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In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. It takes a locally integrable function ''f'' : R''d'' → C and returns another function ''Mf'' that, at each point ''x'' ∈ R''d'', gives the maximum average value that ''f'' can have on balls centered at that point. More precisely, : where ''B''(''x'', ''r'') is the ball of radius ''r'' centred at ''x'', and |''E''| denotes the ''d''-dimensional Lebesgue measure of ''E'' ⊂ R''d''. The averages are jointly continuous in ''x'' and ''r'', therefore the maximal function ''Mf'', being the supremum over ''r'' > 0, is measurable. It is not obvious that ''Mf'' is finite almost everywhere. This is a corollary of the Hardy–Littlewood maximal inequality ==Hardy–Littlewood maximal inequality== This theorem of G. H. Hardy and J. E. Littlewood states that ''M'' is bounded as a sublinear operator from the ''Lp''(R''d'') to itself for ''p'' > 1. That is, if ''f'' ∈ ''Lp''(R''d'') then the maximal function ''Mf'' is weak ''L''1-bounded and ''Mf'' ∈ ''Lp''(R''d''). Before stating the theorem more precisely, for simplicity, let denote the set . Now we have: Theorem (Weak Type Estimate). For ''d'' ≥ 1 and ''f'' ∈ ''L''1(R''d''), there is a constant ''Cd'' > 0 such that for all λ > 0, we have: With the Hardy–Littlewood maximal inequality in hand, the following ''strong-type'' estimate is an immediate consequence of the Marcinkiewicz interpolation theorem: Theorem (Strong Type Estimate). For ''d'' ≥ 1, 1 < ''p'' ≤ ∞, and ''f'' ∈ ''Lp''(R''d''), In the strong type estimate the best bounds for ''Cp,d'' are unknown.〔 However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following: Theorem (Dimension Independence). For 1 < ''p'' ≤ ∞ one can pick ''Cp,d'' = ''Cp'' independent of ''d''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hardy–Littlewood maximal function」の詳細全文を読む スポンサード リンク
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