Rough Singular Integrals with Kernels Supported by Submanifolds of Finite Type

Abstract

We establish the L p boundedness of singular integrals on product domains with rough kernels in L(log L) 2 and are supported by subvarieties.

Rough singular integrals on product spaces

“…Subsequently, the L p (1 < p < ∞) boundedness of T Φ,Ω was established under conditions much weaker than Ω ∈ L q (S n−1 ) [1,6]. In particular, Al-Qassem et al [1] established the L p boundedness of T Φ,Ω under the condition that the function Ω belongs to the block space B introduced by Jiang and Lu in (see [14]). In fact, they proved the following theorem.…”

“…For more details, we advise the readers to consult [1,3,4,6,7,8,10], among others. A particular result that we will need to prove our results is the following result in [4] which is an extension of a result of Duoandikoetxea in [7].…”

Lp(ℝn) boundedness for higher commutators of singular integrals with rough kernels belonging to certain block spaces

“…where w = 2 [log(|I| −1 )] , |I| < e −2 and [·] denotes the greatest integer function. Lemma 2.1 (See [3].) Letb be a function defined as in (2.2).…”