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The Sharpness of Condition for Solving the Jump Problem

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

The Sharpness of Condition for Solving the Jump Problem

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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 i\gamma 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 $\displaystyle \nu > \frac{d}{2}$, being the jump function defined in γ and satisfying a Hölder condition with exponent $0<\nu\leq 1$, cannot be weakened on the whole class of d-summable curves.