Multipliers of Hardy Spaces Associated with Generalized Hermite Expansions

The purpose of this paper is to study coefficient multipliers of the Hardy spaces H p ( R ) associated with Hermite expansions. The main results are that, if a sequence { λ n } n = 0 ∞ satisfies the condition ∑ k = n 2 n | λ k | q = O n q 2 7 6 - 1 p , then { λ n } is a multiplier of H p ( R ) into the sequence space ℓ q associated with Hermite expansions for (i) p = 1 , 2 ≤ q < ∞ ; (ii) 0 < p < 1 ≤ q < ∞ . As a consequence, a Paley-type inequality is obtained; that is, for a Hadamard sequence { n k } satisfying n k + 1 / n k ≥ ρ > 1 and for f ∈ H p ( R ) , 0 < p ≤ 1 , the coefficients a n ( f ) of its Hermite expansion satisfy ∑ k = 1 ∞ n k 7 6 - 1 p | a n k ( f ) | 2 < ∞ . The results in the paper are proved in a more general case, that is, for the generalized Hermite functions which are defined by H 2 k ( λ ) ( x ) = c k L k ( λ - 1 / 2 ) ( x 2 ) e - x 2 2 | x | λ , H 2 k - 1 ( λ ) ( x ) = c k k - 1 / 2 x L k - 1 ( λ + 1 / 2 ) ( x 2 ) e - x 2 2 | x | λ , where c k = 2 k ! / Γ ( k + λ + 1 / 2 ) 1 / 2 . Note that H n ( x ) = H n ( 0 ) ( x ) ( n ≥ 0 ) are the usual Hermite functions.