[Paper Review] $L^p$-results for fractional integration and multipliers for the Jacobi transform

Troels Roussau Johansen|arXiv (Cornell University)|Aug 17, 2011

Advanced Harmonic Analysis Research28 references3 citations

TL;DR

This paper establishes near-optimal $L^p$-$L^q$ estimates for Riesz transforms (fractional integrals) and multiplier operators associated with the Jacobi transform, using precise asymptotic expansions of Jacobi functions $\varphi^{(\alpha,\beta)}_{\lambda}$ for $\Re\alpha > \frac{1}{2}$, $\Re\alpha > \Re\beta > -\frac{1}{2}$. It extends classical Hörmander-type multiplier theorems to Jacobi analysis via transference principles on hypergroups, unifying results for rank one symmetric spaces, Damek–Ricci spaces, and Heckman–Opdam transforms.

ABSTRACT

We use precise asymptotic expansions for Jacobi functions $ϕ^{(α,β)}_λ$ parameters $α$, $β$ satisfying $α>1/2$, $α>β>-1/2$, to generalizing classical Hörmander-type multiplier theorem for the spherical transform on a rank one Riemannian symmetric space (by Clerc/Stein and Stanton/Tomas) to the framework of Jacobi analysis. In particular, multiplier results for the spherical transform on Damek--Ricci spaces are subsumed by this approach, and it yields multiplier results for the hypergeometric `Heckman--Opdam transform' associated with a rank one root system. We obtain near-optimal $L^p-L^q$ estimates for the integral operator associated with the convolution kernel $m_a:λ\mapsto(λ^2+ρ^2)^{-a/2}$, $a>0$.

Motivation & Objective

• To generalize classical Hörmander-type multiplier theorems from spherical transforms on rank one symmetric spaces to the broader framework of Jacobi analysis.
• To derive near-optimal $L^p$-$L^q$ bounds for the Riesz potential operator associated with the kernel $m_a(\lambda) = (\lambda^2 + \rho^2)^{-a/2}$, $a > 0$.
• To unify multiplier results across diverse settings, including rank one Riemannian symmetric spaces, Damek–Ricci spaces, and $BC$ root system Heckman–Opdam transforms.
• To establish a transference principle for Jacobi convolution operators by leveraging the hypergroup structure, overcoming limitations of classical convolution methods.

Proposed method

• Utilizes precise asymptotic expansions of Jacobi functions $\varphi^{(\alpha,\beta)}_{\lambda}$ for complex parameters satisfying $\Re\alpha > \frac{1}{2}$, $\Re\alpha > \Re\beta > -\frac{1}{2}$.
• Applies a transference principle for hypergroups, adapted from [14], to relate norm estimates on the Jacobi hypergroup to those on Euclidean space.
• Employs Laplace-type integral representations and Bessel function estimates to control growth and decay of spectral derivatives.
• Derives error estimates for truncated series expansions of $\varphi_{\lambda}(t)$ via bounds on coefficients $a_m(t)$ and Gamma function asymptotics.
• Uses $L^p$-boundedness of the Euclidean Hörmander–Mikhlin multiplier theorem as a key tool, transferred back to the Jacobi setting via the hypergroup structure.
• Establishes holomorphic extension of multipliers into a suitable strip in $\mathbb{C}$, a necessary condition for multiplier theorems.

Experimental results

Research questions

• RQ1 Can classical Hörmander-type multiplier theorems for spherical transforms be extended to the Jacobi transform framework?
• RQ2 What are the sharp $L^p$-$L^q$ mapping properties of Riesz potentials $m_a(\lambda) = (\lambda^2 + \rho^2)^{-a/2}$ in the Jacobi setting?
• RQ3 How can transference principles from Euclidean Fourier analysis be adapted to the non-commutative, non-abelian structure of Jacobi hypergroups?
• RQ4 To what extent do asymptotic expansions of Jacobi functions $\varphi^{(\alpha,\beta)}_{\lambda}$ enable uniform treatment of diverse symmetric spaces and root systems?

Key findings

• Near-optimal $L^p$-$L^q$ estimates are obtained for the Riesz potential operator associated with $m_a(\lambda) = (\lambda^2 + \rho^2)^{-a/2}$, $a > 0$, extending results from [27] to the Jacobi setting.
• The multiplier theorem for the Jacobi transform is established under the condition $\Re\alpha > \frac{1}{2}$, $\Re\alpha > \Re\beta > -\frac{1}{2}$, with multipliers extending holomorphically into a suitable strip in $\mathbb{C}$.
• The method yields uniform results across rank one symmetric spaces, Damek–Ricci spaces, and $BC$ root system Heckman–Opdam transforms, subsuming prior results.
• Error estimates for the spectral expansion of $\varphi_{\lambda}(t)$ are shown to decay as $|E_{M+1}(\lambda t)| \lesssim t^{2(M+1)}|\lambda|^{-(\Re\alpha + M + 1)}$, with uniform bounds on the constant $c_M$.
• Derivatives of $\varphi_{\lambda}(t)$ with respect to $\lambda$ satisfy $\left|\frac{d^n}{d\lambda^n}\varphi_{\lambda}(t)\right| \lesssim (1+t)^{n+1}e^{(|\Im\lambda| - \Re\rho)t}$, ensuring control in the spectral parameter.
• The proof technique, based on asymptotic expansions and hypergroup transference, provides an alternative to Abel transform methods, particularly useful in settings where the Abel transform is poorly behaved.