Maximal averages along a planar vector field depending on one variable (Q2847029)

Let \(v : \mathbb{R}^2 \rightarrow [0,1]\) be a vector field depending on one variable with \(v(x,y)=v(x)\). For any rectangle \(R\), denote by \(L(R)\) the length of \(R\), by \(W(R)\) the width of \(R\), and by \(\theta(R)\) the interval of width \(\frac{W(R)}{L(R)}\) centered at the slope of the long side of \(R\). We now set NEWLINE\[NEWLINEV(R) = \{(x,y) \in R : v(x,y) \in \theta(R) \} NEWLINE\]NEWLINE and for \(0 < w \leq 1\) and \(0 < \delta \leq 1\) we set NEWLINE\[NEWLINE\mathcal{R}_{\delta} = \{ R : |V(R)| \geq \delta|R|, W(R) =w \}. NEWLINE\]NEWLINE Then for \(f \in \mathbb{R}^2\) define a maximal operator \(M_{\mathcal{R}_{\delta}}\) for any collection of rectangles \(\mathcal{R}_{\delta}\) by NEWLINE\[NEWLINE M_{\mathcal{R}_{\delta}}f(x) = \sup_{x \in R \in \mathcal{R}_{\delta}} \frac{1}{|R|}\int_R f(y)\, dy. NEWLINE\]NEWLINENEWLINENEWLINEThe author proves (essentially) sharp \(L^2\) estimates for a maximal operator \(M_{\mathcal{R}_{\delta}}\) related to vector fields \(v\) depending only one variable. The main results are as follows:NEWLINENEWLINETheorem: If \(v : \mathbb{R}^2 \rightarrow [0,1]\) depends on one variable, i.e. \(v(x,y)=v(x)\), then for \(f \in L^2(\mathbb{R}^2)\) NEWLINE\[NEWLINE\|M_{\mathcal{R}_{\delta}}f\|_{2} \;\leq \;C \big( \log\frac{1}{\delta} \big)^{\frac{3}{2}}\|f\|_{2}.NEWLINE\]NEWLINENEWLINENEWLINECorollary: Under the same hypotheses as the theorem, when \(p \in (1,2)\), we have for \(f \in L^p(\mathbb{R}^2)\) NEWLINE\[NEWLINE\|M_{\mathcal{R}_{\delta}}f\|_{p} \;\leq \;C \big( \log\frac{1}{\delta} \big)^{3(1-\frac{1}{p})} \, \frac{1}{\delta^{\frac{2}{p}-1}} \, \|f\|_{p}.NEWLINE\]NEWLINE It is also pointed out that it is not clear that this theorem generalizes to the situation of rectangles with arbitrary width.