Sharpness of Morrey’s inequality

Thm: Suppose {n<p\leq \infty}, 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<p\leq \infty} 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<n}, 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)}.

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