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.
Statement 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
formula_1
where "B""r"("x") denotes the open ball of radius "r" with center "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),
formula_2
In general, a maximal operator "M" is said to be of strong type ("p", "p") if
formula_3
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".