Marcinkiewicz exponents and integrals over non‐rectifiable paths

Math Methods in App Sciences

2018

In this paper, we obtain a Cauchy‐type integral representation to solve certain Beltrami equations and apply it for research of the Riemann boundary value problem for that equations on nonrectifiable contours. While more or less studied for piecewise smooth contours, for the case of nonrectifiable curve, this is a pioneer result. There is only one relatively simple type of Beltrami equation that is solved under such condition (see below). Our next goal is to achieve solutions for more types of the Beltrami equations on nonrectifiable curves.

“…Let us describe briefly the scheme of solving the jump problem from Kats and Katz. () If j ∈ H (Γ, ν ), then it is representable as sum

φ = true \sum_{k = 1}^{n} φ_{k}

, where

supp φ_{k} \subset normalΓ

is sufficiently small and contains a point t k such that φ k ∈ H (Γ, ν ( t k )). Let

φ_{k}^{w}

be a Whitney extension of φ k on the whole complex plane (see, for instance, Stein).…”

Section: Integration Over Nonrectifiable Pathsmentioning

confidence: 99%

“…() This problem for nonrectifiable curves was solved for the first time by Kats. () Then the conditions of its solvability were improved by Katz() (see also previous studies()).…”

Section: Introductionmentioning

confidence: 99%

Riemann boundary value problem on nonrectifiable curves for certain Beltrami equations

Math Methods in App Sciences

2018

“…There exist publications dealing with this subject . The present paper continues the work , but unlike the last one, we define the generalized integration as interaction of germs with supports on non‐rectifiable curves (in , it is defined as distribution). We consider certain its properties and applications.…”

mentioning

confidence: 83%

“…Clearly, this mapping

W^{1} (double-struckC ∖ normal Γ) \times C_{0}^{1} \mapsto double-struckC

is continuous in intrinsic sense. In the paper , it is considered for fixed F as mapping

C^{\infty} (double-struckC) \mapsto double-struckC

, that is, as a distribution.…”

Section: Germs and Integrationsmentioning

confidence: 99%

Interactions of germs with applications

2017

Math. Meth. Appl. Sci.

“…This approach was developed in papers and certain others. Its contemporary state is described in previous studies . All these works concerns mainly the integration of Hölder functions.…”

Section: Introductionmentioning

confidence: 99%

Curvilinear integrals of discontinuous functions over nonrectifiable paths and Riemann boundary‐value problem

Math Methods in App Sciences

2019

Self Cite

We study a generalization of the concept of curvilinear integral for the case where path of integration is nonrectifiable, and integrand has discontinuities. Then we apply that integrals for solving of the Riemann boundary‐value problem for domains with nonrectifiable boundaries and discontinuous boundary data. The research is supported by RFBR (grant 18‐31‐00060) and is performed according to a special programme of the Russian government supporting research at Kazan Federal University. Mathematical Methods in the Applied Sciences journal encourages authors to share the data and other artifacts supporting the results in the paper by archiving it in an appropriate public repository. Authors may provide a data availability statement, including a link to the repository they have used, in order that this statement can be published in their paper. Shared data should be cited.