Positivity 13 (2009), 497–518
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/030497-22, published online October 28, 2008
Boundedness and compactness
of positive integral operators on cones
of homogeneous groups
Usman Ashraf, Muhammad Asif and Alexander Meskhi
Abstract. Necessary and suﬃcient conditions on a weight function v guaran-
teeing the boundedness/compactness of integral operators with positive ker-
nels deﬁned on cones of homogeneous groups from L
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.
Keywords. Operators with positive kernels, potentials, homogeneous groups,
trace inequality, weights, singular numbers of kernel operators.
In this paper boundedness/compactness criteria from L
(E) are estab-
lished for the operator
k(x, y)f(y)dy, x ∈ E, (1)
with positive kernel k, where 1 <p,q<∞ or 0 <q≤ 1 <p<∞, E
are certain cones in homogeneous groups and k satisﬁes the conditions which in
one-dimensional case are similar to those of .
A full characterization of pair of weights (v, w) governing the boundedness
of integral operators with positive kernels from L
been established in (seealso, Ch.3). Criteria guaranteeing the bounded-
ness/compactness of the operator
(x − t)
f(t)dt, x > 0,
The work was partially supported by the INTAS grant No. 05-1000008-8157 and the Georgian
National Foundation Grant No. GNSF/ST06/3-010.