Math. Nachr. 242 (2002), 27 – 37 Inversion of Bessel Potentials with the Aid of Weighted Wavelet Transforms By Ilham A. Aliev *) and Melih Eryigit of Antalya (Received November 28, 2000; revised version March 14, 2001; accepted April 3, 2001) Abstract. A special type of weighted wavelet transforms is introducedand the relevant Calder´on reproducing formula for functions f ∈ L p (IR n ) is proved. By making use of these wavelet–type transforms a new inversion formula of the classical Bessel potentials is obtained. 1. Introduction The classical Bessel potentials are defined in terms of Fourier transforms by (J α f ) ∧ (x)= ( 1+ |x| 2 ) -α/2 f ∧ (x) (x ∈ IR n ,α> 0) , (1.1) and are interpreted as negative fractional powers of the differential operator I -△  △ = ∑ n k=1 ∂ 2 ∂x 2 k is the Laplacian, I is the identity operator  . These potentials, being a powerful technical tool in harmonic analysis and its applications, were investigated by N. Aronszajn, K. Smith, A. Calder´ on, F. Mulla, P. Szeptycki, R. Adams, E. Stein, T. Flett, P. Lizorkin, S. Samko and many other mathematicians (see [1, 3, 4, 11, 12, 13, 14] for further information and references). An important problem concerning the Bessel potentials is to obtain explicit inversion for them. A number of approaches to this problem is known (Nogin [6], Rubin [7, 8]). In this paper we develop a new approach to the aforementioned problem and invert Bessel potentials by means of the so–called weighted wavelet transforms. Note that wavelet–type representation of n–dimensional Riesz potentials and of Riesz fractional derivatives and their spherical analogies was developed by B. Rubin [9, 10, 11]. Inver- sion of parabolic potentials by means of anisotropic wavelet transforms was obtained in [2]. 2000 Mathematics Subject Classification. Primary: 26A33, 44A35; Secondary: 42C40. Keywords and phrases. Bessel potentials, fractional derivative, weighted wavelet transform, Calder´on–type reproducing formula. *) Corresponding author/ialiev@pascal.sci.akdeniz.edu.tr c WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0025-584X/02/24208-0027 $ 17.50+.50/0