Maximal functions and smoothness spaces in !$!L_{p}(ℝ^{d})

We study smoothness spaces generated by maximal functions related to the local approximation errors of integral operators. It turns out that in certain cases these smoothness classes coincide with the spaces ${\displaystyle C^{\alpha }\_p\left({ℝ}^d\right)}$, 0 < p≤∞, introduced by DeVore and Sharpley [DS] by means of the so-called sharp maximal functions of Calderón and Scott. As an application we characterize the ${\displaystyle C^{\alpha }\_p\left({ℝ}^d\right)}$ spaces in terms of the coefficients of wavelet decompositions.