On the theory of multilinear Littlewood–Paley $g$-function

Let m \ge 2 and define the multilinear Littlewood–Paley g-function by ¶ g(\vec{f})(x)=\bigg(\int_{0}^{\infty} \bigg| \frac{1}{t^{mn}}\int_{(\mathbb{R}^n)^m} \psi\bigg(\frac{y_1}{t},\dots,\frac{y_m}{t}\bigg) \prod_{j=1}^mf_j(x-y_j)dy_{j}\bigg|^2 \frac{dt}{t} \bigg)^{1/2}. ¶ In this paper, we establis...

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Bibliographic Details
Published in:Journal of the Mathematical Society of Japan 2015, Vol.67 (2), p.535-559
Main Authors: XUE, Qingying, PENG, Xiuxiang, YABUTA, Kôzô
Format: Article
Language:English
Subjects:
42B25
47G10
end-point weighted estimates
iterated commutators
multilinear Littlewood–Paley g-function
multiple weights
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Summary:Let m \ge 2 and define the multilinear Littlewood–Paley g-function by ¶ g(\vec{f})(x)=\bigg(\int_{0}^{\infty} \bigg| \frac{1}{t^{mn}}\int_{(\mathbb{R}^n)^m} \psi\bigg(\frac{y_1}{t},\dots,\frac{y_m}{t}\bigg) \prod_{j=1}^mf_j(x-y_j)dy_{j}\bigg|^2 \frac{dt}{t} \bigg)^{1/2}. ¶ In this paper, we establish the strong L^{p_1}(w_1)\times \dots \times L^{p_m}(w_m) to L^p(\nu_{\vec{\omega}}) boundedness and weak type L^{p_1}(w_1)\times \dots \times L^{p_m}(w_m) to L^{p,\infty}(\nu_{\vec{\omega}}) estimate for the multilinear g-function. The weighted strong and end-point estimates for the iterated commutators of g-function are also given. Here \nu_{\vec{\omega}} = \prod_{i = 1}^m\omega_i^{{p}/{p_i}} and each w_i is a nonnegative function on \mathbb{R}^n.
ISSN:0025-5645
1881-2333
DOI:10.2969/jmsj/06720535