Multipliers of Hardy Spaces Associated with Generalized Hermite Expansions

Abstract

The purpose of this paper is to study coefficient multipliers of the Hardy spaces $H^p({\mathbb R})$ associated with Hermite expansions. The main results are that, if a sequence $\{\lambda _n\}_{n=0}^{\infty }$ satisfies the condition $\sum _{k=n}^{2n}|\lambda _k|^q=O\left( n^{\frac{q}{2}\left( \frac{7}{6}-\frac{1}{p}\right) }\right)$, then $\{\lambda _n\}$ is a multiplier of $H^p({\mathbb R})$ into the sequence space $\ell ^q$ associated with Hermite expansions for (i) $p=1$, $2\le q<\infty$; (ii) $0<p<1\le q<\infty$. As a consequence, a Paley-type inequality is obtained; that is, for a Hadamard sequence $\{n_k\}$ satisfying $n_{k+1}/n_k\ge \rho >1$ and for $f\in H^p({\mathbb R})$, $0<p\le 1$, the coefficients $a_n(f)$ of its Hermite expansion satisfy $\sum _{k=1}^{\infty }n_k^{\frac{7}{6}-\frac{1}{p}}|a_{n_k}(f)|^2<\infty$. The results in the paper are proved in a more general case, that is, for the generalized Hermite functions which are defined by ${\mathcal {H}}_{2k}^{(\lambda )}(x)=c_kL_k^{(\lambda -1/2)}(x^2)e^{-\frac{x^2}{2}}|x|^{\lambda }$, ${\mathcal {H}}_{2k-1}^{(\lambda )}(x)=c_k k^{-1/2}xL_{k-1}^{(\lambda +1/2)}(x^2)e^{-\frac{x^2}{2}}|x|^{\lambda }$, where $c_k=\left( 2k!/\Gamma (k+\lambda +1/2)\right) ^{1/2}$. Note that ${\mathcal {H}}_n(x)={\mathcal {H}}_n^{(0)}(x)$ ($n\ge 0$) are the usual Hermite functions.