$L^2$-boundedness of the $n$-th Calderón commutator on Lipschitz graphs

Technical Summary: L2-Boundedness of the n-th Calderón Commutator on Lipschitz Graphs

Problem Statement
This paper investigates the $L^2$-boundedness of the $n$-th Calderón commutator, denoted $T_{A,n}$, associated with a Lipschitz graph $\Gamma = \{x + iA(x) : x \in \mathbb{R}\}$. The operator is defined via the principal value integral:
$$T_{A,n}f(x) = \text{p.v.} \int_{\mathbb{R}} \left( \frac{A(y) - A(x)}{y - x} \right)^n \frac{f(y)}{y - x} \, dy.$$
These operators arise from the formal expansion of the Cauchy kernel on Lipschitz graphs. The central problem is to establish quantitative bounds for the operator norm $\|T_{A,n}\|_{L^2 \to L^2}$ in terms of the order $n$ and the Lipschitz constant $\|A'\|_\infty$.

While the boundedness of the Cauchy integral was established by Coifman, McIntosh, and Meyer, the quantitative dependence on $n$ remains critical for summing the series expansion of the Cauchy kernel. Previous work by Verdera and Mateu (unpublished in full proof) suggested a linear growth bound of the form $C(1+n)\|A'\|_\infty^n$. However, it was unclear if this linear dependence on $n$ was sharp or if additional regularity on $A$ could yield sublinear growth.

Methodology
The authors employ two distinct analytical frameworks to derive their estimates:

1. Symmetrization and the Local $T(1)$ Theorem:
   For the general Lipschitz case, the authors utilize a symmetrization strategy introduced by Verdera. They analyze the testing condition $\|T_{A,n,\varepsilon}\chi_I\|_{L^2(I)}^2$ by decomposing the integration domain into a symmetric set $S_\varepsilon$ and an error set $E_\varepsilon$.

   • The error term is controlled directly using the Lipschitz property.
   • The symmetric term is rewritten using an algebraic identity involving a polynomial $W_n(\alpha, \beta, \gamma)$, revealing an underlying positivity structure (specifically for odd $n$) related to Menger curvature.
   • This approach allows the application of the local $T(1)$ theorem for antisymmetric Calderón-Zygmund operators.

2. Hörmander Conditions and Interpolation:
   To obtain sharper bounds under stronger regularity assumptions, the authors bypass the standard $T(1)$ theorem. Instead, they establish a Hörmander-type condition for the kernel $K_n(x,y)$ with constants growing logarithmically in $n$ (i.e., $O(\log n)$).

   • They prove that for odd $n$, the kernel satisfies a positivity property that facilitates better control.
   • Using the Fefferman-Stein interpolation theorem between $H^1 \to L^1$ and $L^\infty \to \text{BMO}$, they relate the $L^2$ bound to the local testing constants and the Hörmander constant.
   • A key structural reduction involves an auxiliary function $F_n(t, x) = \frac{1}{\|A'\|_\infty^n} \left( \frac{A(x+t) - A(x)}{t} \right)^n$. The problem is reduced to estimating the $L^2$ norm of the half-order fractional derivative $\partial_t^{1/2} F_n$.

Key Contributions and Results

• General Lipschitz Case (Theorem 2.1):
  The paper provides a complete proof of the estimate:
  $$\|T_{A,n}\|_{L^2 \to L^2} \leq C n \|A'\|_\infty^n.$$
  This formalizes a claim by Verdera and Mateu, confirming linear growth in $n$ for general Lipschitz functions. The proof relies on the symmetrization strategy and the local $T(1)$ theorem.

• Improved Bounds under Dini Regularity (Theorem 2.2):
  If the derivative $A'$ satisfies a Dini condition (i.e., $\int_0^1 \frac{\omega_{A'}(t)}{t} dt < \infty$), the growth in $n$ is improved to a sublinear rate:
  $$\|T_{A,n}\|_{L^2 \to L^2} \leq C \left( 1 + \sqrt{n} C_\omega + \log n \right) \|A'\|_\infty^n.$$
  This result demonstrates that slightly stronger regularity than Lipschitz continuity yields a significant improvement in the dependence on $n$.

• Improved Bounds under Besov Regularity (Theorem 2.4):
  The authors establish that if $A'$ belongs to the logarithmic Besov space $B^{0,1}_{1,1}(\mathbb{R})$, the bound improves to:
  $$\|T_{A,n}\|_{L^2 \to L^2} \leq C \sqrt{n} \|A'\|_\infty^n \left( 1 + \frac{\|A'\|_{B^{0,1}_{1,1}}}{\|A'\|_\infty} \right)^{1/2}.$$
  This space includes compactly supported functions in Sobolev spaces $H^s(\mathbb{R})$ for $0 < s < 1$ and functions of bounded variation (BV). The proof utilizes the reduction to the fractional derivative of the auxiliary function $F_n$ and Littlewood-Paley decomposition.

• Incomparability of Regularity Conditions (Section 8):
  The paper constructs counterexamples to demonstrate that the Dini condition (Theorem 2.2) and the fractional Sobolev/Besov condition (Corollary 6.3/Theorem 2.4) are incomparable.

  • There exists a function in $C^{1,\text{Dini}}$ whose derivative is not in $H^{1/2}$.
  • There exists a function with derivative in $H^{1/2}$ (and thus in $B^{0,1}_{1,1}$) that does not satisfy the Dini condition.

Significance and Claims
The paper claims to rigorously complete the proof of the linear bound (1.3) previously indicated by Verdera and Mateu, thereby making the claim in [V1] mathematically sound. Furthermore, it investigates the optimality of this linear growth. The authors note that while the linear dependence on the Lipschitz constant is known to be sharp (via David's example), the linear dependence on $n$ in the general case may not be optimal, as lower bounds in examined examples only reflect the exponential factor $\|A'\|_\infty^n$.

Consequently, the primary significance of the work lies in demonstrating that sublinear growth in $n$ is achievable under mild additional regularity assumptions on the derivative of the Lipschitz function. By providing estimates of order $\sqrt{n}$ under Dini or Besov conditions, the paper refines the understanding of how the smoothness of the graph influences the quantitative behavior of higher-order Calderón commutators. The work also clarifies the relationship between different regularity classes (Dini vs. Sobolev/Besov) in the context of these singular integral operators.