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2012 The Sharpness of Condition for Solving the Jump Problem
R. A. Blaya, T. M. García, J. B. Reyes
Commun. Math. Anal. 12(2): 26-33 (2012).


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.


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R. A. Blaya. T. M. García. J. B. Reyes. "The Sharpness of Condition for Solving the Jump Problem." Commun. Math. Anal. 12 (2) 26 - 33, 2012.


Published: 2012
First available in Project Euclid: 16 March 2012

zbMATH: 1262.30032
MathSciNet: MR2905129

Primary: 30E25
Secondary: 28A80

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

Rights: Copyright © 2012 Mathematical Research Publishers

Vol.12 • No. 2 • 2012
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