Abstract
We give a complete characterisation of the spaces (B)over dot(p,q)(alpha) and (F)over dot(p,q)(alpha) by using a non-smooth kernel satisfying near minimal conditions. The tools used include a Stromberg-Torchinsky type estimate for certain maximal functions and the concept of a distribution of finite growth, inspired by Stein. In addition, our exposition also makes essential use of a number of refinements of the well-known Calderon reproducing formula. The results are then applied to obtain the characterisation of these spaces via a fractional derivative of the Poisson kernel. Moreover, our results offer an approach to deal with the calculus modulo polynomials in homogeneous function spaces, a subtle problem raised recently by Triebel.