In mathematics, the SteinâStrömberg theorem or SteinâStrömberg inequality is a result in measure theory concerning the HardyâLittlewood 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.
Let λn denote n- dimensional Lebesgue measure on n-dimensional Euclidean space Rn and let M denote the HardyâLittlewood maximal operator: for a function f : Rn â R, Mf : Rn â R is defined by
where Br(x) denotes the open ball of radius r with center x. Then, for each p > 1, there is a constant Cp > 0 such that, for all natural numbers n and functions f â Lp(Rn; R),
In general, a maximal operator M is said to be of strong type (p, p) if
for all f â Lp(Rn; R). Thus, the SteinâStrömberg theorem is the statement that the HardyâLittlewood maximal operator is of strong type (p, p) uniformly with respect to the dimension n.
In mathematics, the SteinâStrömberg theorem or SteinâStrömberg inequality is a result in measure theory concerning the HardyâLittlewood 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.
Let λn denote n- dimensional Lebesgue measure on n-dimensional Euclidean space Rn and let M denote the HardyâLittlewood maximal operator: for a function f : Rn â R, Mf : Rn â R is defined by
where Br(x) denotes the open ball of radius r with center x. Then, for each p > 1, there is a constant Cp > 0 such that, for all natural numbers n and functions f â Lp(Rn; R),
In general, a maximal operator M is said to be of strong type (p, p) if
for all f â Lp(Rn; R). Thus, the SteinâStrömberg theorem is the statement that the HardyâLittlewood maximal operator is of strong type (p, p) uniformly with respect to the dimension n.