Coefficient multipliers in the Hardy space associated with Jacobi expansions

Abstract

In this paper a multiplier theorem in the Hardy space H 1 ( T ) H^1(\mathbb {T}) associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence { λ n } n = − ∞ ∞ \left \{\lambda _n\right \}_{n=-\infty }^{\infty } is a multiplier from H 1 ( T ) H^1(\mathbb {T}) into the sequence space ℓ 1 ( Z ) \ell ^1(\mathbb {Z}) associated with Jacobi expansions of exponential type, if \[ sup N ∑ k = 1 ∞ ( ∑ k N > | j | ≤ ( k + 1 ) N | λ j | ) 2 > ∞ . \sup _N\sum _{k=1}^{\infty }\left (\sum _{kN>|j|\le (k+1)N}|\lambda _j|\right )^2>\infty . \] This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space H 1 ( T ) H^1(\mathbb {T}) , and for λ n = ( | n | + 1 ) − 1 \lambda _n=(|n|+1)^{-1} , a Hardy type inequality for Jacobi expansions is immediate which has ever been proved by Kanjin and Sato [Math. Inequal. Appl. 7 (2004), pp. 551–555].