## 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)

### MSC:

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

### Citations:

Zbl 1141.46313; Zbl 1225.46021
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### References:

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