Haar functions in weighted Besov and Triebel–Lizorkin spaces

The paper deals with Haar wavelet bases in function spaces of Besov and Triebel–Lizorkin type with local Muckenhoupt weights. We show that Haar wavelets can be used to characterize such function spaces as far as absolute value of smoothness parameter is small enough and weights fulfill some conditions. The result is based on mapping properties of linear operators involving characteristic functions of dyadic cubes in related spaces and on local means characterization of weighted Besov and Triebel–Lizorkin spaces.