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HölderRegularity
Hölder regularity is a concept in mathematics that describes the smoothness of functions, curves, and surfaces. It generalizes the idea of a function being differentiable or having continuous derivatives. The concept is named after the German mathematician Otto Hölder. Here's a technical definition along with the necessary formulas:
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Hölder Condition: A function
$f: \mathbb{R} \rightarrow \mathbb{R}$ is said to satisfy the Hölder condition of order$\alpha$ (where$0 < \alpha \leq 1$ ) on an interval$I$ if there exists a constant$C \geq 0$ such that for all$x, y$ in$I$ , the following inequality holds:
Here,
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Hölder Continuity: A function is Hölder continuous of order
$\alpha$ on$I$ if it satisfies the Hölder condition of that order. This is a stronger form of continuity than mere uniform continuity. For$\alpha = 1$ , Hölder continuity is the same as Lipschitz continuity. -
Hölder Space: The space of all functions that satisfy the Hölder condition of order
$\alpha$ over a domain$D$ is called a Hölder space, denoted as$C^{k,\alpha}(D)$ , where$k$ is an integer denoting the number of continuous derivatives the function has. In this notation,$C^0, \alpha(D)$ refers to functions that are just Hölder continuous, while$C^{k,\alpha}(D)$ refers to functions whose$k$ -th derivatives are Hölder continuous of order$\alpha$ . -
Importance in Analysis: Hölder regularity is crucial in many areas of analysis, particularly in the study of partial differential equations and potential theory. Functions with higher Hölder regularity (larger
$\alpha$ ) are smoother.
In essence, Hölder regularity provides a scale of "smoothness" for functions, with different degrees of regularity depending on the value of