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.
Citation
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.
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