On Certain Integral Operators of Fractional Type

In this paper we study integral operators of the form $$T,fleft( x right) = int {k_1 left( {x - a_1 y} right)k_2 left( {x - a_2 y} right)...k_m left( {x - a_m y} right)fleft( y right)dy} ,$$ $$k_i left( y right) = sumlimits_{j in Z} {2^{frac{{jn}}{{q_i }}} } varphi _{i,j} left( {2^j y} right),,1 leqq q_i < infty ,frac{1}{{q_1 }} + frac{1}{{q_2 }} + ... + frac{1}{{q_m }} = 1 - r,$$ $0 leqq r < 1$ , and $varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:L^p left( {R^n } right) to T:L^q left( {R^n } right)$ for $1 < p < frac{1}{r}$ and $frac{1}{q} = frac{1}{p} - r$ .