Distorted Hankel integral operators

For $\text{a},\text{b}>0$ and for a locally integrable function (or, more generally, a distribution) $\text{f}$ on $(0,\text{be})$, we study integral ooperators ${\frak G}^{\a,\b}\f$on$L^2(\R+)$definedby$\big({\frak G}^{\a,\b}\f f\big)(x)=\int{\R_+}\f\big(x^\a+y^\b\big)f(y)dy$. We describe the bounded and compact operators ${\mathfrak{G}}_{\text{f}}^{\text{a},\text{b}}$ and operators ${\mathfrak{G}}_{\text{f}}^{\text{a},\text{b}}$ of Schatten–von Neumann class ${\text{bS}}_p$. We also study continuity properties of the averaging projection ${\text{Q}}_{\text{a},\text{b}}$ onto the operators of the form ${\frak G}^{\a,\b}\f$\mathrm{.}Inparticular,weshowthatif$\a\le\b$and$\b>1$,then${\frak G}^{\a,\b}\f$isboundedon$\bS_p$ifandonlyif$2\b(\b+1)^{-1}<p<2\b(\b-1)^{-1}$.