounded open set and suppose \(\partial U\) is in \(C^1\). Then there exists a constant \(C\) depending on \(U\), \(n\), \(k\) and \(p\) such that

\begin{equation*} \lVert u\rVert_{L^{q}(U)}\le C \lVert u\rVert_{W^{k,p}(U)} \end{equation*}

for all \(u\in W^{k,p}(U)\) and \(\frac{1}{q}=\frac{1}{p}-\frac{k}{n}\). [1, 5.6 Theorem 6]

Proof idea Link to heading

Apply iteratively (0x6745c729) .

Remarks
  • Let \(0\le l < k\). According to the prove we have \(W^{k,p}(U)\hookrightarrow W^{l,q}(U)\) with \(\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}\).

References Link to heading

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