[Paper Review] Failure of necessity of the energy condition

Eric T. Sawyer, Chun‐Yen Shen+1 more|arXiv (Cornell University)|Jul 20, 2016

Geometric Analysis and Curvature Flows18 references3 citations

TL;DR

This paper constructs a counterexample showing that the energy condition is not necessary for two-weight $L^2$ boundedness of a Calderón-Zygmund operator with cancellation, even when the kernel satisfies ellipticity. The authors define a flattened Hilbert kernel that is elliptic but not gradient-elliptic, and construct a pair of weights on the real line where the operator is bounded but the energy condition fails, thereby demonstrating the necessity of gradient ellipticity for the energy condition to be required in the two-weight $T1$ theorem.

ABSTRACT

We give an example of a pair of weights (u,v) on the line, and an elliptic convolution singular integral operator H on the line, such that H_u is bounded from L^2(u) to L^2(v), yet the measure pair (u,v) fails to satisfy the backward energy condition. The key to the construction is that the kernel K of H has flat spots where d/dx K(x) = 0. Conversely, we show that if H is gradient elliptic, i.e. d/dx K(x) =< c < 0, then the energy conditions are necessary for boundedness of H, and by our theorem in arXiv:1603.04332v2, the T1 theorem holds for H.

Motivation & Objective

• To challenge the necessity of the energy condition in two-weight $T1$ theorems for singular integrals.
• To construct a counterexample where the operator is bounded but the energy condition fails.
• To clarify the role of gradient ellipticity in the necessity of energy conditions.
• To demonstrate that the energy condition is not required when the kernel lacks gradient ellipticity, even if it satisfies standard ellipticity.

Proposed method

• Construct a smooth, flattened version of the Hilbert transform kernel $K_{lat}(x)$ that satisfies $|K_{lat}(x)| \gtrsim 1/|x|$ but not $|K_{lat}'(x)| \gtrsim 1/|x|^2$.
• Define a singular integral operator $H_{lat}$ using this kernel, which is elliptic but not gradient-elliptic.
• Construct a pair of weights $(\widehat{\sigma}, \widehat{\omega})$ on $\mathbb{R}$ using a self-similar Cantor-type measure construction.
• Use dyadic tree structures and self-similarity to analyze testing conditions and energy conditions on intervals.
• Show that the forward and backward testing conditions hold uniformly, and the $A_2$ condition is finite.
• Prove that the backward energy condition fails for $(\widehat{\sigma}, \widehat{\omega})$, yet the operator remains bounded from $L^2(\widehat{\sigma})$ to $L^2(\widehat{\omega})$.

Experimental results

Research questions

• RQ1 Is the energy condition necessary for two-weight $L^2$ boundedness of Calderón-Zygmund operators with cancellation?
• RQ2 Can a bounded operator fail to satisfy the energy condition while still satisfying the $A_2$ and testing conditions?
• RQ3 Does the absence of gradient ellipticity in the kernel invalidate the necessity of the energy condition?
• RQ4 Can a non-gradient-elliptic kernel support a two-weight $T1$ theorem without the energy condition?
• RQ5 What is the minimal kernel regularity required for energy conditions to be necessary?

Key findings

• The operator $H_{\flat,\sigma}$ is bounded from $L^2(\widehat{\sigma})$ to $L^2(\widehat{\omega})$ for the constructed weights $(\widehat{\sigma}, \widehat{\omega})$.
• The measure pair $(\widehat{\sigma}, \widehat{\omega})$ satisfies the $A_2$ condition and both forward and backward testing conditions uniformly.
• The backward energy condition fails for $(\widehat{\sigma}, \widehat{\omega})$, despite the boundedness of the operator.
• The kernel $K_{\flat}$ is elliptic but not gradient-elliptic, as $K_{\flat}'(x) = 0$ on a family of intervals where $K_{\flat}$ is flat.
• The failure of the energy condition occurs precisely because the kernel lacks gradient ellipticity, showing that this condition is essential for the necessity of energy conditions.
• The result implies that the energy condition is not universally necessary in two-weight $T1$ theorems, but only when gradient ellipticity is assumed.