{"history":[],"article":{"id":"313f1986-f521-4365-8d3b-0799f4b0d596","title":"Muckenhoupt weights","slug":"muckenhoupt-weights","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the class of **Muckenhoupt weights**\nAp\nconsists of those weights ω for which the [Hardy–Littlewood maximal operator](https://bliptext.com/articles/hardy-littlewood-maximal-operator) is bounded on\nLp(dω)\n. Specifically, we consider functions\n f \non\nRn\nand their associated [maximal functions](https://bliptext.com/articles/maximal-function)\nM( f )\ndefined as\nwhere\nBr(x)\nis the ball in\nRn\nwith radius r and center at x. Let\n1 ≤ p < ∞\n, we wish to characterise the functions\nω : Rn → [0, ∞)\nfor which we have a bound\nwhere C depends only on p and ω.\nThis was first done by [Benjamin Muckenhoupt](https://bliptext.com/articles/benjamin-muckenhoupt).\n\n# Definition\nFor a fixed\n1 < p < ∞\n, we say that a weight\nω : Rn → [0, ∞)\nbelongs to\nAp\nif ω is locally integrable and there is a constant C such that, for all balls B in\nRn\n, we have\nwhere\nis the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) of B, and q is a real number such that:\n1⁄p + 1⁄q = 1\n.\nWe say\nω : Rn → [0, ∞)\nbelongs to\nA1\nif there exists some C such that\nfor almost every\nx ∈ B\nand all balls B.\n\n# Equivalent characterizations\nThis following result is a fundamental result in the study of Muckenhoupt weights.\n1 < p < ∞\n. A weight ω is in\nAp\nif and only if any one of the following hold.\nLp(ω(x)dx)\n, that is\n f \non\nRn\n, and all balls B:\nEquivalently:\n1 < p < ∞\n, then\nif and only if both of the following hold:\nThis equivalence can be verified by using [Jensen's Inequality](https://bliptext.com/articles/jensen-s-inequality).\n\n# Reverse Hölder inequalities and \n\nA{{sub|∞}}\nThe main tool in the proof of the above equivalence is the following result.\nThe following statements are equivalent\nω ∈ Ap\nfor some\n1 ≤ p < ∞\n.\n0 < δ, γ < 1\nsuch that for all balls B and subsets\nE ⊂ B\n,\nimplies\nω(E) ≤ δ ω(B)\n.\n1 < q\nand c (both depending on ω) such that for all balls B we have:\nWe call the inequality in the third formulation a reverse Hölder inequality as the reverse inequality follows for any non-negative function directly from [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality).\nIf any of the three equivalent conditions above hold we say ω belongs to\nA∞\n.\n\n# Weights and BMO\nThe definition of an\nAp\nweight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly.\nThis property can be phrased equivalently in terms of how much the logarithm of the weight oscillates:\nw ∈ Ap, (p ≥ 1),\nthen\nlog(w) ∈ BMO\n(i.e.\nlog(w)\nhas [bounded mean oscillation](https://bliptext.com/articles/bounded-mean-oscillation)).\n f ∈ BMO\n, then for sufficiently small\nδ > 0\n, we have\neδf ∈ Ap\nfor some\np ≥ 1\n.\nThis equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the [John–Nirenberg inequality](https://bliptext.com/articles/john-nirenberg-inequality).\nNote that the smallness assumption on\nδ > 0\nin part (b) is necessary for the result to be true, as\n−log\n, but:\nis not in any\nAp\n.\n\n# Further properties\nHere we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:\nw ∈ Ap\n, then\nw dx\ndefines a [doubling measure](https://bliptext.com/articles/doubling-measure): for any ball B, if\n2B\nis the ball of twice the radius, then\nw(2B) ≤ Cw(B)\nwhere\nC > 1\nis a constant depending on w.\nw ∈ Ap\n, then there is\nδ > 1\nsuch that\nwδ ∈ Ap\n.\nw ∈ A∞\n, then there is\nδ > 0\nand weights such that.\n\n# Boundedness of singular integrals\nIt is not only the Hardy–Littlewood maximal operator that is bounded on these weighted\nLp\nspaces.\nIn fact, any [Calderón-Zygmund singular integral operator](https://bliptext.com/articles/singular-integrals) is also bounded on these spaces.\nLet us describe a simpler version of this here.\nSuppose we have an operator T which is bounded on\nL2(dx)\n, so we have\nSuppose also that we can realise T as convolution against a kernel K in the following sense: if\n f , g\nare smooth with disjoint support, then:\nFinally we assume a size and smoothness condition on the kernel K:\nThen, for each\n1 < p < ∞\nand\nω ∈ Ap\n, T is a bounded operator on\nLp(ω(x)dx)\n. That is, we have the estimate\nfor all\n f \nfor which the right-hand side is finite.\n\n## A converse result\nIf, in addition to the three conditions above, we assume the non-degeneracy condition on the kernel K: For a fixed unit vector\nu0\nwhenever with\n−∞ < t < ∞\n, then we have a converse.\nIf we know\nfor some fixed\n1 < p < ∞\nand some ω, then\nω ∈ Ap\n.\n\n# Weights and quasiconformal mappings\nFor\nK > 1\n, a K-[quasiconformal mapping](https://bliptext.com/articles/quasiconformal-mapping) is a homeomorphism\n f : Rn →Rn\nsuch that\nwhere\nDf (x)\nis the [derivative](https://bliptext.com/articles/derivative) of\n f \nat x and\nJ( f , x) = det(Df (x))\nis the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant).\nA theorem of Gehring states that for all K-quasiconformal functions\n f : Rn →Rn\n, we have\nJ( f , x) ∈ Ap\n, where p depends on K.\n\n# Harmonic measure\nIf you have a simply connected domain\nΩ ⊆ C\n, we say its boundary curve\nis K-chord-arc if for any two points\nz, w\nin\nΓ\nthere is a curve\nγ ⊆ Γ\nconnecting z and w whose length is no more than\nK\n. For a domain with such a boundary and for any\nz0\nin\nΩ\n, the [harmonic measure](https://bliptext.com/articles/harmonic-measure)\nw( ⋅ ) = w(z0, Ω, ⋅)\nis absolutely continuous with respect to one-dimensional [Hausdorff measure](https://bliptext.com/articles/hausdorff-measure) and its [Radon–Nikodym derivative](https://bliptext.com/articles/radon-nikodym-derivative) is in\nA∞\n. (Note that in this case, one needs to adapt the definition of weights to the case where the underlying measure is one-dimensional Hausdorff measure).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T10:28:28.757Z","updated_at":"2025-01-26T10:28:28.757Z","search_vector":"'muckenhoupt':1 'weight':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the class of **Muckenhoupt weights**\nAp\nconsists of those weights ω for which the [Hardy–Littlewood maximal operator](https://bliptext.com/articles/hardy-littlewood-maximal-operator) is bounded on\nLp(dω)\n. Specifically, we consider functions\n f \non\nRn\nand their associated [maximal functions](https://bliptext.com/articles/maximal-function)\nM( f )\ndefined as\nwhere\nBr(x)\nis the ball in\nRn\nwith radius r and center at x. Let\n1 ≤ p 0\n, we have\neδf ∈ Ap\nfor some\np ≥ 1\n.\nThis equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the [John–Nirenberg inequality](https://bliptext.com/articles/john-nirenberg-inequality).\nNote that the smallness assumption on\nδ > 0\nin part (b) is necessary for the result to be true, as\n−log\n, but:\nis not in any\nAp\n.\n\n# Further properties\nHere we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:\nw ∈ Ap\n, then\nw dx\ndefines a [doubling measure](https://bliptext.com/articles/doubling-measure): for any ball B, if\n2B\nis the ball of twice the radius, then\nw(2B) ≤ Cw(B)\nwhere\nC > 1\nis a constant depending on w.\nw ∈ Ap\n, then there is\nδ > 1\nsuch that\nwδ ∈ Ap\n.\nw ∈ A∞\n, then there is\nδ > 0\nand weights such that.\n\n# Boundedness of singular integrals\nIt is not only the Hardy–Littlewood maximal operator that is bounded on these weighted\nLp\nspaces.\nIn fact, any [Calderón-Zygmund singular integral operator](https://bliptext.com/articles/singular-integrals) is also bounded on these spaces.\nLet us describe a simpler version of this here.\nSuppose we have an operator T which is bounded on\nL2(dx)\n, so we have\nSuppose also that we can realise T as convolution against a kernel K in the following sense: if\n f , g\nare smooth with disjoint support, then:\nFinally we assume a size and smoothness condition on the kernel K:\nThen, for each\n1 1\n, a K-[quasiconformal mapping](https://bliptext.com/articles/quasiconformal-mapping) is a homeomorphism\n f : Rn →Rn\nsuch that\nwhere\nDf (x)\nis the [derivative](https://bliptext.com/articles/derivative) of\n f \nat x and\nJ( f , x) = det(Df (x))\nis the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant).\nA theorem of Gehring states that for all K-quasiconformal functions\n f : Rn →Rn\n, we have\nJ( f , x) ∈ Ap\n, where p depends on K.\n\n# Harmonic measure\nIf you have a simply connected domain\nΩ ⊆ C\n, we say its boundary curve\nis K-chord-arc if for any two points\nz, w\nin\nΓ\nthere is a curve\nγ ⊆ Γ\nconnecting z and w whose length is no more than\nK\n. For a domain with such a boundary and for any\nz0\nin\nΩ\n, the [harmonic measure](https://bliptext.com/articles/harmonic-measure)\nw( ⋅ ) = w(z0, Ω, ⋅)\nis absolutely continuous with respect to one-dimensional [Hausdorff measure](https://bliptext.com/articles/hausdorff-measure) and its [Radon–Nikodym derivative](https://bliptext.com/articles/radon-nikodym-derivative) is in\nA∞\n. (Note that in this case, one needs to adapt the definition of weights to the case where the underlying measure is one-dimensional Hausdorff measure).","revisionCount":0},"hasMore":false,"nextCursor":null}

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