Hölder space ($C^{0,\alpha }$)

A bounded function belongs to $C^{0,\alpha }({\mathbb{R}}^n)$ when

Write $\alpha =r+\beta$ with an integer $r\ge 0$ and $0<\beta \le 1$, using $r=\alpha -1$ at integer $\alpha$. For $f\in C^{r,\beta }$, subtract its degree-$r$ Taylor polynomial at $x_0$. For $r\ge 1$, the integral Taylor remainder with $f^{(r)}(t)-f^{(r)}(x_0)$ is bounded using the Hölder seminorm; for $r=0$ this is exactly Hölder continuity. The resulting bound is the displayed estimate. At integer order, the convention is $C^{r,1}$; a $C^{r+1}$ function on a compact closed interval also obeys this estimate.

Localized wavelets of support diameter $O(2^{-j})$ and $L^1$ norm $O(2^{-j/2})$, with enough vanishing moments, satisfySubtract a Taylor polynomial annihilated by the wavelet, apply the Hölder-Taylor remainder bound on its support of a function, and multiply by its $L^1$ norm. The same argument applies to localized boundary wavelets with the same cancellation; finitely many coarse scaling functions are handled separately.

The full norm is $|u{|}_{k,\alpha ;U}=|u{|}_{k;U}+[D^{k}u{]}_{\alpha ;U}$, combining the derivative supremum norm with the top-derivative Hölder seminorm. At order zero it is the sum of the supremum norm and the Hölder seminorm.

The Hölder seminorm of exponent $0<\alpha \le 1$ isIt measures Hölder continuity without including the supremum norm of the function itself.

If $f_j\to f$ uniformly and the Hölder seminorms are uniformly bounded, each fixed difference quotient converges. Taking the supremum over point pairs gives $[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$. The supremum norms converge, so the full Hölder norm has the same lower semicontinuity of a Hölder seminorm.

A Hölder class bounds derivatives through order $\lceil \beta \rceil -1$ and requires the highest derivative to be Hölder continuous with exponent $\beta -\lceil \beta \rceil +1$ and constant $L$.

A function is Hölder continuous with exponent $0<\alpha \le 1$ when $|f(x)-f(y)|\le C|x-y{|}^{\alpha }$ for some constant $C$.

If a real function on an interval obeys $|f(x)-f(y)|\le C|x-y{|}^{\alpha }$ for finite $C$ and $\alpha >1$, its difference quotients tend to zero everywhere. The mean value theorem makes it constant. Equivalently, divide a segment into $N$ pieces to obtain $|f(x)-f(y)|\le C|x-y{|}^{\alpha }{N}^{1-\alpha }\to 0$.

For $f\in L^2(0,1)$ and $0<\alpha <1/2$, the integral $g(x)={\int }_0^{1}f(y)|x-y{|}^{-\alpha }dy$ is well defined for every real $x$. Scaling the squared kernel difference over the real line and using the Cauchy-Schwarz inequality give $|g(x+h)-g(x)|\le C_{\alpha }\Vert f{\Vert }_2|h{|}^{1/2-\alpha }$.

If a holomorphic function on a half-rectangle satisfies $|f^{\prime }(x+iy)|\le C{y}^{-1+\alpha }$ with $0<\alpha \le 1$, then it extends to an $\alpha$-Hölder continuous function on the closed half-rectangle. Integrate vertically to height $r=|z-w|$, horizontally at that height, and vertically back; each contribution is $O(r^{\alpha })$.

On a ball of radius $R$, lower derivative norms of $u\in C^{l,\alpha }$ are bounded by an arbitrarily small multiple of the top Hölder seminorm plus a multiple of $\Vert u{\Vert }_{C^0}$. One endpoint form is

On a bounded regular domain, a bounded subset of $C^{k,\alpha }$ is relatively compact in $C^{l,\beta }$ whenever $k+\alpha >l+\beta$. This is an Arzela-Ascoli theorem argument applied successively to the derivatives.

If $s>n/2+\alpha$ with $0<\alpha \le 1$, then Fourier inversion, $|e^{it}-1|\le C_{\alpha }|t{|}^{\alpha }$, and Cauchy-Schwarz giveThe weighted frequency integral converges precisely because $s-\alpha >n/2$.