Riesz transform and its related inequalities for degenerate elliptic operators of Grushin type

Technical Summary: Riesz Transform and Related Inequalities for Degenerate Elliptic Operators of Grushin Type

Problem Statement
This paper investigates the $L^p$ boundedness of the Riesz transform and the validity of the reverse Riesz inequality for a class of degenerate elliptic operators of Grushin type. The operator $L$ is defined on $\mathbb{R}^{n+m}$ by:
$$L = -\nabla_x \cdot (|x|^{2\alpha}\nabla_x) + |x|^{2\beta}\Delta_y,$$
where $n, m \ge 1$, $0 < \alpha < 1$, and $\beta \ge 0$. The associated intrinsic gradient is $\nabla_L = (|x|^\alpha \nabla_x, |x|^\beta \nabla_y)$. The degeneracy occurs on the singular set $\{x=0\}$.

The central questions are:

1. Riesz Transform Boundedness: For which $p \in (1, \infty)$ does the inequality $\|\nabla_L L^{-1/2} f\|_p \le C_p \|f\|_p$ hold?
2. Reverse Riesz Inequality: For which $p$ does the reverse inequality $\|L^{1/2} f\|_p \le C_p \|\nabla_L f\|_p$ hold?

The analysis is complicated by the anisotropic geometry of the Grushin space and the behavior of the Friedrichs realization near the singular set, which varies significantly depending on the dimension $n$ and the degeneracy parameter $\alpha$.

Methodology
The proof strategy relies on establishing reverse Hölder inequalities for gradients of $L$-harmonic functions, leveraging the abstract equivalence between these inequalities and Riesz transform boundedness (Coulhon, Jiang, Koskela, Sikora). The methodology is divided into distinct regimes based on dimension and degeneracy:

1. Kernel Constructions and Explicit Analysis:

   • Friedrichs Realization: The authors rigorously characterize the admissible boundary conditions at the singular set $\{x=0\}$. For $0 < \alpha < 1/2$, the form domain admits a common trace, leading to flux-matching conditions. For $1/2 \le \alpha < 1$, the singular set has zero capacity, separating the domain into two half-lines with zero-flux conditions on each.
   • Poisson and Green Kernels: The authors construct explicit Poisson and Green kernels adapted to the Friedrichs extension. This involves Fourier transforming in the $y$-variable and decomposing in spherical harmonics (for $n \ge 2$) or separating even/odd parts (for $n=1$).
   • Modified Bessel Functions: The radial part of the operator reduces to modified Bessel equations. The analysis requires precise estimates of ratios of modified Bessel functions ($I_\nu$ and $K_\nu$).
     • In the $n \ge 2$ case, the order $\nu_\ell$ of the Bessel functions depends on the angular frequency $\ell$ and tends to infinity. The authors develop uniform-in-order estimates for these Bessel ratios, which are critical for controlling the singular set interaction.
     • In the $n=1$ case, the problem reduces to fixed-order Bessel analysis on half-lines, but with distinct boundary conditions (zero-flux for even components, zero-trace for odd components).

2. Reverse Riesz Inequality via Harmonic Annihilation:

   • For the reverse inequality, particularly in the strongly degenerate regime ($n=1, \alpha \ge 1/2$) where global Poincaré inequalities fail, the authors employ a harmonic annihilation method.
   • This technique isolates the leading harmonic term in the kernel of the Riesz transform (which causes unboundedness in certain regimes) and eliminates it via integration by parts in the bilinear form. The remaining term is controlled using a Grushin-adapted Hardy inequality.

3. Hodge Projector Argument:

   • In the strongly degenerate one-dimensional case ($n=1, 1/2 \le \alpha < 1$), the lack of a global Poincaré inequality prevents the direct application of standard reverse Hölder characterizations. The authors combine the reverse Riesz inequality with a Hodge projector boundedness result (Shen's criterion) to recover the full range of Riesz transform boundedness.

Key Contributions and Results

• Full-Range Boundedness for $n \ge 2$:
  Theorem 1.1 establishes that for $n \ge 2$ and $0 < \alpha < 1$, the Riesz transform is bounded on $L^p$ for the full range $1 < p < \infty$. This is achieved by proving the endpoint reverse Hölder estimate $(RH_\infty)$ near the singular set using uniform Bessel estimates.

• Sharp Range and Endpoint Obstruction for Weakly Degenerate $n=1$:
  For $n=1$ and $0 < \alpha < 1/2$, Theorem 1.2 proves that the Riesz transform is bounded if and only if $1 < p < \alpha^{-1}$.

  • Mechanism: The odd component of the solution satisfies a zero-trace condition at the origin, behaving like $\text{sgn}(x)|x|^{1-2\alpha}$. Its intrinsic gradient scales as $|x|^{-\alpha}$, creating a singularity that obstructs boundedness for $p \ge \alpha^{-1}$.
  • Sharpness: The paper constructs a counterexample demonstrating the failure of the reverse Hölder inequality at the endpoint.

• Reverse Riesz Inequality for All Regimes:
  Theorem 1.3 establishes the reverse Riesz inequality for all $n, m \ge 1$.

  • For $n \ge 2$, it holds for all $1 < p < \infty$.
  • For $n=1$, it holds for $p \in (1, (1-\alpha)^{-1}) \cup ((1-\alpha)^{-1}, \infty)$. The excluded exponent $p = (1-\alpha)^{-1}$ corresponds to the Hardy threshold.
  • Crucially, in the strongly degenerate case ($n=1, \alpha \ge 1/2$), the reverse inequality holds for all $1 < p < \infty$ despite the failure of the global Poincaré inequality, thanks to the harmonic annihilation technique.

• Full-Range Boundedness for Strongly Degenerate $n=1$:
  Theorem 1.4 proves that for $n=1$ and $1/2 \le \alpha < 1$, the Riesz transform is bounded for the full range $1 < p < \infty$.

  • Mechanism: The Friedrichs realization separates the two half-lines, selecting the zero-flux branch on each. While the global Poincaré inequality fails, the separated structure allows for a strengthened local reverse Hölder estimate. Combining this with the reverse Riesz inequality and the Hodge projector boundedness yields the full range result.

Significance and Claims
The paper claims to provide a complete description of the $L^p$ boundedness of the Riesz transform for Grushin operators across all parameter regimes.

• Transition in Behavior: A key finding is the "striking transition" in the behavior of the singular set. In the weakly degenerate one-dimensional case, the singular set acts as a barrier that creates a specific obstruction ($p < \alpha^{-1}$). In the strongly degenerate case, the singular set effectively separates the space, but the zero-flux condition on the separated components restores full-range boundedness.
• Technical Novelty: The work introduces uniform-in-order estimates for modified Bessel functions in the context of Grushin operators, which are necessary for the higher-dimensional case. It also adapts the harmonic annihilation method to handle degenerate operators where the singular set has zero capacity, bypassing the need for global Poincaré inequalities in the reverse Riesz analysis.
• Limitations: The paper notes that the endpoint cases $p = \alpha^{-1}$ (for $n=1, \alpha < 1/2$) and $p = (1-\alpha)^{-1}$ (for the reverse inequality) remain open or are known to be the exact thresholds of failure, but the paper does not resolve the boundedness at these exact points.

The results highlight that the boundedness of the Riesz transform is sensitive not just to the heat kernel geometry, but specifically to the local endpoint behavior of harmonic functions at the singular set and the interplay between the Friedrichs realization and the intrinsic gradient.