## Communications in Mathematical Analysis

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

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

R. A. Blaya, T. M. García, and J. B. Reyes

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR2905129

**Zentralblatt MATH identifier**

1262.30032

**Subjects**

Primary: 30E25: Boundary value problems [See also 45Exx]

Secondary: 28A80: Fractals [See also 37Fxx]

**Keywords**

Non-rectifiable curve Fractional dimension jump problem Cauchy integral analytic functions

#### Citation

Blaya, R. A.; Reyes, J. B.; García, T. M. The Sharpness of Condition for Solving the Jump Problem. Commun. Math. Anal. 12 (2012), no. 2, 26--33. https://projecteuclid.org/euclid.cma/1331929869