Abstract:
The thesis is devoted to the weighted criteria for integral operators with positive ker-
nels in variable exponent Lebesgue and amalgam spaces. Similar results for multiple
kernel operators defined with respect to a Borel measure in the classical Lebesgue
spaces are also obtained. More precisely, we established necessary and sufficient
conditions on a weight function v governing the boundedness/compactness of the
weighted positive kernel operator Kv f (x) = v(x)
x
0
k(x, y)f (y)dy from Lp(·) (R+ ) to
Lq(·) (R+ ) under the local log-H ̈lder continuity condition and the decay condition at
o
infinity on the exponents p and q. In the case when Kv is bounded but not compact,
two-sided estimates of the measure of non-compactness (essential norm) for Kv are
obtained in terms of the weight v and kernel k. Criteria guaranteeing the boundedness
/compactness of weighted kernel operators defined on R+ (resp. on R) in variable ex-
ponent amalgam spaces are found. The kernel operators under consideration involve,
x
for example, the Riemann-Liouville transform Rα f (x) =
0
f (t)
dt,
(x−t)1−α
0 < α < 1.
Necessary and sufficient conditions ensuring weighted estimates for maximal and po-
tential operators in variable exponent amalgam spaces are also established under the
local log-H ̈lder continuity condition on exponent of spaces. Further, we establish
o
criteria on measures governing the boundedness of integral operators with product
positive kernels defined with respect to a Borel measure in the classical Lebesgue
spaces. Finally, we point out that Fefferman-Stein type inequality for the multi-
ple Riemann-Liouville transform defined with respect to a product Borel measure is
derived.
