Estimates for compositions of maximal operators with singular integrals (Q2862961)

The maximal operator being considered is NEWLINE\[NEWLINE \Delta^*[f] (x) = \sup_k | \Delta_k[ f ](x) |, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Delta_k[f] = \sum_{\substack{\omega \in \Omega: |\omega| = 2^{-k} \\ \omega \cap \Xi \neq \emptyset}} \widehat{\phi_{\omega}}* f, NEWLINE\]NEWLINE and \(\Omega\) is the collection of all dyadic subintervals of \(R, \{ \phi_{\omega} \}\) is a collection of smooth functions each adapted to \(\Omega\), i. e. they are supported on \(\omega\) and NEWLINE\[NEWLINE D_M = \sup_{\omega \in \Omega} |\omega|^M \| \phi_{\omega}^{(M)} \|_{\infty} NEWLINE\]NEWLINE is finite, where \( \phi_{\omega}^{(M)}\) denotes the \(M\)-th derivative of \(\phi_{\omega}\) with M depending on \(\epsilon\) (to be defined below). \(\Xi\) is a finite collection of real numbers.NEWLINENEWLINEIt has been shown \textit{F. Nazarov} et al. [Math. Res. Lett. 17, No. 3, 529--545 (2010; Zbl 1256.42023)] that for \(1 < q \leq 2, r>2\) NEWLINE\[NEWLINE \| \Delta^*[f]\|_q \leq C_{q, r} (1 + \log |\Xi|)|\Xi|^{1/q - 1/r} \left( D_M + \sup_{\xi \in \Xi} \left | \left | \sum_{\omega \in \Omega: |\omega| = 2^{-k} } \phi_{\omega}(\xi)\right| \right |_{V^r_k} \right) \| f\|_{L^q}, NEWLINE\]NEWLINE where \(V^r_k\) is the \(L^r\) variation norm on the intervals of length \(2^{-k}\). In this paper the author considers a finite set \(\Upsilon\) of disjoint (not necessarily dyadic) subintervals of \(R\) and let \(\{ \psi_v \}_{v \in \Upsilon}\) be a collection of functions such that each \(\psi_v\) is supported on \(v\). He writes NEWLINE\[NEWLINE \Psi[ f ] = \sum_{v \in \Upsilon} (\psi_v \widehat{f})^{\check{\;} }. NEWLINE\]NEWLINE \textit{R. Coifman} et al. [C. R. Acad. Sci., Paris, Sér. I 306, No. 8, 351--354 (1988; Zbl 0643.42010)] proved that for \(r \geq 1, \frac1q - \frac12 < \frac1r, \epsilon>0\), NEWLINE\[NEWLINE \|\Psi[ f ]\|_q \leq C_{q, r, \epsilon} | \Upsilon|^{\frac1q - \frac12 + \epsilon} \sup_{v \in \Upsilon} \|\psi_v\|_{V^r} \| f\|_q. NEWLINE\]NEWLINE Combining the two results gives an estimate involving \(| \Xi|^{\frac1q - \frac1r + \epsilon} | \Upsilon|^{\frac1q - \frac12 + \epsilon}\). The author improves the estimate to \((| \Xi | + | \Upsilon|)^{\frac1q - \frac1r + \epsilon} \). His main result isNEWLINENEWLINETheorem 1.1. Suppose \(1 < q < 2, 2 < r <2q, \epsilon>0\), then NEWLINE\[NEWLINE\begin{multlined} \|\Delta^*\Psi[ f ] \|_q \leq C_{q, r, \epsilon} (|\Xi| + |\Upsilon|)^{1/q - 1/r + \epsilon} \\ \times \left( D_M + \sup_{\xi \in \Xi} \left | \left | \sum_{\omega \in \Omega: |\omega| = 2^{-k} } \phi_{\omega}(\xi)\right| \right |_{V^r_k} \right) \sup_{v \in \Upsilon} \| \psi_v \|_{V^r} \| f\|_{L^q}. \end{multlined}NEWLINE\]NEWLINE The author considers this inequality in the hope of applying it to the return times conjecture for the truncated Hilbert transform. The application requires a result similar to the above, which has been proved for the Walsh analogue of \(\Delta^*\Psi\). His result is a step towards a pointwise convergence result, but the author remarks that it is not clear if the result is strong enough. He would like to replace \(\psi_{v}\) by the characteristic function of \(v\), but his current proof does not allow this for \(q <2\).