On parabolic potantials associated with the generalized translation

In this thesis, firstly, the generalized Gauss-Weierstrass semigroup resulting from the generalized shift associated with the Laplace-Bessel differential operator was defined and then the parabolic Riesz and parabolic Bessel potentials resulting from the generalized shift were defined and the single integrals of these potentials were expressed by means of the generalized Gauss-Weierstrass semigroup. In the findings section of the thesis, firstly, the boundedness of the parabolic Riesz and parabolic Bessel potentials in weighted Lebesgue spaces was proved. Then, the convergence properties of the potentials in weighted Lebesgue spaces were investigated.Finally, the Calderon type reproduction formula was obtained by using the appropriate wavelet transform.