Abstract
In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators M Ω , ϕ , h , ρ ( r ) is considered. Under the condition that Ω is a function in L q ( S n − 1 ) with q ∈ ( 1 , 2 ] , appropriate estimates of the aforementioned operators from Triebel–Lizorkin spaces to L p spaces are obtained. By these estimates and an extrapolation argument, we establish the boundedness of such operators when the kernel function Ω belongs to the block space B q 0 , ν − 1 ( S n − 1 ) or in the space L ( log L ) ν ( S n − 1 ) . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.
Highlights
Throughout this work, we assume that Rn (n ≥ 2) is the n-dimensional Euclidean space and x 0 = x/| x | for x ∈ Rn \ {0}
We assume that sense that the exponent 1/2 in L(logL)1/2 (Sn−1) is the unit sphere in Rn, which is equipped with the normalized Lebesgue surface measure dσ
For ρ = τ + iυ (τ, υ ∈ R with τ > 0), let KΩ,h be the kernel on Rn defined by KΩ,h (u) = |u|ρ−n Ω(u0 )h(|u|), where h is a measurable function on R+ and Ω is a homogeneous function of degree zero on Rn with
Summary
As a matter of fact, he found that the last result is still true for all p ∈ (1, ∞) under the conditions that Ω ∈ L(log L)(Sn−1 ), 1 < r < ∞ and h ∈ Γmax{r0 ,2} (R+ ), where Γs (R+ ) is the collection of all measurable functions h : [0, ∞) → C satisfying khkΓs (R+ ) = sup. For s ≥ 1, we let Ls (R+ ) denote the set of all measurable functions h : [0, ∞) → C that satisfy the condition. We let N s (R+ ) denote the set of all measurable functions h : [0, ∞) → C that satisfy the condition. For ν > 0, let L(logL)ν (Sn−1 ) denote the space of all measurable functions Ω on Sn−1 that satisfy kΩk L(logL)ν (Sn−1 ) =.
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