Maximal functions and related weight classes
The authors have made deep contributions to the area of \(A_ p\)-weights and maximal functions with this paper. Foremost, the classical result of Muckenhoupt on the boundedness of the Hardy-Littlewood maximal operator in \(L^ p(w)\) is extended to the case \(p = \infty\) by the introduction of the geometrical maximal operator. The necessary notations and definitions are as follows. First, for \(g \in L^ q_{\text{loc}} (\mathbb{R}^ n)\), \(q>0\), the \(q\)-maximal function of \(g\) is \(M_ q g(x) = \sup_{Q \ni x} (\int- _ Q | g(t) |^ qdt)^{1/q}\). (The symbol \(\int-_ Q f(x) dx\) denotes \({1 \over | Q |} \int_ Q f(x)dx)\). The authors define the geometrical maximal operator, \(M_ 0g\), as \[ M_ 0g = \sup_{Q \ni x} \exp \left(\int-_ Q \ln | g(t) | dt \right). \] The \(A_ p\)-constant of a weight function \(w(x)\) is \[ A_ p (w) = \sup_ Q \int-_ Q w(x)dx \left(\int-_ Qw^{{-1 \over p - 1}} (x)dx \right)^{p - 1}. \] One definition of the \(A_ \infty\)- constant of \(w\) is \(A_ \infty (w) = \sup_ Q \int-_ Q w(x) dx (\exp \int-_ Q \ln {1 \over w(x)} dx)\). Since it is well-known that \(A_ \infty = \cup_{p > 1} A_ p\) and that \(A_ p (w)\) is a decreasing function of \(p\), it is also natural to consider \(\overline A_ \infty (w) = \lim_{p \to \infty} A_ p(w)\) as a possible definition of the \(A_ \infty\)-constant of \(w\). Here the authors prove that if \(w \in A_ \infty\), then \(A_ \infty (w) = \overline A_ \infty (w)\), thus answering a question of \textit{R. Johnson} [Contemp. Math. 107, 93-99 (1990; 709.42013)]. The next key result of the paper motivates the definition of the geometrical maximal operator: If \(f \in L^ \alpha_{\text{loc}} (\mathbb{R}^ n)\) for some \(\alpha > 0\), then \(\lim_{p \to \infty} M_{1/p} f(x) = M_ 0 f(x)\) for all \(x\). Muckenhoupt's theorem can be written in the form: A weight function \(w \in A_ p\) if and only if \[ m_ p (w) = \sup_{\| f \|_{L(w) = 1}} \int_{\mathbb{R}^ n} M_{{1 \over p}} f(x) w(x) dx < \infty. \] The authors extend this to the case \(p = \infty\) with the Theorem: The weight \(w \in A_ \infty\) if and only if \[ m_ \infty (w) = \sup_{\| f \|_{L(w) = 1}} \int_{\mathbb{R}^ n} M_ 0 f(x) w(x) dx < \infty. \] Estimates for \(m_ \infty (w)\) in terms of \(A_ \infty (w)\) are also given. In the latter half of the paper, the authors specialize to the case of nonnegative, nonincreasing \(L^ 1 (w)\)-functions on \((0, \infty)\). (Denote this class of functions by \(L_ d (w).)\) They extend recent results of \textit{M. A. Arino} and \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 320, No. 2, 727-735 (1990; Zbl 0716.42016)] on weights in the \(B_ p\)-class and develop a number of analogues of certain \(A_ p\)-theorems for the class \(B_ p\). In particular, they prove \(B_ \infty = \cup_{p > 0} B_ p\) and establish that \(M_ 0\) is a bounded operator on \(L_ d (w)\) if and only if \(w \in B_ \infty\).
scientific article; zbMATH DE number 722039