WEIGHTED Lp-BOUNDS FOR THE CARLESON TYPE MAXIMAL OPERATOR WITH KERNEL SATISFYING MILD REGULARITY

Abstract

This paper is concerned with the Carleson type maximal operator ${\mathcal{𝒯}}^{\ast }$ defined by

$${\mathcal{𝒯}}^{\ast }f(x)=\underset{\lambda }{\text{sup }}\left|{\int }_{{ℝ}^n}{e}^{i{P}_{\lambda }(y)}K(y)f(x-y)dy\right|,$$

where $P_{\lambda }(y)={\sum }_{2\le \left|\alpha \right|\le d}{\lambda }_{\alpha }{y}^{\alpha }$ is the polynomial in ${ℝ}^n$ with real coefficients $\lambda :={({\lambda }_{\alpha })}_{2\le \left|\alpha \right|\le d}$. Under the assumption that the kernel function $K$ satisfies an $L^r$-Hörmander condition with , the authors show that ${\mathcal{𝒯}}^{\ast }$ is bounded on the weighted Lebesgue spaces $L^p(\omega )$ for and $\omega \in A_{p∕r^{\prime }}$, which improves and generalizes the previous results obtained by Stein and Wainger (Math. Res. Lett. 8:5-6 (2001), 789–800), and Ding and Liu (Proc. Amer. Math. Soc. 140:8 (2012), 2739–2751).