Stein-Strömberg theorem

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 mathematicians Elias M. Stein and Jan-Olov Strömberg.

tatement of the theorem

Let "λ"^"n" denote "n"-dimensional Lebesgue measure on "n"-dimensional Euclidean 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

:

where "B"_"r"("x") denotes the open ball of radius "r" with centre "x". Then, for each "p" > 1, there is a constant "C"_"p" > 0 such that, for all natural numbers "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