mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Stein–Strömberg theorem or Stein–Strömberg inequality is a result in

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

concerning the Hardy–Littlewood maximal operator. The result is foundational in the study of the problem of

differentiation of integrals In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. ...

. The result is named after the

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

s Elias M. Stein and Jan-Olov Strömberg.

Statement of the theorem

Let ''λ''^''n'' denote ''n''-

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

on ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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_ \frac1 \int_ ,  f(y) ,  \, \mathrm \lambda^ (y),

where ''B''_''r''(''x'') denotes the open ball of

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

''r'' with center ''x''. Then, for each ''p'' > 1, there is a constant ''C''_''p'' > 0 such that, for all

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s ''n'' and functions ''f'' ∈ ''L''^''p''(R^''n''; R), :

\,  Mf \, _ \leq C_ \,  f \, _.

In general, a maximal operator ''M'' is said to be of strong type (''p'', ''p'') if :

\,  Mf \, _ \leq C_ \,  f \, _

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

* * {{DEFAULTSORT:Stein-Stromberg theorem Inequalities Theorems in measure theory Operator theory