Weighted maximal inequalities for the Haar system
Let \(h=(h_n)_{n\geq0}\) be the collection of Haar functions on \([0, 1)\), given by \[ \begin{aligned} h_0&=\chi_{[0, 1)}, \\ h_1&=\chi_{[0, 1/2)}-\chi_{[1/2, 1)}, \\ h_2&=\chi_{[0, 1/4)}-\chi_{[1/4,1/2)}, \\ h_3&=\chi_{[1/2, 3/4)}-\chi_{[3/4,1)}, \\ h_4&=\chi_{[0, 1/8)}-\chi_{[1/8,1/4)} \\ h_5&=\chi_{[1/4, 3/8)}-\chi_{[3/8,1/2)}, \\ h_6&=\chi_{[1/2, 5/8)}-\chi_{[5/8,3/4)} \\ h_7&=\chi_{[3/4, 7/8)}-\chi_{[7/8,1)} \end{aligned} \] and so on. A weight \(\omega\) is called a dyadic \(A_p\) weight with \(p\in(1, \infty)\), if \[ [\omega]_{A_p}=\sup_I\left(\frac1{|I|}\int_I\omega\right)\left(\frac1{|I|}\int_I \omega^{-1/(p-1)}\right)^{p-1}<\infty; \] and \(\omega\) is called a dyadic \(A_1\) weight, if \[ [\omega]_{A_1}=\sup_I\left(\frac1{|I|}\int_I\omega\right)/{\mathop{\operatorname{essinf}}_I}\omega<\infty. \] Here in the both conditions, the suprema are taken over all dyadic subintervals \(I\) of \([0, 1)\). This paper studies the weighted maximal \(L^p\)-inequalities for the Haar system \(h=(h_n)_{n\geq0}\), with the optimal dependence on the characteristics of the weights involved. The author shows that for \(p\in(1, \infty)\), if \(\omega\) is a dyadic \(A_p\) weight, \(N\) is a non-negative integer, \(a_0, a_1,\dots, a_N\) are real numbers and \(\varepsilon_0, \varepsilon_1,\dots, \varepsilon_N\) is a sequence of signs, then \[ \left\|\max_{0\leq n\leq N}\left|\sum_{k=0}^n \varepsilon_ka_kh_k\right|\right\|_{L^p(\omega)} \leq 2^{1+1/p}C_p[\omega]_{A_p}^{\max\{1, 1/(p-1)\}}\left\|\sum_{k=0}^N a_kh_k\right\|_{L^p(\omega)}. \] Moreover, the exponent \(\max\{1, 1/(p-1)\}\) is the best possible. The author also shows that for any parameters \(p,q\in[1, \infty)\), there is a constant \(C_{p,q}\) depending only on \(p\) and \(q\) which has the following property: \[ \left\|\max_{0\leq n\leq N}\left|\sum_{k=0}^n \varepsilon_ka_kh_k\right|\right\|_{L^q(\omega)} \leq C_{p,q}[\omega]_{A_p}\left\|\max_{0\leq n\leq N}\left|\sum_{k=0}^n a_kh_k\right|\right\|_{L^q(\omega)}. \] Moreover, the linear dependence on the \(A_p\) characteristics is optimal for each \(p\). The proofs exploit the Bellman function method by deducing the validity of the above two weighted norm inequalities from the existence of certain special functions which enjoy appropriate size conditions and concavity.
scientific article; zbMATH DE number 6894850