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Homogeneous operators and homogeneous integral operators
  • Zhirayr Avetisyan,
  • Alexey Karapetyants
Zhirayr Avetisyan
Southern Federal University

Corresponding Author:jirayrag@gmail.com

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Alexey Karapetyants
Southern Federal University
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We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure - measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples - classes of homogeneous integral operators on various metric spaces, and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article - various specific cases - illustrate general statements and results given in the paper and at the same time are of interest in their own way.
10 Oct 2021Submitted to Mathematical Methods in the Applied Sciences
11 Oct 2021Submission Checks Completed
11 Oct 2021Assigned to Editor
05 Nov 2021Reviewer(s) Assigned
31 Jan 2022Review(s) Completed, Editorial Evaluation Pending
03 Feb 2022Editorial Decision: Revise Minor
20 Feb 20221st Revision Received
22 Feb 2022Assigned to Editor
22 Feb 2022Submission Checks Completed
22 Feb 2022Reviewer(s) Assigned
18 May 2022Review(s) Completed, Editorial Evaluation Pending
21 May 2022Editorial Decision: Revise Minor
23 May 20222nd Revision Received
24 May 2022Submission Checks Completed
24 May 2022Assigned to Editor
08 Jun 2022Reviewer(s) Assigned
17 Jun 2022Review(s) Completed, Editorial Evaluation Pending
26 Jun 2022Editorial Decision: Accept
15 Jan 2023Published in Mathematical Methods in the Applied Sciences volume 46 issue 1 on pages 811-829. 10.1002/mma.8549