On a supremum operator

Denote by germ{M}^{downarrow} the set of all non-negative, non-increasing functions on [0,infty). Let Phi(x,y) be a measurable non-increasing function on {(x,y):xge yge 0} and define the supremum operator R by Rvarphi(t)coloneq operatornamewithlimits{roman{esssup},}limits_{yin[0,infty)}Phi(y,t)varphi(y),quad varphiingerm{M}^{downarrow}. The paper deals with the weighted L_p-L_q boundedness of the operator R, i.e. with the inequality left(int_0^infty[Rvarphi(t)]^qw(t) dtright)^{1/q}le Cleft(int_0^inftyvarphi^p(t)v(t) dtright)^{1/p}, quadvarphiingerm{M}^{downarrow}, with non-negative locally integrable weight functions v and w, and a constant Cge 0 independent of varphi. Using the technique developed by A. Gogatishvili, B. Opic and L. Pick [Collect. Math. {bf 57} (2006), no.~3, 227--255; [msn] MR2264321 (2007g:26019) [/msn]], the least possible constant C is determined in the case 0