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

TL;DR

This paper constructs a variable exponent Lebesgue space where the Cauchy-Leray-Fantappiè operator for complex ellipsoids is unbounded, showing the sharpness of the logarithmic continuity condition for the exponent function.

Contribution

It provides the first explicit counterexample demonstrating unboundedness of the operator on non-strictly convex domains with variable exponents.

Findings

The operator is unbounded on the constructed space.

The logarithmic continuity condition for $p( ext{·})$ is necessary.

Unboundedness occurs near boundary points where convexity fails.

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.