Hölder

Hölder is a German surname that is most prominently associated with mathematical analysis, particularly through Otto Hölder (1851–1931). Hölder was a German mathematician and logician whose work contributed to analysis and function theory, and several foundational concepts bear his name.

One of Hölder’s most enduring legacies is Hölder’s inequality, a cornerstone of integration theory. In its classical form, it states that for conjugate exponents p and q (1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1) and functions f ∈ L^p, g ∈ L^q, the integral of their product is bounded by the product of their norms: |∫ f g| ≤ ∥f∥_p ∥g∥_q. This inequality extends to sequences and more general settings and underpins many results in analysis.

Hölder also introduced the concept of Hölder continuity, a quantitative form of uniform continuity. A function f defined on a metric space is Hölder continuous with exponent α ∈ (0,1] if there exists a constant C such that |f(x) − f(y)| ≤ C|x − y|^α for all x, y. When α = 1 the condition is called Lipschitz continuity.

Function spaces named after Hölder include the Hölder spaces C^{k,α}, consisting of k-times continuously differentiable functions whose k-th derivatives are Hölder continuous with exponent α (0 < α ≤ 1). These spaces are Banach spaces equipped with the Hölder norm and are central in analysis and partial differential equations.

In summary, Hölder’s influence spans inequalities, continuity concepts, and specialized function spaces, all anchored to Otto Hölder’s work in mathematics.