The sharpness of condition for solving the jump problem

Abstract:

Let γ be a non-rectifiable closed Jordan curve in C, which is merely assumed to be d-summable (1 < d < 2) in the sense of Harrison and Norton [7]. We are interested in the so-called jump problem over γ, which is that of finding an analytic function in C having a prescribed jump across the curve. The goal of this note is to show that the sufficient solvability condition of the jump problem given by ν > d/2 , being the jump function defined in and satisfying a Hölder condition with exponent ν, 0 < ν ≤ 1, cannot be weakened on the whole class of dsummable curves.

Año de publicación:

2012

Keywords:

• Cauchy integral
• Non-rectifiable curve
• Fractional dimension
• jump problem
• Analytic functions

Fuente:

scopus