## Communications in Mathematical Analysis

### The Sharpness of Condition for Solving the Jump Problem

#### Abstract

Let $\gamma$ be a non-rectifiable closed Jordan curve in $\mathbb{C}$, 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 $\gamma$, which is that of finding an analytic function in $\mathbb{C}$ 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 $\displaystyle \nu > \frac{d}{2}$, being the jump function defined in $\gamma$ and satisfying a Hölder condition with exponent $\nu$, $0<\nu\leq 1$, cannot be weakened on the whole class of $d$-summable curves.

#### Article information

Source
Commun. Math. Anal., Volume 12, Number 2 (2012), 26-33.

Dates
First available in Project Euclid: 16 March 2012

https://projecteuclid.org/euclid.cma/1331929869

Mathematical Reviews number (MathSciNet)
MR2905129

Zentralblatt MATH identifier
1262.30032

Subjects