A note on Marcinkiewicz integrals supported by submanifolds

Abstract

In the present paper, we establish the boundedness and continuity of the parametric Marcinkiewicz integrals with rough kernels associated to polynomial mapping as well as the corresponding compound submanifolds, which is defined by on the Triebel–Lizorkin spaces and Besov spaces when and for some . Our main results represent significant improvements and natural extensions of what was known previously.

Integral operators of Marcinkiewicz type on Triebel–Lizorkin spaces

“…The following results obtained by Liu in [21] are two extension of the famous result on the L p ( q (L r )) boundedness of the Hardy-Littlewood maximal operator.…”

“…The following results obtained by Liu in are two extension of the famous result on the boundedness of the Hardy–Littlewood maximal operator. Lemma (i)Let be the Hardy–Littlewood maximal operator on .…”

“…Let be a lacunary sequence with satisfying for any . Let be a collection of functions with the following properties: We now introduce a important lemma which followed form . Lemma Let , and be two nonsingular linear transformations.…”

“…During the last several years, a considerable amount of attention has been given to investigate the boundedness for various kinds of integral operators on the above function spaces. For examples, see , , , , for singular integrals and , , , , for Marcinkiewicz integrals. In this paper we continue to focus on this topic.…”

“…As applications, the bounds for main operators of this paper on Besov spaces are obtained. We would like to remark that some ideas in our proofs are taken from , , and the main novelty in this paper is to give a standard approach on the bounds for Marcinkiewicz integral operators on Triebel–Lizorkin spaces and Besov spaces. It should be pointed out that the methods and techniques used in main proofs are more delicate and complex than that of .…”