Weighted inequalities for vector-valued anisotropic maximal functions (Q1071206)

Some classical results on Muckenhoupt weights are extended to the case where the maximal function is defined by means of anisotropic parallelepipeds instead of cubes. The method is based on a modification of the covering lemma of Calderon-Zygmund. Main result: Let \(a=(a_ 1,...,a_ n)\in R^ n_+,\) \(x\in R^ n\), \(t\in R_+\), and \(E(x,t)=\{z\in R^ n;\quad | z_ i-x_ i| \leq t^{a_ i}/2,\quad i=1,...,n\},\quad E=\{E(x,t);\quad x\in R^ n,\quad t\in R_+\}.\) For a measurable function \(f: R^ n\times Y\to R\), let \[ M_{(1)}f(x,y)=\sup_{t>0}(1/| E(x,t)|)\int_{E(x,t)}| f(z,y)| \quad dz,\quad x\in R^ n,\quad y\in Y. \] If \(1\leq p\leq q<\infty\), \(q>1\), and \(w\in A_ p(E)\), then there exists a positive constant c such that \[ \mu_ w\{x\in R^ n;\quad (\int_{Y}[M_{(1)}f(x,y)]^ q dv)^{1/q}>s\}\leq cs^{- p}\int_{R^ n}(\int_{Y}| f(x,y)|^ q dv)^{p/q} w(x)dx, \] for every \(s>0\) and every function f.