JACOBI TRANSFORM OF (ν, γ, p)-JACOBI–LIPSCHITZ FUNCTIONS IN THE SPACE L(R,∆(α,β)(t)dt)

Using a generalized translation operator, we obtain an analog of Younis’ theorem [Theorem 5.2, Younis M. S. Fourier transforms of Dini–Lipschitz functions, Int. J. Math. Math. Sci., 1986] for the Jacobi transform for functions from the (ν, γ, p)-Jacobi–Lipschitz class in the space L(R,∆(α,β)(t)dt).

The main aim of this paper is to establish an analog of Theorem 1 for the Jacobi transform in the space L p (R + , ∆ (α,β) (t)dt). For this purpose, we use a generalized translation operator which was defined by Flensted-Jensen and Koornwinder [5].
In order to confirm the basic and standard notation, we briefly overview the theory of Jacobi operators and related harmonic analysis. The main references are [1,4,6].
Let λ ∈ C, α ≥ β ≥ −1/2, and α = 0. The Jacobi function φ λ of order (α, β) is the unique even C ∞ -solution of the differential equation where ρ = α + β + 1, D α,β is the Jacobi differential operator defined as and ∆ ′ (α,β) (x) is the derivative of ∆ (α,β) (x). The Jacobi functions φ λ can be expressed in terms of Gaussian hypergeometric functions as where the Gaussian hypergeometric function is defined as with a, b, z ∈ C, c / ∈ −N, a 0 = 1, and a m = a(a + 1) · · · (a + m − 1). The function z → F (a, b, c, z) is the unique solution of the differential equation which is regular at 0 and equals 1 there. From [7, Lemmas 3.1-3.3], we obtain the following statement. Lemma 1. The following inequalities are valid for a Jacobi function φ λ (t) (λ, t ∈ R + ): be the space of p-power integrable functions on R + endowed with the norm .
Now, we define the Jacobi transform for all functions f on R + and complex numbers λ for which the right-hand side is well defined. The Jacobi transform reduces to the Fourier transform when α = β = −1/2. We have the following inversion formula [6].
From [3], we have the Hausdorff-Young inequality where 1/p + 1/q = 1 and C 2 is a positive constant. The generalized translation operator T h of a function f ∈ L p α,β (R + ) is defined as where K is an explicity known kernel function such that and K(x, y, z) = 0 elsewhere and B = cosh 2 x + cosh 2 y + cosh 2 z − 1 2 cosh x cosh y cosh z .
Thus, there exists C 4 such that and this completes the proof.