On Maximal Functions over Circular Sectors with Rotation Invariant Measures

Given a rotation invariant measure in Rn, we define the maximal operator over circular sectors. We prove that it is of strong type (p, p) for p > 1 and we give necessary and sufficient conditions on the measure for the weak type (1, 1) inequality. Actually we work in a more general setting containing the above and other situations. Let X be a topological space and μ be a Borel measure on X. By Lμ(X) we will denote the space of all real valued measurable functions on X such that ‖f‖p,μ = ( ∫ X |f(x)| dμ(x)) < ∞ if 1 ≤ p < ∞, and ‖f‖∞,μ = inf{a : μ({x ∈ X : |f(x)| > a}) = 0} < ∞, if p = +∞. By Lμ,loc(X) we mean the set of all real valued measurable functions on X which are integrable on compact sets. An operator T defined on Lμ(X) is said to be of strong type (p, p), 1 ≤ p ≤ +∞, if there exists a positive constant Ap such that for every f ∈ Lμ(X), ‖Tf‖p,μ ≤ Ap‖f‖p,μ. The operator T is of weak type (p, p), 1 ≤ p < +∞, if there exists a constant Bp such that for every f ∈ Lμ(X) and λ > 0, μ({x ∈ X : |Tf(x)| > λ}) ≤ Bp‖f‖p,μλ. Let y ∈ Sn−1, the unit sphere in R, α ∈ (0, π) and 0 ≤ r ≤ R ≤ +∞. The set S = S(y, α, r,R) = {x ∈ R : arg(x, y) < α and r < |x| < R}, where arg(x, y) is the angle between x and y, is called a circular sector in R. If G is a Borel subset of Sn−1, G(r,R) will denote the set {ρx : x ∈ G, r < ρ < R} and σr will mean the surface measure on the sphere of radious r. Let μ be a non-negative, rotation invariant, Radon measure. If f ∈ Lμ,loc(R) and x 6= 0 we define (1) M μf(x) = sup x∈S 1 μ(S) ∫ S |f(y)|dμ(y) where the sup is taken over all bounded sectors S in R with non-zero measure containing x. Set M μf(0) = 0. It is not hard to prove that if G is a Borel set in Sn−1, 0 ≤ r ≤ R ≤ +∞ and f ∈ Lμ(R) then ∫ G(r,R) f(y)dμ(y) = 1 σ1(Sn−1) ∫