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Weighted endpoint estimates for singular integral operators associated with Zygmund dilations. (English) Zbl 1430.42018

The authors study multi-parameter singular integral operators which commute with Zygmund dilations: \(\rho_{s,t}(x_1,x_2,x_3)=(sx_1,tx_2,stx_3)\). For this purpose, they develop the theory of a weighted multi-parameter Hardy space \(H^p_{\zeta,w}(\mathbb R^3)\) and prove the boundedness for these operators on \(H^p_{\zeta,w}(\mathbb R^3)\), for certain \(p \le 1\), which provide endpoint estimates for those singular integral operators studied in [F. Ricci and E. M. Stein, Ann. Inst. Fourier 42, No. 3, 637–670 (1992; Zbl 0760.42008)] and [R. Fefferman and J. Pipher, Am. J. Math. 119, No. 2, 337–369 (1997; Zbl 0877.42004)].
To achieve their goals, they use a “standard strategy”, but adapted to the structure of Zygmund dilations. More specifically, they construct a Calderón reproducing formula associated with Zygmund dilations, then discretize the operators according to it, apply almost orthogonal estimates, and implement various stopping time arguments. Some interpolation results, for the Hardy spaces, are also obtained. One difficulty they have to overcome is that it is not known whether there exists an atomic decomposition for \(H^p_{\zeta,w}(\mathbb R^3)\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
47G10 Integral operators
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