The Riesz-Herz equivalence for capacitary maximal functions. (English) Zbl 1281.42017

The authors consider the maximal operator \[ M_Cf(x)=\sup_{Q\ni x}\frac1{C(Q)}\int_Q|f(x)|\,\text{d}x, \] associated to a capacity \(C\) on \(\mathbb R^n\) (see [J. Cerdà, Contemporary Mathematics 445, 45–59 (2007; Zbl 1141.46313)] and [J. Cerdà, J. Martín and P. Silvestre, Collect. Math. 62, No. 1, 95–118 (2011; Zbl 1225.46021)]). It is a generalization of the classical Hardy-Littlewood maximal operator \(M\) (with \(C(Q)=|Q|\), the \(n\)-dimensional volume of a cube \(Q\)) or its fractional version \(M_\alpha\) (with \(C(Q)=|Q|^{1-\alpha/n}\), \(0\leq\alpha<n\)).
The authors extend the well-known Riesz-Herz equivalence \((Mf)^*(t)\approx f^{**}(t)\), \(t\geq0\), (where \(f^*\) denotes the non-increasing rearrangement of \(f\) and \(f^{**}(t)=\frac1t\int_0^tf^*\)) written also as \(t(Mf)^*(t)\approx K(t,f,L^1,L^\infty)\) (where \(K\) is the Peetre \(K\)-functional), for \(M_C\) instead of \(M\). Namely, they prove the equivalence \[ t(M_Cf)^*_C(t)\approx K(t,f,L^1,\mathcal L^{1,C}),\quad t\geq0, \] where \(f^*_C(t)=\inf\big\{\lambda>0; C\big(\{x; |f(x)|>\lambda\}\big)\leq t\big\}\) (the non-increasing rearrangement with respect to the capacity \(C\)) and \(\mathcal L^{1,C}\) is the Morrey-type space of functions \(f\) satisfying the condition \(\|M_Cf\|_{L^\infty}<\infty\). As a byproduct, the authors obtain a description of the norm of the interpolation space \((L^1,\mathcal L^{1,C})_{(p-1)/p,p}\), \(p\in(1,\infty)\).
Reviewer: Petr Gurka (Praha)


42B25 Maximal functions, Littlewood-Paley theory
46B70 Interpolation between normed linear spaces
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