On Certain Estimates for Parabolic Marcinkiewicz Integrals Related to Surfaces of Revolution on Product Spaces and Extrapolation

Abstract

In this paper, appropriate Lp bounds for particular classes of parabolic Marcinkiewicz integrals along surfaces of revolution on product spaces are obtained. These bounds allow us to use Yano’s extrapolation argument to obtain the Lp boundedness of the aforesaid integral operators under weak conditions on the kernels. These conditions on the kernels are the best possible among their respective classes. In this work, several previously known results on Marcinkiewicz integrals are fundamentally improved and extended.

On the Functions of Marcinkiewicz Integrals along Surfaces of Revolution on Product Domains via Extrapolation

“…Consequently, by interpolating between (24) and (25), we obtain (23), which, in turn, finishes the proof of Theorem 1.…”

“…The L p boundedness of the operator M φ,ψ Ω,h under different conditions on the functions φ, ψ, Ω, and h was discussed by many authors (one can consult [19,[23][24][25][26]).…”

On Certain Rough Marcinkiewicz Integral Operators with Grafakos-Stefanov Kernels on Product Spaces

“…The study of the boundedness of M ℧ started by Ding in [11] and then continued by many authors. For a sample of past studies and for more information about the applications and development of the operator M ℧ , we refer the readers to consult [14][15][16][17][18][19][20][21] and the references therein.…”