Weighted Norm Inequalities for Rough Singular Integral Operators

By Kangwei Li, C. P'erez, Israel P. Rivera-R'ios, Luz Roncal. Journal of Geometric Analysis, 2017

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In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $$T_\Omega $$TΩ with $$\Omega \in L^\infty (\mathbb {S}^{n-1})$$Ω∈L∞(Sn-1) and the Bochner–Riesz multiplier at the critical index $$B_{(n-1)/2}$$B(n-1)/2. More precisely, we prove qualitative and quantitative versions of Coifman–Fefferman type inequalities and their vector-valued extensions, weighted $$A_p-A_\infty $$Ap-A∞ strong and weak type inequalities for $$1<p<\infty $$1<p<∞, and $$A_1-A_\infty $$A1-A∞ type weak (1, 1) estimates. Moreover, Fefferman–Stein type inequalities are obtained, proving in this way a conjecture raised by the second-named author in the 1990s. As a corollary, we obtain the weighted $$A_1-A_\infty $$A1-A∞ type estimates. Finally, we study rough homogenous singular integrals with a kernel involving a function $$\Omega \in L^q(\mathbb {S}^{n-1})$$Ω∈Lq(Sn-1), $$1<q<\infty $$1<q<∞, and provide Fefferman–Stein inequalities too. The arguments used for our proofs combine several tools: a recent sparse domination result by Conde–Alonso et al. (Anal PDE 10(5):1255–1284, 2017), results by the first author (Collect Math 68:129–144, 2017), suitable adaptations of Rubio de Francia algorithm, the extrapolation theorems for $$A_{\infty }$$A∞ weights (Cruz-Uribe et al. in J Funct Anal 213:412–439, 2004, Curbera et al. in Adv Math 203:256–318, 2006), and ideas contained in previous works by Seeger (J Am Math Soc 9:95–105 1996) and Fan and Sato (Tohoku Math J 53:265–284, 2001).

Subfields: Applied Mathematics

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Sparse bounds for maximal rough singular integrals via the Fourier transform (2017, result): “ate kTΩ,δkLp(w) . [w] p′ Ap .We also observe that the proof of the mixed estimate (1.5) actuallyyieldsthefollowingestimateforthenon-maximallytruncatedoperators,improvingthe previous estimate given in [28] kTΩ,δfkLp(w) . [w] 1 p Aq [w] 1 p′ A∞ kfkLp(w). Finally, we emphasize that (1.7) also yields a ”