Sharp bounds for \(t\)-Haar multipliers on \(L^2\) (Q2875845)

For \(r\in\mathbb R\), \(m,\;n\in\mathbb N\), and a weight \(w:\;\mathbb R\to[0,\infty)\), the \(t\)-Haar multiplier of complexity \((m,n)\) associated to \(w\), \(T_{t,w}^{m,n}\), is defined formally by, for all \(x\in\mathbb R\), NEWLINE\[NEWLINET_{t,w}^{m,n}f(x):= \sum_{L\in\mathcal D}\sum_{I\in \mathcal D_m(L),J\in \mathcal D_n(L)} c_{I,J}^L\frac{w^t(x)}{(M_Lw)^t}\langle f,h_I\rangle h_J(x), NEWLINE\]NEWLINE where \(|c_{I,J}^L|\leq \sqrt {|I||J|}/|L|\), \(\mathcal D\) denotes the set of all dyadic intervals, \(|I|\) the length of interval \(I\), \(\mathcal D_m(L)\) denotes the dyadic subintervals of \(L\) with length \(2^{-m}|L|\), \(h_I\) is an \(L^2\)-normalized Haar function associated to \(|I|\), and \(\langle f,g\rangle\) denotes the \(L^2\)-inner product of \(f\) and \(g\). Let \(A_q^d\) be the set of Muckenhoupt dyadic weights and \([w]_{A_q^d}\) the \(A_q^d\) constant of \(w\). Let \(C_{2t}^d\) be the set of weights \(w\) satisfying NEWLINE\[NEWLINE[w]_{C_{2t}^d}:=\sup_{I\in\mathcal D}\left[\frac 1{|I|}\int_Iw^{2t}(x)\,dx\right] \left[\frac1{|I|}\int_Iw(x)\,dx\right]^{-2t}<\infty.NEWLINE\]NEWLINE In this paper, the authors discuss sharp bounds of \(t\)-Haar multipliers and one of these results is as follows: Assume that \(t\in\mathbb R\), \(w\in C_{2t}^d\) and there is \(q\in(1,\infty)\) such that \(w^{2t}\in A_q^d\). Then there exists a positive constant \(K_q\), depending only on \(q\), such that NEWLINE\[NEWLINE\left\|T_{t,w}^{m,n}f\right\|_{L^2(dx)} \leq K_q(m+n+2)^{5/2}[w]_{C_{2t}^d}^{1/2}[w^{2t}]_{A_q^d}^{1/2}\|f\|_{L^2(dx)}.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 1285.00036].