{"history":[],"article":{"id":"40252b7d-c4a8-4443-8979-d86197523b62","title":"Hölder condition","slug":"h-lder-condition","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a real or complex-valued function\nf\non d-dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space) satisfies a **Hölder condition**, or is **Hölder continuous**, when there are real constants\nC ≥ 0\n,\nα > 0\n, such that\nfor all x and y in the domain of\nf\n. More generally, the condition can be formulated for functions between any two [metric spaces](https://bliptext.com/articles/metric-space).\nThe number \\alpha is called the exponent of the Hölder condition.\nA function on an interval satisfying the condition with\nα > 1\nis [constant](https://bliptext.com/articles/constant-function) (see proof below).\nIf\nα = 1\n, then the function satisfies a [Lipschitz condition](https://bliptext.com/articles/lipschitz-condition).\nFor any\nα > 0\n, the condition implies the function is [uniformly continuous](https://bliptext.com/articles/uniformly-continuous).\nThe condition is named after [Otto Hölder](https://bliptext.com/articles/otto-h-lder).\nWe have the following chain of inclusions for functions defined on a closed and bounded interval of the real line with\na < b\nwhere\n0 < α ≤ 1\n.\n\n# Hölder spaces\nHölder spaces consisting of functions satisfying a Hölder condition are basic in areas of [functional analysis](https://bliptext.com/articles/functional-analysis) relevant to solving [partial differential equations](https://bliptext.com/articles/partial-differential-equations), and in [dynamical systems](https://bliptext.com/articles/dynamical-system).\nThe Hölder space\nCk,α(Ω)\n, where\nΩ\nis an open subset of some Euclidean space and k ≥ 0 an integer, consists of those functions on Ω having continuous [derivatives](https://bliptext.com/articles/derivative) up through order k and such that the k-th partial [derivatives](https://bliptext.com/articles/derivative) are Hölder continuous with exponent α, where\n0 < α ≤ 1\n. This is a [locally convex topological vector space](https://bliptext.com/articles/locally-convex-topological-vector-space).\nIf the Hölder coefficient\nis finite, then the function\nf\nis said to be ''(uniformly) Hölder continuous with exponent α in\nΩ\n.'' In this case, the Hölder coefficient serves as a [seminorm](https://bliptext.com/articles/seminorm).\nIf the Hölder coefficient is merely bounded on [compact](https://bliptext.com/articles/compact-set) subsets of\nΩ\n, then the function\nf\nis said to be ''locally Hölder continuous with exponent α in\nΩ\n.''\nIf the function\nf\nand its derivatives up to order k are bounded on the closure of Ω, then the Hölder space can be assigned the norm\nwhere β ranges over [multi-indices](https://bliptext.com/articles/multi-index-notation) and\nThese seminorms and norms are often denoted simply and or also and in order to stress the dependence on the domain of\nf\n. If\nΩ\nis open and bounded, then is a [Banach space](https://bliptext.com/articles/banach-space) with respect to the norm.\n\n# Compact embedding of Hölder spaces\nLet Ω be a bounded subset of some Euclidean space (or more generally, any totally bounded metric space) and let 0 < α < β ≤ 1 two Hölder exponents.\nThen, there is an obvious inclusion map of the corresponding Hölder spaces:\nwhich is continuous since, by definition of the Hölder norms, we have:\nMoreover, this inclusion is compact, meaning that bounded sets in the\n‖ · ‖0,β\nnorm are relatively compact in the\n‖ · ‖0,α\nnorm.\nThis is a direct consequence of the [Ascoli-Arzelà theorem](https://bliptext.com/articles/ascoli-arzel-theorem).\nIndeed, let\n(un)\nbe a bounded sequence in\nC0,β(Ω)\n. Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that\nun → u\nuniformly, and we can also assume\nu = 0\n. Then\nbecause\n\n# Examples\n0 < α ≤ β ≤ 1\nthen all Hölder continuous functions on a bounded set Ω are also Hölder continuous.\nThis also includes\nβ = 1\nand therefore all [Lipschitz continuous](https://bliptext.com/articles/lipschitz-continuity) functions on a bounded set are also\nC0,α\nHölder continuous.\nf(x) = xβ\n(with\nβ ≤ 1\n) defined on serves as a prototypical example of a function that is\nC0,α\nHölder continuous for\n0 < α ≤ β\n, but not for\nα > β\n. Further, if we defined\nf\nanalogously on [0,\\infty), it would be\nC0,α\nHölder continuous only for\nα = β\n.\nConsider the case x < y where.\nThen, so the difference quotient converges to zero as |x-y| \\to 0.\nHence f' exists and is zero everywhere.\nMean-value theorem now implies f is constant.\n[Q.E.D.](https://bliptext.com/articles/q-e-d)\n**Alternate idea:** Fix x < y and partition [x,y] into where.\nThen as n\\to \\infty, due to \\alpha > 1.\nThus f(x) = f(y).\n[Q.E.D.](https://bliptext.com/articles/q-e-d)\nf(0) = 0\nand by\nf(x) = 1/log(x)\notherwise is continuous, and therefore uniformly continuous by the [Heine-Cantor theorem](https://bliptext.com/articles/heine-cantor-theorem).\nIt does not satisfy a Hölder condition of any order, however.\nC := 2\n.\n[0, 1]2\ncan be constructed to be 1/2–Hölder continuous.\nIt can be proved that when the image of a \\alpha-Hölder continuous function from the unit interval to the square cannot fill the square.\nu\nsatisfies then\nu\nis Hölder continuous with exponent α.\nu(x)\nsatisfies for a fixed λ with\n0 < λ < 1\nand all sufficiently small values of r, then\nu\nis Hölder continuous.\nC\n, depending only on p and n, such that: where Thus if\nu ∈ W1, p(Rn)\n, then\nu\nis in fact Hölder continuous of exponent γ, after possibly being redefined on a set of measure 0.\n\n# Properties\nH\n, connected by α–Hölder continuous arcs with\nα > 1/2\n, is a linear subspace.\nThere are closed additive subgroups of\nH\n, not linear subspaces, connected by 1/2–Hölder continuous arcs.\nAn example is the additive subgroup\nL2(R, Z)\nof the Hilbert space\nL2(R, R)\n.\nf\non a metric space X admits a [Lipschitz approximation](https://bliptext.com/articles/modulus-of-continuity) by means of a sequence of functions\n(fk)\nsuch that\nfk\nis k-Lipschitz and Conversely, any such sequence\n(fk)\nof Lipschitz functions converges to an α–Hölder continuous uniform limit\nf\n.\nf\non a subset X of a normed space E admits a [uniformly continuous extension](https://bliptext.com/articles/modulus-of-continuity) to the whole space, which is Hölder continuous with the same constant\nC\nand the same exponent α.\nThe largest such extension is:\nL\n, where.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:08:40.176Z","updated_at":"2025-01-26T07:08:40.176Z","search_vector":"'condit':2 'hölder':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a real or complex-valued function\nf\non d-dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space) satisfies a **Hölder condition**, or is **Hölder continuous**, when there are real constants\nC ≥ 0\n,\nα > 0\n, such that\nfor all x and y in the domain of\nf\n. More generally, the condition can be formulated for functions between any two [metric spaces](https://bliptext.com/articles/metric-space).\nThe number \\alpha is called the exponent of the Hölder condition.\nA function on an interval satisfying the condition with\nα > 1\nis [constant](https://bliptext.com/articles/constant-function) (see proof below).\nIf\nα = 1\n, then the function satisfies a [Lipschitz condition](https://bliptext.com/articles/lipschitz-condition).\nFor any\nα > 0\n, the condition implies the function is [uniformly continuous](https://bliptext.com/articles/uniformly-continuous).\nThe condition is named after [Otto Hölder](https://bliptext.com/articles/otto-h-lder).\nWe have the following chain of inclusions for functions defined on a closed and bounded interval of the real line with\na β\n. Further, if we defined\nf\nanalogously on [0,\\infty), it would be\nC0,α\nHölder continuous only for\nα = β\n.\nConsider the case x 1.\nThus f(x) = f(y).\n[Q.E.D.](https://bliptext.com/articles/q-e-d)\nf(0) = 0\nand by\nf(x) = 1/log(x)\notherwise is continuous, and therefore uniformly continuous by the [Heine-Cantor theorem](https://bliptext.com/articles/heine-cantor-theorem).\nIt does not satisfy a Hölder condition of any order, however.\nC := 2\n.\n[0, 1]2\ncan be constructed to be 1/2–Hölder continuous.\nIt can be proved that when the image of a \\alpha-Hölder continuous function from the unit interval to the square cannot fill the square.\nu\nsatisfies then\nu\nis Hölder continuous with exponent α.\nu(x)\nsatisfies for a fixed λ with\n0 1/2\n, is a linear subspace.\nThere are closed additive subgroups of\nH\n, not linear subspaces, connected by 1/2–Hölder continuous arcs.\nAn example is the additive subgroup\nL2(R, Z)\nof the Hilbert space\nL2(R, R)\n.\nf\non a metric space X admits a [Lipschitz approximation](https://bliptext.com/articles/modulus-of-continuity) by means of a sequence of functions\n(fk)\nsuch that\nfk\nis k-Lipschitz and Conversely, any such sequence\n(fk)\nof Lipschitz functions converges to an α–Hölder continuous uniform limit\nf\n.\nf\non a subset X of a normed space E admits a [uniformly continuous extension](https://bliptext.com/articles/modulus-of-continuity) to the whole space, which is Hölder continuous with the same constant\nC\nand the same exponent α.\nThe largest such extension is:\nL\n, where.","revisionCount":0},"hasMore":false,"nextCursor":null}

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