Heat and Poisson Semigroups for Fourier-neumann Expansions

Given α > −1, consider the second order differential operator in (0,∞) Lαf ≡ ( x 2 d 2 dx2 + (2α+ 3)x d dx + x + (α+ 1) ) (f), which appears in the theory of Bessel functions. The purpose of this paper is to develop the corresponding harmonic analysis taking Lα as the analogue to the classical Laplacian. Namely we study the boundedness properties of the heat and Poisson semigroups. These boundedness properties allow us to obtain some convergence results that can be used to solve the Cauchy problem for the corresponding heat and Poisson equations. Given α > −1, we shall consider the second order operator on functions defined on (0,∞) Lαf ≡ ( x d2 dx2 + (2α + 3)x d dx + x + (α+ 1) ) (f). This operator appears in the theory of Bessel functions (see [10]). It is selfadjoint with respect to the measure dμα(x) = x 2α+1 dx. It is well known that the functions j n (x) = √ 2(α+ 2n+ 1) Jα+2n+1(x)x −α−1, n = 0, 1, 2, . . . where Jν stands for the Bessel function of the first kind of order ν, are eigenfunctions of the operator Lα. In fact Lαj α n = (α+ 2n + 1) j n , n = 0, 1, . . . Date: October 31, 2005. 2000 Mathematics Subject Classification. Primary 42C10; Secondary 35K05, 35J05.