The rate of convergence of Steklov means on metric measure spaces and Hausdorff dimension (Q650398)

Suppose that \((X,d,\mu)\) is a metric space with metric \(d\) and regular Borel measure \(\mu\). Let \(B(x,r)= \{y\in X: d(x,y)< r\}\) be the ball centered at the point \(x\in X\) of radius \(r> 0\). Also, let \[ f_B= -\hskip-.9em\int_B f d\mu={1\over \mu(B)} \int_B fd\mu \] be the mean value of the function \(f\in L^1(B)\) over the ball \(B\subset X\) (the Steklov mean). By \(L^p= L^p(X)\), \(1\leq p<\infty\), the Lebesgue space generated by the measure \(\mu\) is denoted. Consider the maximal function \[ S_\alpha f(x)= \sup_{B\ni x} r^{-\alpha}_B -\hskip-.9em\int_B |f(y)- f_B|dy, \] where \(r_B\) is the radius of the ball \(B\), and the supremum is taken over all balls \(B\), \(r_B\in (0,1)\) containing the point \(x\in X\). Let \[ C^p_\alpha= C^p_\alpha(X)= \{f\in L^p:\| f\|_{C^p_\alpha}=\| f\|_{L^p}+\| S_\alpha f\|_{L^p}< \infty\}. \] In this note the rate of convergence of Steklov means for functions from the classes \(C^p_\alpha\) is studied similar to some previous results such as in [Math. Notes 85, No. 4, 584--589 (2009); translation from Mat. Zametki 85, No. 4, 616--621 (2009; Zbl 1182.46023)] by the second author.