Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds

Abstract

We establish certain delicate L p {L}^{p} bounds for a class of generalized Marcinkiewicz integral operators along submanifolds with rough kernels. These bounds allow us to use Yano’s extrapolation method to prove the L p {L}^{p} boundedness of the aforementioned integral operators under very weak assumptions on the kernels. Our results in this article improve and generalize many previously known results.

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

“…Suppose that Φ(t, ) = φ 1 (t) + φ 2 ( ), where φ j (•) (j = 1, 2) is either a generalized polynomial or is in C 2 (R + ), increasing and convex function with φ j (0) = 0. Then, there is a constant C p such that the estimate (7) holds for all |1/p − 1/2| < min{1/2, 1/γ }.…”

“…Thereafter, the study of the operator M φ Ω,h has been studied by many researchers. For a sample of known results relevant to our study, the readers are referred to consult [2][3][4][5][6][7][8][9].…”

“…Then, there is a constant C p such that the estimate (7) holds for all |1/p − 1/2| < min{1/2, 1/γ }.…”