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arXiv · 2511.05997

An example of a space $L^{p(\cdot)}$ on which the Cauchy-Leray-Fantappi\`{e} operator for complex ellipsoid is not bounded

Aleksandr Rotkevich
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Abstract

We construct an example of a Lebesgue space with variable exponent on which Cauchy-Leray-Fantappi\`{e} operator associated with a complex ellipsoid is not bounded. This result extends previous counterexamples for the unit ball and demonstrates that the logarithmic continuity condition for the exponent function $p(\cdot)$ is sharp even for non-strictly convex domains. The proof is based on an explicit construction of test functions supported near points where the boundary fails to be strictly convex.

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Aleksandr Rotkevich. 2025-11-08. An example of a space $L^{p(\cdot)}$ on which the Cauchy-Leray-Fantappi\`{e} operator for complex ellipsoid is not bounded. https://arxiv.org/abs/2511.05997

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