Boundedness and compactness of positive integral operators on cones of homogeneous groups

Abstract

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.

Kernel operators on the upper half-space: boundedness andcompactness criteria

“…Using Minkowski integral inequality and taking into account the main results of this paper and [3], it can be checked that ifk ∈V p and k ∈ V p , where k is defined by (8), then the boundedness/compactness of…”

“…for all x ∈ E . Using Minkowski integral inequality and taking into account the main results of this paper and [3], it can be checked that ifk ∈V p and k ∈ V p , where k is defined by (8), then the boundedness/compactness of K from L p (E) to L q (E) implies the boundedness/compactness ofK from L p (E) to L q (Ê) . Furthermore, if q ≤ p ′ , andk ∈V p , then k ∈ V p , where k is defined by (8).…”

“…The problems studied in this paper can be considered as a natural continuation of the investigation carried out in [3] (see also [21], Ch. 3), where the authors derived the similar results for the operator…”