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
In general, a maximal operator M is said to be of strong type (p, p) if
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.
- Stein, Elias M.; Strömberg, Jan-Olov (1983). "Behavior of maximal functions in Rn for large n". Ark. Mat. 21 (2): 259–269. doi:10.1007/BF02384314. MR 727348
- Tišer, Jaroslav (1988). "Differentiation theorem for Gaussian measures on Hilbert space". Trans. Amer. Math. Soc. 308 (2): 655–666. doi:10.2307/2001096. MR 951621