Stein-Strömberg theorem

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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 mathematicians Elias M. Stein and Jan-Olov Strömberg.

[edit] Statement of the theorem

Let λn denote n-dimensional Lebesgue measure on n-dimensional Euclidean space Rn and let M denote the Hardy-Littlewood maximal operator: for a function f : Rn → R, Mf : Rn → R is defined by

Mf(x) = \sup_{r > 0} \frac1{\lambda^{n} \big( B_{r} (x) \big)} \int_{B_{r} (x)} | f(y) | \, \mathrm{d} \lambda^{n} (y),

where Br(x) denotes the open ball of radius r with centre x. Then, for each p > 1, there is a constant Cp > 0 such that, for all natural numbers n and functions f ∈ Lp(Rn; 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 ∈ Lp(Rn; 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.

[edit] References