Local and weighted Marcinkiewicz exponents with applications

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.

“…() 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

“…It remains to study points ±i. The calculations in [8] show that m − (Γ ; +i) = 1 − β − 1 (β + 1)α , m + (Γ ; +i) = 2 β + 1 , and, vice versa,…”

Section: Proofs and Commentariesmentioning

confidence: 98%

“…Recently the second author introduced new metric characteristics for non-rectifiable curves and call them Marcinkiewicz exponents (see [7][8][9]). These characteristics are local, what allows us to extend technique of works [4,5] on the Riemann surfaces.…”

Section: Introductionmentioning

confidence: 99%

New solvability criterion for jump problem

2016

Bol. Soc. Mat. Mex.

Self Cite

“…In what follows, we need two local characteristics of curve Γ. The first family of characteristics was introduced in papers) and called the Marcinkiewicz exponents. Let r > 0, p > 0, B ( t ; r ): = { z :| z − t | < r },

B^{\pm} false(t; r false) : = B false(t; r false) ⋂ D^{\pm}

I_{p}^{\pm} false(t; r false) = \iint_{B^{\pm} false(t; r false)} \frac{d x d y}{{dist}^{p} false(x + i y, normalΓ false)}

.…”

Section: Jump Discontinuitiesmentioning

confidence: 99%

“…The properties of these characteristics are described in papers false[2−truedm‾γ,1false], where γ is any neighborhood of t in Γ, and

\overset{dm}{true} γ

is upper Minkowski dimension (see, for instance, Falconer) of γ .…”

Section: Jump Discontinuitiesmentioning

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.