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| In [[mathematics]], the '''Stein–Strömberg theorem''' or '''Stein–Strömberg inequality''' is a result in [[measure theory]] concerning the [[Hardy–Littlewood maximal operator]]. The result is foundational in the study of the problem of [[differentiation of integrals]]. The result is named after the [[mathematician]]s [[Elias M. Stein]] and [[Jan-Olov Strömberg]].
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| ==Statement of the theorem==
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| Let ''λ''<sup>''n''</sup> denote ''n''-[[dimension]]al [[Lebesgue measure]] on ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> and let ''M'' denote the Hardy–Littlewood maximal operator: for a function ''f'' : '''R'''<sup>''n''</sup> → '''R''', ''Mf'' : '''R'''<sup>''n''</sup> → '''R''' is defined by
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| :<math>Mf(x) = \sup_{r > 0} \frac1{\lambda^{n} \big( B_{r} (x) \big)} \int_{B_{r} (x)} | f(y) | \, \mathrm{d} \lambda^{n} (y),</math> | |
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| where ''B''<sub>''r''</sub>(''x'') denotes the [[open ball]] of [[radius]] ''r'' with center ''x''. Then, for each ''p'' > 1, there is a constant ''C''<sub>''p''</sub> > 0 such that, for all [[natural number]]s ''n'' and functions ''f'' ∈ ''L''<sup>''p''</sup>('''R'''<sup>''n''</sup>; '''R'''),
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| :<math>\| Mf \|_{L^{p}} \leq C_{p} \| f \|_{L^{p}}.</math>
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| In general, a maximal operator ''M'' is said to be of '''strong type''' (''p'', ''p'') if
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| :<math>\| Mf \|_{L^{p}} \leq C_{p, n} \| f \|_{L^{p}}</math> | |
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| for all ''f'' ∈ ''L''<sup>''p''</sup>('''R'''<sup>''n''</sup>; '''R'''). Thus, the Stein–Strömberg theorem is the statement that the Hardy–Littlewood maximal operator is of strong type (''p'', ''p'') uniformly with respect to the dimension ''n''.
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| ==References==
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| * {{cite journal
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| | last = Stein
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| | first = Elias M.
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| | authorlink = Elias M. Stein
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| | coauthors = Strömberg, Jan-Olov
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| | title = Behavior of maximal functions in '''R'''<sup>''n''</sup> for large ''n''
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| | journal = Ark. Mat.
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| | volume = 21
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| | year = 1983
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| | issue = 2
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| | pages = 259–269
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| | doi = 10.1007/BF02384314
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| }} {{MathSciNet|id=727348}}
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| * {{cite journal
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| | last = Tišer
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| | first = Jaroslav
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| | title = Differentiation theorem for Gaussian measures on Hilbert space
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| | journal = Trans. Amer. Math. Soc.
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| | volume = 308
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| | year = 1988
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| | issue = 2
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| | pages = 655–666
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| | doi = 10.2307/2001096
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| }} {{MathSciNet|id=951621}}
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| {{DEFAULTSORT:Stein-Stromberg theorem}}
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| [[Category:Inequalities]]
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| [[Category:Theorems in measure theory]]
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| [[Category:Operator theory]]
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Мy name is Calvin Aгriaga. I life in Bayonne (Francе).
my page :: LiveJasmin Credits Hack