In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if and are nonnegative measurable real functions vanishing at infinity that are defined on -dimensional Euclidean space , then where and are the symmetric decreasing rearrangements of and , respectively. The decreasing rearrangement of is defined via the property that for all the two super-level sets Web参考: Real Analysis, SteinSingular Integrals and Differentiability Properties of Functions, Elias M. Stein关于另一种形式的极大函数(规定球心), 见Folland, Real Analysis 3.4, 视频播 …
Hardy–Littlewood–Sobolev Inequalities with the Fractional …
WebDec 7, 2015 · In words, it means that the maximal function is not much larger than $ f $. (Stein and Shakarchi Real Analysis p.101). The lefthand side is the measure of the set of values at which the maximal function is larger than a particular real number. fox news started by african american
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This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from the L (R ) to itself for p > 1. That is, if f ∈ L (R ) then the maximal function Mf is weak L -bounded and Mf ∈ L (R ). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x … See more In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis. See more While there are several proofs of this theorem, a common one is given below: For p = ∞, the inequality is trivial (since the average of a … See more • Rising sun lemma See more The operator takes a locally integrable function f : R → C and returns another function Mf. For any point x ∈ R , the function Mf returns the maximum of a set of reals, namely the set of average values of f for all the balls B(x, r) of any radius r at x. Formally, See more It is still unknown what the smallest constants Cp,d and Cd are in the above inequalities. However, a result of Elias Stein about spherical maximal functions can be used to … See more WebHARMONIC ANALYSIS PIOTR HAJLASZ 1. Maximal function For a locally integrable function f2L1 loc (R n) the Hardy-Littlewood max- imal function is de ned by Mf(x) = sup r>0 Z B(x;r) jf(y)jdy; x2Rn: The operator Mis not linear but it is subadditive. http://lxbwk.njournal.sdu.edu.cn/CN/10.6040/j.issn.1671-9352.0.2024.353 fox news starter pack