Necessary and sufficient conditions on a weight function v guaranteeing the boundedness/compactness of integral operators with positive kernels defined on cones of homogeneous groups from L p to L q v are established, where 1 < p, q < ∞ or 0 < q ≤ 1 < p < ∞. Behavior of singular numbers for these operators is also studied. Mathematics Subject Classification (2000). Primary 26A33, 42B25; Secondary 43A15, 46B50, 47B10, 47B34.

Abstract. Necessary and sufficient conditions on a weight governing the trace inequality for the Riemann-Liouville transform with variable parameter R α(x) in L p(x) spaces are established provided that p and q satisfy the log-Hölder continuity condition. Weighted criteria for theare also derived.Mathematics subject classification (2010): 46E30, 47B34.

We establish necessary and sufficient conditions on a weight v governing the trace inequalitywhere E is a cone on a homogeneous group,Ê := E ×R+ andK is a positive kernel operator defined onÊ . Compactness criteria for this operator are also established.