Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Aleksandr Rotkevich

TL;DR

The paper constructs a counterexample demonstrating that the Cauchy-Leray-Fantappi\`e operator on the complex ellipsoid $\mathcal{E}=\{ |z_1|^{2m_1}+\cdots+|z_n|^{2m_n}<1\}$ is not bounded on the variable-exponent Lebesgue space $L^{p(\cdot)}(\partial\mathcal{E})$ when the exponent fails the logarithmic continuity condition. It extends prior unit-ball counterexamples to domains with degenerate boundary geometry by explicitly analyzing the kernel $w(\xi,z)$ and constructing test functions supported near boundary points where strict convexity fails. The authors establish precise kernel estimates on thin boundary patches $W_\alpha$ and $V_\alpha$, showing that the image of these test functions can be forced to grow in a way that defeats $L^{p(\cdot)}$-boundedness. The key technical contribution is a careful, explicit construction of an exponent function $p(\cdot)$ via a modulating function $\psi$, together with a disjoint-patch accumulation of test functions that yields $h\in L^{p(\cdot)}(\partial\mathcal{E})$ but $Kh\notin L^{p(\cdot)}(\partial\mathcal{E})$, thus proving sharpness of the logarithmic continuity condition even for non-uniformly convex domains.

Abstract

We construct an example of a Lebesgue space with variable exponent on which Cauchy-Leray-Fantappiè 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.

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

TL;DR

The paper constructs a counterexample demonstrating that the Cauchy-Leray-Fantappi\`e operator on the complex ellipsoid is not bounded on the variable-exponent Lebesgue space when the exponent fails the logarithmic continuity condition. It extends prior unit-ball counterexamples to domains with degenerate boundary geometry by explicitly analyzing the kernel and constructing test functions supported near boundary points where strict convexity fails. The authors establish precise kernel estimates on thin boundary patches and , showing that the image of these test functions can be forced to grow in a way that defeats -boundedness. The key technical contribution is a careful, explicit construction of an exponent function via a modulating function , together with a disjoint-patch accumulation of test functions that yields but , thus proving sharpness of the logarithmic continuity condition even for non-uniformly convex domains.

Abstract

We construct an example of a Lebesgue space with variable exponent on which Cauchy-Leray-Fantappiè 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 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.

Paper Structure

This paper contains 5 sections, 5 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Principal notations
  3. Boundedness of Cauchy--Leray--Fantappiè integral operator for complex ellipsoid
  4. Estimates of the kernel of the Cauchy--Leray--Fantappiè integral operator and construction of the test functions
  5. The construction of the exponent such that the operator $K$ is not bounded in $L^{p(\cdot)}(\partial \mathcal{E}).$

Key Result

Theorem 1

Suppose $\mathcal{E}$ is a complex ellipsoid and $p:\partial \mathcal{E}\to (1,\infty)$ is a logarithmic continuous (or logarithmic Hölder) function, that is then the operator $K$ in eq1 maps $L^{p(\cdot)}(\partial\mathcal{E})$ continuously into itself.

Theorems & Definitions (10)

  • Theorem 1
  • Definition 2: Boundary patches.
  • Lemma 3: Measure of boundary patches
  • proof
  • Lemma 4: Estimate of the kernel on boundary patches
  • proof
  • Lemma 5: Estimate of the image of test functions
  • proof
  • Theorem 6
  • proof