Journal of the Mathematical Society of Japan

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

Qingying XUE, Xiuxiang PENG, and Kôzô YABUTA

Full-text: Open access

Abstract

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 $\mathbb{R}^n$.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 2 (2015), 535-559.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1429624593

Digital Object Identifier
doi:10.2969/jmsj/06720535

Mathematical Reviews number (MathSciNet)
MR3340185

Zentralblatt MATH identifier
1325.42024

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 47G10: Integral operators [See also 45P05]

Keywords
multilinear Littlewood–Paley $g$-function end-point weighted estimates iterated commutators multiple weights

Citation

XUE, Qingying; PENG, Xiuxiang; YABUTA, Kôzô. On the theory of multilinear Littlewood–Paley $g$-function. J. Math. Soc. Japan 67 (2015), no. 2, 535--559. doi:10.2969/jmsj/06720535. https://projecteuclid.org/euclid.jmsj/1429624593


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