- Stein-Strömberg theorem
In
mathematics , the Stein-Strömberg theorem or Stein-Strömberg inequality is a result inmeasure theory concerning theHardy-Littlewood maximal operator . The result is foundational in the study of the problem ofdifferentiation of integrals . The result is named after themathematician sElias M. Stein andJan-Olov Strömberg .tatement of the theorem
Let "λ""n" denote "n"-
dimension alLebesgue measure on "n"-dimensionalEuclidean space R"n" and let "M" denote the Hardy-Littlewood maximal operator: for a function "f" : R"n" → R, "Mf" : R"n" → R is defined by:Mf(x) = sup_{r > 0} frac1{lambda^{n} ig( B_{r} (x) ig)} int_{B_{r} (x)} | f(y) | , mathrm{d} lambda^{n} (y),
where "B""r"("x") denotes the
open ball ofradius "r" with centre "x". Then, for each "p" > 1, there is a constant "C""p" > 0 such that, for allnatural number s "n" and functions "f" ∈ "L""p"(R"n"; R),:Mf |_{L^{p leq C_{p} | f |_{L^{p.
In general, a maximal operator "M" is said to be of strong type ("p", "p") if
:Mf |_{L^{p leq C_{p, n} | f |_{L^{p
for all "f" ∈ "L""p"(R"n"; 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".
References
* cite journal
last = Stein
first = Elias M.
authorlink = Elias M. Stein
coauthors = Strömberg, Jan-Olov
title = Behavior of maximal functions in R"n" for large "n"
journal = Ark. Mat.
volume = 21
year = 1983
issue = 2
pages = 259--269
issn = 0004-2080
doi = 10.1007/BF02384314 MathSciNet|id=727348
* cite journal
last = Tišer
first = Jaroslav
title = Differentiation theorem for Gaussian measures on Hilbert space
journal = Trans. Amer. Math. Soc.
volume = 308
year = 1988
issue = 2
pages = 655–666
issn = 0002-9947
doi = 10.2307/2001096 MathSciNet|id=951621
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