On the pointwise domination of a function by its maximal function (Q1660100)

The author studies when the pointwise inequality \[ |f(x)|\le Mf(x) \tag{1} \] is satisfied for \(\mu\)-a.e.~\(x\). Here \(M\) is the centered Hardy-Littlewood maximal operator with respect to the measure \(\mu\) in the metric measure space \((X,d,\mu)\), that is \[ M f(x)=\sup_{\overline{B}}\frac{1}{\mu(\overline{B})}\int_{\overline{B}}f(y)\text{d}\mu(y), \] where the supremum is taken over all closed balls \(\overline{B}=\overline{B}(x,r)=\{y\in X; d(x,y)\le r\}\) such that \(\mu(\overline{B})>0\). It is shown that inequality \((1)\) is equivalent, under quite general assumptions, to the condition \[ \limsup_{r\downarrow0}\frac{1}{\mu(B(x,r))} \int_{B(x,r)}\mathbf{1}_F(y)\text{d}\mu(y) =\mathbf{1}_F(x) \] for all closed and totally bounded sets \(F\subset\operatorname{supp}\mu\) and a.e.~\(x\in X\) (here \(B(x,r)=\{y\in X; d(x,y)<r\}\)).