Discrete Harmonic Analysis Associated with Ultraspherical Expansions

• Published in: Potential analysis 2020-08, Vol.53 (2), p.523-563
• Main Authors: Betancor, Jorge J., Castro, Alejandro J., Fariña, Juan C., Rodríguez-Mesa, L.
• Format: Article
• Language: English
• Subjects: Eigenvectors; Finite differences; Fourier analysis; Functional Analysis; Geometry; Harmonic analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Orthogonal functions; Potential Theory; Probability Theory and Stochastic Processes; Transplantation
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-019-09777-9

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted ℓ p -boundedness properties of maximal operators and Littlewood-Paley g -functions defined by Poisson and heat semigroups generated by the difference operator Δ λ f ( n ) : = a n λ f ( n + 1 ) − 2 f ( n ) + a n − 1 λ f ( n − 1 ) , n ∈ ℕ , λ > 0 , where a n λ : = { ( 2 λ + n ) ( n + 1 ) / [ ( n + λ ) ( n + 1 + λ ) ] } 1 / 2 , n ∈ ℕ , and a − 1 λ : = 0 . We also prove weighted ℓ p -boundedness properties of transplantation operators associated with the system { φ n λ } n ∈ ℕ of ultraspherical functions, a family of eigenfunctions of Δ λ . In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.