Pseudo-differential operator involving generalized Hankel-Clifford transformation (Q2806098)

The authors consider an extension of generalized Hankel-Clifford transformations \( h_{1,\alpha, \beta} \) and \( h_{2,\alpha, \beta} \) studied in [\textit{S. P. Malgonde} and \textit{S. R. Bandewar}, Proc. Indian Acad. Sci., Math. Sci. 110, No. 3, 293--304 (2000; Zbl 0964.46024)].NEWLINENEWLINESince the extension is obtained by the means of a symbol \(a (x,y) \in C^\infty ((0,\infty) \times (0,\infty) ) \), the resulting transformations are called pseudo-differential operators (p.d.o). The symbol decays as NEWLINE\[NEWLINE (1+x)^q | D^r _y D^s _x a(x,y) | \leq C (1+y)^{l-s} NEWLINE\]NEWLINE for a fixed real number \(l\) and for any non-negative integers \(q,r,s\), where \(C\) depends on \(l,q,r,s\). Then the p.d.o \(h_{1,\alpha, \beta,a}\) is given by NEWLINE\[NEWLINE (h_{1,\alpha, \beta,a} \varphi ) (x) = x^{-\alpha-\beta} \int_0 ^\infty C_{\alpha, \beta} (xy) a(x,y) (h_{1,\alpha, \beta} \varphi ) (y) \,dy, NEWLINE\]NEWLINE and similarly for \(h_{2,\alpha, \beta,a} \).NEWLINENEWLINEThe authors prove the continuity of the action of \(h_{1,\alpha, \beta,a}\) and \(h_{2,\alpha, \beta,a}\) on appropriate spaces of test function spaces and give an easy integral representation formula.