L^p-results for fractional integration and multipliers for the Jacobi transform

2011/08/17 by Troels Roussau Johansen, Johansen, Troels Roussau
Mathematics · #20N20 #33C05 #34E05 #42A45 (secondary) #44A35 (primary) #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1108.3478

openalex publication_date 2011/08/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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↦(λ²+ρ²)^-a/2, a>0.

Related

• On a Class of Non-Integrable Multipliers for the Jacobi Transform
• Riemann-Liouville Operator in Weighted L_p Spaces via the Jacobi Series Expansion
• Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein g_k-functions and the Laplace type multipliers
• Fractional Integration and Fractional Differentiation for d-dimensional Jacobi Expansions
• Discrete Harmonic Analysis associated with Jacobi expansions II: the Riesz transform