An extension of the notion of the order of a distribution

Abstract

In this article we review the notion of the order of a distribution and extend it to the case of positive real numbers. We suggest to use the name Holder distributions for such distributions. The first part of the paper concerns itself with functional-analytic properties of the Holder test function spaces and its duals. Of particular interest are the $${C}^{r,\alpha +}_c(\Omega )$$ and the $${\mathcal {D}}'_{(r+\alpha )+}(\Omega )$$ spaces which have notably better properties such as reflexivity, compared to the classical Holder spaces. We also give a few examples and some Fourier-analytic properties of distributions of fractional order, and at the end, we note how one can extend classical results where estimates of the order of distributions appear, such as giving a bound on the order of convolution of distributions.