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Kernel Operators in Some New Function Spaces

Thesis Info

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Author

Zaighum, Muhammad Asad

Supervisor

Alexander Meskhi

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2009

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2740/1/2905S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726566511

Similar


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.
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