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\(A_ p\) and approach regions. (English) Zbl 0871.42016

García-Cuerva, José (ed.) et al., Fourier analysis and partial differential equations. Proceedings of the conference held in Miraflores de la Sierra, Madrid, Spain, June 15–20, 1992. Boca Raton, FL: CRC Press. Studies in Advanced Mathematics. 311-315 (1995).
For a family \(\Omega=\{\Omega_x\}_{x\in\mathbb R}\) of non-empty measurable sets in \(\mathbb R_+^{n+1}=\{(x,r); x\in\mathbb R^n, r>0\}\) satisfying \((y,t)\in\Omega_x\), \(s>t\), the maximal operator \(M_\Omega\) is defined by \[ M_\Omega f(x)=\sup_{(y,t)\in\Omega_x}\frac1{|B(y,t)|}\int_{B(y,t)}|f(\xi)|d\xi \] (where \(B(y,t)=\{\xi\in\mathbb R^n; |\xi-y|<t\}\), \(|B|=\int_B d\xi\), \(f\in L^1_{loc}(\mathbb R^n)\)). In Math. Nachr. 172, 249-260 (1995; Zbl 0842.42009) the authors characterized the classes of weights \(A_p^\Omega\), \(p\geq1\), i.e., classes of those \(w\in L^1_{\text{loc}}(\mathbb R^n)\), \(w\geq0\), for which \(M_\Omega: L^p(w)\to L^{p,\infty}(w)\) is bounded, and they proved that whenever the cross section of the approach region is (in some sense) comparable to a ball then the \(A_p^\Omega\)-class is exactly the classical \(A_p\)-class (of B. Muckenhoupt).
In the present paper the authors show a converse result, that is if \(A_p^\Omega\) equals \(A_p\) then \(\Omega\) is essentially a cone.
For the entire collection see [Zbl 0847.00037].
Reviewer: P.Gurka (Praha)

MSC:

42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0842.42009
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