Products and transforms of white-noise functionals (in general setting) (Q1344941)

Let \((E)_{p, \beta} = \{\varphi \in (E)^* : \Gamma (\beta) \varphi \in (E)_ p\}\) be a set of Sobolev spaces, \(p \in R\) and \(\beta > 0\), where \((E)^*\) is the space of Hida distributions and \(\Gamma (\beta)\) is the second quantization of the scalar multiplier \(\beta \cdot\). Let \(\varphi \sim (f_ n)\) and \(\psi \sim (g_ n)\) be two elements of \((E)^*\). Then \[ \varphi \psi \sim (h_ n), \quad h_ l = \sum_{k = 0}^ \infty k! \sum_{m + n = l} {m + k \choose k} {n + k \choose k} f_{m + k} \widehat \otimes_ k g_{n + k} \] is called a Wiener product of \(\varphi\) and \(\psi\). It holds \[ \| \varphi \psi \|_{q, \alpha} \leq \bigl( 1 - C(p,q), \beta_ 1, \beta_ 2, \alpha) \bigr)^{- 1/2} \| \varphi \|_{p, \beta_ 1} \| \psi \|_{q, \beta_ 2},\qquad | q | \leq p,\;\beta_ 1, \beta_ 2 > 0,\;\beta_ 1 \beta_ 2 > p^{p + q}. \] Another product \(h_ n = \sum_{k_ j = n}\) \(f_ n \widehat \otimes g_ j\), called Wick product \(\varphi : \psi\) of \(\varphi, \psi\) is defined. It holds \[ \| \varphi : \psi \|_{p, \alpha} \leq \| \varphi \|_{p, \beta_ 1} \cdot \| \psi \|_{p, \beta_ 2}, \quad \beta_ 1, \beta_ 2 > \alpha > 0,\;\beta_ 1^{-2} + \beta_ 2^{-2} = a^{-2}. \] The author also considers scaling transforms and translations with corresponding differentiations.