Abstract
Let μ be a nonnegative Radon measure on ℝd which only
satisfies the following growth condition that there exists a
positive constant C such that μ(B(x,r))≤Crn for all
x∈ℝd, r>0 and some fixed n∈(0,d]. In this paper, the
authors prove that for suitable indexes ρ and λ, the
parametrized gλ∗ function ℳλ∗,ρ is bounded
on Lp(μ) for p∈[2,∞) with the assumption that the kernel
of the operator ℳλ∗,ρ
satisfies some Hörmander-type condition, and is bounded from
L1(μ) into weak L1(μ) with the assumption that the kernel
satisfies certain slightly stronger Hörmander-type condition. As a
corollary, ℳλ∗,ρ with the kernel satisfying the above stronger
Hörmander-type condition is bounded on Lp(μ) for p∈(1,2).
Moreover, the authors prove that for suitable indexes ρ and
λ,ℳλ∗,ρ is bounded from L∞(μ) into RBLO(μ)
(the space of regular bounded lower oscillation functions) if the
kernel satisfies the Hörmander-type condition, and from the Hardy
space H1(μ) into L1(μ) if the kernel satisfies the above
stronger Hörmander-type condition. The corresponding properties
for the parametrized area integral ℳSρ are also established in this paper.