Communications in Mathematical Analysis

The Sharpness of Condition for Solving the Jump Problem

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

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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

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

First available in Project Euclid: 16 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30E25: Boundary value problems [See also 45Exx]
Secondary: 28A80: Fractals [See also 37Fxx]

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


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.

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