A Class of Rough Generalized Marcinkiewicz Integrals on Product Domains

Abstract

In this article, suitable estimates for a class of rough generalized Marcinkiewicz integrals on product spaces are established. By these estimates, together with employing Yano’s extrapolation technique, we obtain the boundedness of the aforementioned integral operators under weak conditions on singular kernels. A number of known previous results on Marcinkiewicz as well as generalized Marcinkiewicz operators over a symmetric space are essentially improved or extended.

Estimates for Generalized Parabolic Marcinkiewicz Integrals with Rough Kernels on Product Domains

“…(v) For s ∈ {κ, η} with β s,1 = β s,2 = • • • = β s,s = 1, our results are the same as that obtained in [13]. Throughout the rest of the paper, the letter A stands for a positive constant which is independent of the essential variables and its value not necessary the same at each occurrence.…”

“…refers to a special class of block spaces introduced in [12]. Very recently, the result in [10] was improved in [13] in which the authors proved that if…”

“…,h and the results in [13] regarding the boundedness of the generalized parametric Marcinkiewicz operator M…”

“…Consequently, the proof of the lemma is finished in the case p ∈ ( µ µ+ −1 , µ) if we interpolate (12) with (13).…”

“…for all p ∈ ( , ∞) if µ ≥ . A proof of this Lemma can be constructed by following a similar argument as that employed in the proof of Lemma 3 and following similar argument as that used in the proofs of Theorems 4-5 in [13] (with minor modifications). We omit the details.…”