How to find the norms defined on the Hölder space
The Hölder Space $C^{k,\gamma}(\bar{U})$ consisting of the all $u \in C^k(\bar{U})$
for which the norm
$$\|u\|_{C^{k,\gamma}(\bar{U})}:= \sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})}+\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})}$$
is finite
how to find $|u\|_{C^{k,\gamma}(\bar{U})}$
For example $u(x,y,z)=x^2+y^2+z^2$ then how to find $|u\|_{C^{k,\gamma}(\bar{U})}$
What does this mean $\sum_{|\alpha|\le k} \|D^\alpha u \|_{C(\bar{U})},\sum_{|\alpha|=k} [D^\alpha u]_{C^{0,\gamma}(\bar{U})}$ ?
Can you please explain how to find norms of $||u||_{C(\bar{U})}$
and $[u]_{C^{0,\gamma(\bar{U})}}$
I am stating studying these things and I'm getting confusing
And can some on suggest me where can i get examples such kinds
Thank you so much
Best Answer
Your question is vague, but lets compute say $\|u\|_{C^{1,1/2}([0,1]^3)}$. Let $\mathbf x = (x,y,z)$, so that $u(\mathbf x) = |\mathbf x|^2$.
First, we need to compute derivatives of order 1. Its $\nabla u(\mathbf x) = 2\mathbf x$. So $$ \sum_{|\alpha| \le 1}\|D^\alpha f\|_{C^0} = \sup_{\mathbf x\in[0,1]^3} |\mathbf x|^2 + \sup_{x\in[0,1]}|2x| + \sup_{y\in[0,1]}|2y| + \sup_{z\in[0,1]}|2z| =3+2+2+2 = 9 $$ Next we need $[D^{\alpha} u]_{C^{0,1/2}}$. If $\alpha = (1,0,0)$, then $$ |D^\alpha u(\mathbf x) - D^\alpha u(\mathbf {\tilde x})| = 2| x-\tilde x| $$ so $$[D^\alpha u]_{C^{0,1/2}} = \sup_{\mathbf x,\mathbf {\tilde x} \in[0,1]^3: \mathbf x \ne \mathbf {\tilde x}} \frac{2|x-\tilde x|}{|\mathbf x - \mathbf {\tilde x}|^{1/2}} = \sup_{x,\tilde x \in [0,1]:x\ne \tilde x} 2|x-\tilde x|^{1/2} = 2$$ Symmetry yields the other holder seminorms, and so $$ \|u\|_{C^{1,\alpha}} = 9 + 2 + 2 + 2 = 15$$