Piecewise monotonic doubling measures
The author studies weights (\(A_p\) weights, reverse Hölder weights, and doubling weights) under the restriction of piecewise monotonicity of the density. The \(A_p\) weights control mapping properties of the Hardy-Littlewood maximal operator and of the Hilbert transform. The spaces are defined by \(w\in A_p\) if and only if \[ A^p_p(w)= \sup_Q \Biggl({1\over {|Q|}} \int_Q w\Biggr) \Biggl({1\over {|Q|}} \int_Q w^{-1/(p-1)} \Biggr)^{p-1}< +\infty. \] A measure \(\mu\) is doubling if there is a constant \(C\) such that for each interval \(I\), \(\mu(2I)\leq C\mu(I)\), where \(2I\) is the interval with the same center as \(I\) and twice the length. Doubling measures are not necessarily absolutely continuous. \(A_\infty\) weights are doubling, but not every doubling weight is an \(A_\infty\) weight. The \(A_p\) weights form a nested set with \[ A_1\subseteq A_p\subseteq A_q\subseteq A_\infty= \bigcup_{1\leq p\leq\infty} A_p\subseteq D, \qquad 1\leq p\leq q\leq\infty. \] The author shows that piecewise monotonic doubling measures are absolutely continuous, which means that results can be expressed in terms of the density of the measure. He first characterizes decreasing and increasing doubling weights. Several characterizations are given; I will cite ones in terms of the averaging operator \(A\), where \(Aw(t)= {1\over t}\int^t_0 w(x)dx\). A decreasing weight \(w\) is doubling if and only if there is a constant \(C\) such that \(Aw(t)\leq Cw(t)\), for all \(t>0\) if and only if \(w\in A_1\). An increasing weight \(w\) is doubling if and only if there is a constant \(C\) such that \(CAw(t)\geq w(t)\), for all \(t>0\) if and only if \(w\in A_\infty\). Using these results among others, he reproves results of Guseinov and of Benedetto, Heinig and Johnson characterizing monotone \(A_p\) weights. He also gives necessary and sufficient conditions for monotonic functions to be pointwise multipliers of the class of monotonic weights. He proves that \(\phi\) is a multiplier of decreasing (increasing) \(A_\infty\) weights iff \(\phi\) is a multiplier of general \(A_\infty\) weights (characterized by Johnson and Neugebauer) which is decreasing (increasing), and the same result is valid for \(A_p\).
scientific article; zbMATH DE number 944407
scientific article