Ultrametric pseudodifferential operators and wavelets for the case of non homogeneous measure

A new way of construction of ultrametric spaces of quite general form by completion of directed trees with respect to the metric, defined in the special way, is proposed. We introduce a measure on the constructed ultrametric space, such that the measure of any ball is positive. A family of orthonormal bases of ultrametric wavelets in the space of quadratically integrable with respect to the defined measure functions is introduced. Pseudodifferential operators (PDO) in the defined space are investigated. We prove that these operators are diagonal in the introduced bases of ultrametric wavelets and compute the corresponding eigenvalues.