Convolution operators on groups and multiplier theorems for Hermite and Laguerre expansions

Abstract:

Using harmonic analysis on nilpotent Lie groups the following theorem is proved. Let a sequence $\{ {a_{\text {n}}}\}$ be defined by a function $K \in {C^N}({{\mathbf {R}}_ + })$ such that ${\sup _{\lambda > 0}}|{K^{(j)}}(\lambda ){\lambda ^j}| < \infty ,j = 0,1, \ldots ,N$, for $N$ sufficiently large, putting ${a_{\text {n}}} = K(|{\text {n}}| + m/2)$. Let ${\varphi _{\text {n}}}$ be either Hermite or Laguerre functions. Then the operator \[ \sum \limits _{\text {n}} {(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}} \to \sum \limits _{\text {n}} {{a_{\text {n}}}(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}}} } \] is bounded on ${L^p}\left ( {{\mathbb {R}^m}} \right )$ or ${L^p}(\mathbb {R}_ + ^m)$ respectively, $1 < p < \infty$.