Weighted \(L^p\) boundedness of Carleson type maximal operators (Q2845550)

In this paper the authors establish a weighted estimate for a polynomial-phase Carleson-type operator. Let NEWLINE\[NEWLINE P(x) = \sum_{2\leq|\alpha|\leq d} \lambda_\alpha x^\alpha NEWLINE\]NEWLINE be an arbitrary polynomial on \(\mathbb{R}^n\) of degree \(d\) without constant and linear terms. The Carleson-type maximal operator \(\mathcal{T}^\ast\) is defined as NEWLINE\[NEWLINE \mathcal{T}^\ast f(x) := \sup_{P} \Big|\int_{\mathbb{R}^n} e^{i P(y)} K(y) f(x-y) dy\Big| , NEWLINE\]NEWLINE where we regard \(n\) and \(d\) as fixed. Next, let \(\Omega\) be a measurable function on \(\mathbb{R}^n\setminus\{0\}\), constant on half-lines from the origin, integrable on the unit sphere \(S^{n-1}\), and having mean zero on \(S^{n-1}\). Let us also assume that \(\Omega\) satisfies the \(\mathrm{L}^q\)-Dini condition \(\int_0^1 \frac{\omega_q(\delta)}{\delta}d\delta<\infty\) for some \(1<q\leq\infty\), where \(\omega_q\) is the tangential \(\mathrm{L}^q(S^{n-1})\) modulus of continuity. Suppose that the kernel \(K\) appearing above is of the form \(K(y)=\Omega(y)/|y|^n\). Finally, take an exponent \(p\) such that \(1\leq q'<p<\infty\) and a weight \(w\) from the usual Muckenhoupt class \(\mathrm{A}_{p/q'}\). The authors prove the weighted \(\mathrm{L}^p\) estimate NEWLINE\[NEWLINE \|\mathcal{T}^\ast f\|_{\mathrm{L}^p(w)} \leq C \|f\|_{\mathrm{L}^p(w)} . NEWLINE\]NEWLINE It generalizes the non-weighted estimate for smooth kernels from [\textit{E. M. Stein} and \textit{S. Wainger}, Math. Res. Lett. 8, No. 5--6, 789--800 (2001; Zbl 0998.42007)], when specialized to homogenous kernels of degree \(0\). A particular case of this result, without a general weight and with an additional hypothesis that \(\Omega\) lies in the Hardy space on \(S^{n-1}\), has already been established by the same authors in [\textit{Y. Ding} and \textit{H. Liu}, Tohoku Math. J. (2) 63, No. 2, 255--267 (2011; Zbl 1226.42008)]. It is also interesting to recall a similar weighted estimate for a linear-phase Carleson operator from [\textit{E. Prestini} and \textit{P. Sjölin}, J. Fourier Anal. Appl. 6, No. 5, 457--466 (2000; Zbl 1049.42010)]. It is quite likely that any results of the same type that include both linear and higher-order terms in the polynomial \(P\) would also have to incorporate more involved time-frequency analysis, such as the techniques developed in [\textit{V. Lie}, Geom. Funct. Anal. 19, No. 2, 457--497 (2009; Zbl 1178.42007)].