Sopra un teorema d'intercambio

Trione, Susana Elena

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 59 (1975) no. 5, pp. 357-361

Résumé

Let $\alpha,\beta \in \mathbf{C}$, $\alpha+\beta=n+2h$, $\alpha \ne n+2h$, $\beta \ne n+2h$, $h=0,1,\cdots$. We prove under these conditions, the formula of interchange of the Fourier transformation of convolution of $Pf (H_{\alpha}(P \pm i 0,n) \ast H_{\beta} (P \pm i 0,n))$ into the product of their Fourier trasforms: $$\{ Pf (H_{\alpha}(P \pm i 0,n) \ast H_{\beta} (P \pm i 0, n)) \}^{\Lambda} = \{ H_{\alpha}(P \pm i 0,n) \}^{\Lambda} \cdot \{ H_{\beta}(P \pm i 0,n) \}^{\Lambda}$$ (see, for the definitions of these notations, formulae (1), (1') and Theorem). As an immediate consequence of formula (2) we obtain $$\{ Pf ((P \pm i 0,n)^{\frac{1}{2}t} \ast (P \pm i 0,n)^{\frac{1}{2}s}) \}^{\Lambda} = \{ (P \pm i 0,n)^{\frac{1}{2}t} \}^{\Lambda} \cdot \{ (P \pm i 0,n)^{\frac{1}{2}s} \}^{\Lambda},$$ where $t+s =-n+2h$, $t \ne 2h$, $s \ne 2h$, $h = 0,1,\cdots$. It may be observed that, in the particular case $p=n$, $q = 0$, the distributions $H_{\alpha} (P \pm i 0, n)$ turn out to be the elliptic M. Riesz kernel of which they are "causal" ("anticausal") analogues; and from formula (18), we arrive at $\{ Pf (r^{t} \ast r^{s}) \}^{\Lambda} = \{ r^{t}\}^{\Lambda} \{ r^{s} \}^{\Lambda}$, which is valid for $t+s = - n + 2h$, $t \ne 2h$, $s \ne 2h$, $h = 0,1,\cdots$. The last formula is an extension of formula (VII, 8; 8), p. 271, obtained by L. Schwartz (cf. [6]).