Boundedness of Generalized Parametric Marcinkiewicz Integrals Associated to Surfaces

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 ( l o g L ) ν ( S n - 1 ) . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.

Generalized Parabolic Marcinkiewicz Integral Operators Related to Surfaces of Revolution

“…where φ(x) = φ(−x). Therefore, Equation (15) is satisfied by using Equations (19) and (21). This completes the proof of Lemma 5.…”

“…The idea of the proof of this theorem is taken from [21], which has its roots in [20]. Thanks to Minkowski's inequality, we have that…”

“…For background information and recent advances on the study of the operator M (λ),c Ω,ψ,g , readers may refer to [16,[19][20][21][22], and the references therein.…”

Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

“…By the conclusions from Theorems 1.1 and 1.2 and following the same extrapolation arguments used in refs. [9,20,29,33,34], we have the following: Theorem 1.3. Suppose that Ω satisfies (1.1) and (1.2), Φ is given as in Theorem 1.1 and ∈ ( )…”

“…. For the significance and recent advances on the study of such operators, readers may refer to [16,[19][20][21][22]23].…”