Some remarks about Cauchy integrals and totally real surfaces in C^m

Some remarks about Cauchy integrals and fractal sets

Some aspects of Cauchy integrals on sets with dimension larger than 1 are briefly discussed.

Some remarks about Cauchy integrals

This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.

Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.

Some remarks concerning integrals of curvature on curves and surfaces

In these notes we discuss some relations between complex analysis (derivatives of Cauchy integrals) and curvatures of curves and surfaces. In higher dimensions the Cauchy integrals are based on generalizations of complex analysis using quarternions or Clifford algebras. These topics came up in "Analysis of and on Uniformly Rectifiable Sets" (with Guy David, American Mathematical Society, 1993), and here we describe some of the basic points involved. A presentation based on this paper was made at the AMS Special Session "Surface Geometry and Shape Perception" (Hoboken, 2001).

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function havi…

关于 Tonelli 变分法的一个评注

A remark on Tonelli's calculus of variations

本文用有限维逼近和 Euler–Cauchy 折线收敛，为正定超线性 Lagrangian 下 Tonelli 变分问题极小元的存在性与正则性提供初等证明。