Local and global weighted norm inequalities for the sharp function and the Hardy-Littlewood maximal function (Q1331776)

Let \(f^ \#\) be the sharp function defined by \[ f^ \#(\xi)= \sup{1\over | I|^ 2} \int_ I \int_ I | f(\eta)- f(\zeta)| d\eta d\zeta, \] where supremum is over all cubes \(I\) containing \(\xi\), with sides parallel to the axes, \(Mf\) be a Hardy- Littlewood maximal operator. The main result is the following inequality in weighted mixed norm Lebesgue spaces \(L^{p,q}(\mathbb{R}^ m\times \mathbb{R}^ n, w(x,s))\): \[ \| Mf\|_{p,q,w_ 0,w_ 1}\leq C\| f^ \#\|_{p,q,w_ 0,w_ 1}. \] Here \(0< p, q<\infty\), \(w(x,s)= w_ 0(x) w_ 1(s)\), where \(w_ 0\) and \(w_ 1\) satisfy the Muckenhoupt \(A_ \infty\)-condition. The proofs are based on factorization of \(A_ q\)-weights.