Local Hölder and scaled C-two-alpha norms on balls

Let $n\ge 1$ be an integer and $0<\alpha <1$, let $B_r(a)=\{x\in {\mathbb{R}}^n:\mathrm{\mid }x-a\mathrm{\mid }<r\}$ be a Euclidean ball of radius $r>0$, and let $f:B_r(a)\to \mathbb{R}$ or $\mathbb{C}$. Put $[f{]}_{0,\alpha ;B_r(a)}:=\sup \{\frac{\mathrm{\mid }f(x)-f(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }}:x,y\in B_r(a),x\mathrm{\ne }y\},\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty };B_r(a)}:={\sup }_{x\in B_r(a)}\mathrm{\mid }f(x)\mathrm{\mid },$ $\mathrm{\parallel }f{\mathrm{\parallel }}_{0,\alpha ;B_r(a)}^{\ast }:=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty };B_r(a)}+r^{\alpha }[f{]}_{0,\alpha ;B_r(a)}\mathrm{.}$ Both displayed quantities take values in $[0,+\mathrm{\infty }]$, so a norm can be $+\mathrm{\infty }$; we say that $f$ is $\alpha$-Hölder on $B_r(a)$ when $[f{]}_{0,\alpha ;B_r(a)}<\mathrm{\infty }$.

For $u\in C^2(B_r(a))$, a multi-index $\gamma$, and $j=\mathrm{\mid }\gamma \mathrm{\mid }\le 2$, write $D^{\gamma }u$ for the partial derivative of $C^k$ maps and multi-index derivative notation in Euclidean space in its displayed canonical order, and set $\mathrm{\parallel }u{\mathrm{\parallel }}_{2,\alpha ;B_r(a)}^{\ast }:={\sum }_{j=0}^2{r}^j{\max }_{\mathrm{\mid }\gamma \mathrm{\mid }=j}{\sup }_{x\in B_r(a)}\mathrm{\mid }{D}^{\gamma }u(x)\mathrm{\mid }+r^{2+\alpha }{\max }_{\mathrm{\mid }\gamma \mathrm{\mid }=2}[D^{\gamma }u{]}_{0,\alpha ;B_r(a)},$ where the inner maximum runs over the finitely many multi-indices with the stated order and the $j=0$ term is ${\sup }_{B_r(a)}\mathrm{\mid }u\mathrm{\mid }$. We write $C^{2,\alpha }(B_r(a))$ for the functions $u\in C^2(B_r(a))$ for which this quantity is finite.

• Scaling. If $r>0$, $a\in {\mathbb{R}}^n$, and $v(z):=u(a+rz)$ on $B_1(0)$, then $D^{\gamma }v(z)=r^{\mathrm{\mid }\gamma \mathrm{\mid }}{D}^{\gamma }u(a+rz)$ and therefore $\mathrm{\parallel }v{\mathrm{\parallel }}_{2,\alpha ;B_1(0)}^{\ast }=\mathrm{\parallel }u{\mathrm{\parallel }}_{2,\alpha ;B_r(a)}^{\ast },\mathrm{\parallel }v{\mathrm{\parallel }}_{0,\alpha ;B_1(0)}^{\ast }=\mathrm{\parallel }u{\mathrm{\parallel }}_{0,\alpha ;B_r(a)}^{\ast }\mathrm{.}$ The factors $r^j$ and $r^{2+\alpha }$ are exactly what makes the two sides equal: the norm is computed from the radius of the ball it is taken over, while each derivative of $v$ carries the extra factor $r^j$.
• Local, not global. These are interior ball quantities. They are read off the open ball $B_r(a)$ alone and say nothing about the boundary; in particular no boundary Schauder seminorm, no global $C^{2,\alpha }$ scale and no extension of $u$ beyond $B_r(a)$ are defined here.
• Finiteness. $\mathrm{\parallel }u{\mathrm{\parallel }}_{2,\alpha ;B_r(a)}^{\ast }<\mathrm{\infty }$ holds exactly when $u$, its first derivative field and its second derivative field are bounded on $B_r(a)$ and every second partial derivative is $\alpha$-Hölder there. No third derivative is involved. A finite $\mathrm{\parallel }f{\mathrm{\parallel }}_{0,\alpha ;B_r(a)}^{\ast }$ makes $f$ bounded; whether a continuous extension to the closed ball exists is a separate question, not part of this definition.
• The two seminorms with subscript $0,\alpha$ are used for Hölder sources in the Poincaré-style interior estimate of this page, while $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{2,\alpha }^{\ast }$ is the quantity estimated there.