Assume \(n< p \le \infty\). There exists a constant \(C\) depending only on \(n\) and \(p\) such that

\begin{equation*} \lVert u\rVert_{C^{0,\gamma}(\mathbb{R}^n)}\le C \lVert u\rVert_{W^{1,p}(\mathbb{R}^n)} \end{equation*}

for all \(u\in C^1(\mathbb{R}^n)\), where

\begin{equation*} \gamma=1-\frac{n}{p} \end{equation*}

[1, 5.6 Theorem 4].

See also Link to heading

• Morrey’s inequality on a bounded domain with \(C^1\)-boundary.
• Morrey’s inequality for higher derivatives

References Link to heading

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