## Sharpness of Morrey’s inequality

Thm: Suppose ${n, then there exists a constant ${C}$ such that

$\displaystyle ||u||_{C^{0,\gamma}(\mathbb{R}^n)}\leq C(n,p)||u||_{W^{1,p}(\mathbb{R}^n)}$

for all ${u\in C^1(\mathbb{R}^n)}$, where ${\gamma=1-n/p}$. For functions in ${W^{1,p}(\Omega)}$, we have

Thm: Suppose ${\Omega}$ is bounded domain in ${\mathbb{R}^n}$, and ${\partial\Omega\in C^1}$. Assume ${n and ${u\in W^{1,p}(\Omega)}$, there exists one version ${u^*\in C^{0,\gamma}(\bar{\Omega})}$, for ${\gamma=1-n/p}$, such that

$\displaystyle ||u^*||_{C^{0,\gamma}(\mathbb{R}^n)}\leq C(n,p,\Omega)||u||_{W^{1,p}(\mathbb{R}^n)}$

Remark: This ${\gamma}$ is critical number. For ${\beta\in (\gamma,1]}$, we can choose ${\alpha\in (\gamma,\beta)}$. Consider the function ${u(x)=|x|^\alpha}$ on the unit ball ${B_1}$. Then

$\displaystyle D_iu=\alpha x_i|x|^{\alpha-2}$

${u\in W^{1,p}(B_1)}$ if and only if ${(1-\alpha)p, which is ${\alpha>\gamma}$.

However consider the ${[u]_\beta}$ which is

$\displaystyle [u]_{\beta;B_1}=\sup_{x,y\in B_1}\frac{\big||x|^\alpha-|y|^\beta\big|}{|x-y|^\beta}\geq \sup_{x\in B_1}\frac{|x|^\alpha}{|x|^\beta}=+\infty$

So ${u\not\in C^{0,\beta}(B_1)}$.