On Hardy-Littlewood-Sobolev estimates for degenerate Laplacians

1. Problem Statement

The paper addresses the lack of known $L^p-L^q$ bounds for fractional powers of degenerate Laplacians in weighted spaces.

• Context: Degenerate elliptic operators of the form $-\Delta_\omega$ (where the degeneracy is controlled by a weight $\omega$ in the Muckenhoupt class $A_2(\mathbb{R}^n)$) are well-studied. It is known that the associated heat semigroup is bounded on $L^p_\omega$ and possesses an $H^\infty$ functional calculus.
• The Gap: While the classical Hardy-Littlewood-Sobolev (HLS) theorem establishes $L^p-L^q$ bounds for fractional powers $(-\Delta)^{-\alpha}$ when the weight is trivial ($\omega=1$), it was unknown whether similar bounds exist for the degenerate case $(-\Delta_\omega)^{-\alpha}$.
• The Obstacle: Standard strategies for proving HLS estimates rely on the heat semigroup mapping $L^p$ to $L^q$ ($q > p$). However, for degenerate operators, the semigroup $e^{t\Delta_\omega}$ is generally not bounded from $L^p_\omega$ to $L^q_\omega$ if $q > p$.

2. Methodology

The authors develop a novel approach based on heat kernel size estimates and real interpolation, avoiding the need for translation invariance (which fails in the weighted setting).

• Key Insight: The authors observe that while the semigroup does not map $L^p_\omega \to L^q_\omega$, it does map $L^p_\omega$ to a different weighted space $L^q_{\omega^{q/p}}$ (or more generally $L^q_{\omega^{q/p+\epsilon}}$) provided the weight $\omega$ satisfies an additional Reverse Hölder condition ($\omega \in RH_{q/p}$).
• Heat Kernel Estimates: They utilize Gaussian upper and lower bounds for the heat kernel $K_t(x,y)$ of the degenerate operator. Specifically, the kernel satisfies bounds involving the weight measure $\omega_t(x) = \omega(B(x, \sqrt{t}))$.
• Calderón Representation: They express the fractional power operator via the Calderón reproducing formula:
  $$(-\Delta_\omega)^{-\alpha} f \sim \int_0^\infty t^\alpha e^{t\Delta_\omega} f \frac{dt}{t}$$
• Interpolation Strategy:
  1. Establish "off-diagonal" heat kernel bounds: $\|e^{t\Delta_\omega} f\|_{L^q_{\omega^{q/p}}} \lesssim t^{-\alpha} \|f\|_{L^p_\omega}$.
  2. Use these bounds to prove weak-type estimates for the truncated integral operators.
  3. Apply Stein-Weiss real interpolation (with a change of measure) to upgrade weak-type bounds to strong $L^p-L^q$ bounds.
  4. Extend results to Lorentz spaces and holomorphic functional calculus.

3. Key Contributions and Results

A. Main Theorem (Hardy-Littlewood-Sobolev Estimates)

Theorem 1.1: Let $\omega \in A_2(\mathbb{R}^n)$ and assume $\omega \in RH_{q/p}(\mathbb{R}^n)$ for $1 < p < q < \infty$. Let $\alpha = \frac{n}{2}(\frac{1}{p} - \frac{1}{q})$. Then there exists a constant $C$ such that:
$$\|\omega^{1/p} f\|_{L^q(\mathbb{R}^n)} = \|f\|_{L^q_{\omega^{q/p}}(\mathbb{R}^n)} \leq C \|(-\Delta_\omega)^\alpha f\|_{L^p_\omega(\mathbb{R}^n)}$$

• Interpretation: The fractional power $(-\Delta_\omega)^{-\alpha}$ is a bounded operator from $L^p_\omega$ to $L^q_{\omega^{q/p}}$.
• Endpoint Case ($p=1$): The operator maps $L^1_\omega$ to the weak space $L^{q,\infty}_{dx}$ (with weight $\omega$).

B. Sharpness of Conditions

The authors prove that the Reverse Hölder condition ($\omega \in RH_{q/p}$) is sharp.

• Proposition 3.24: If the semigroup satisfies the $L^p_\omega \to L^q_{\omega^{q/p}}$ bound (or the HLS inequality holds), then $\omega$ must belong to the Reverse Hölder class $RH_{q/p}$. This distinguishes their result from classical weighted estimates where the condition might be different.

C. Extensions to Lorentz Spaces

The paper improves the standard Lebesgue space results to weighted Lorentz spaces using real interpolation:

• Theorem 3.15: The operator maps $L^p_\omega$ to the weighted Lorentz space $L^{q,p}_{\omega}(\omega^{1/p - 1/q})$.
• Theorem 3.17: The operator $\omega^{1/p}(-\Delta_\omega)^{-\alpha}$ maps $L^p_\omega$ to the unweighted Lorentz space $L^{q,p}_{dx}$.
• The authors note that these two results are distinct and neither trivially implies the other in the general weighted Lorentz setting.

D. Functional Calculus and Generalizations

• Holomorphic Functions: The results extend to operators of the form $\psi(-\Delta_\omega)$ where $\psi$ is a holomorphic function with specific decay at 0 and $\infty$ (Corollary 3.22).
• General Degenerate Operators: The methodology applies to general second-order degenerate elliptic operators $L_\omega = -\text{div}(A(x)\nabla \cdot)$ with real coefficients, provided the heat kernel satisfies Gaussian upper bounds.
• Complex Coefficients: For complex coefficients, the results hold for a restricted range of exponents $p, q$ where the semigroup is uniformly bounded, replacing pointwise Gaussian bounds with off-diagonal estimates.

4. Comparison with Existing Literature

• Vs. Muckenhoupt-Wheeden [MW74]: Classical weighted HLS estimates for the unweighted Laplacian $(-\Delta)^{-\alpha}$ require the weight $\tilde{\omega}$ to be in the class $A_{p,q}$. The authors show that their condition ($\omega \in A_2 \cap RH_{q/p}$) is different and, in the context of the degenerate Laplacian, is the natural and sharp condition.
• Vs. Riesz Transforms [CUMR18]: Previous results on the boundedness of Riesz transforms $\nabla(-\Delta_\omega)^{-1/2}$ only cover a limited range of $p$ (typically near 2) and require $n \geq 3$. The current paper covers a broader range of $p$ and $q$ via the fractional power approach.
• Vs. Sobolev-Gagliardo-Nirenberg [LMPT10]: While geometric weighted Sobolev inequalities exist, the authors' approach via the heat semigroup provides a direct functional-analytic proof for the fractional powers without relying on geometric gradient inequalities.

5. Significance

• Foundational: This work establishes the first complete $L^p-L^q$ theory for fractional powers of degenerate Laplacians, filling a significant gap in the theory of weighted PDEs.
• Methodological: It demonstrates that the "off-diagonal" boundedness of the heat semigroup (mapping $L^p_\omega$ to $L^q_{\omega^{q/p}}$) is the correct framework for degenerate operators, replacing the standard $L^p-L^q$ boundedness which fails in this setting.
• Applications: The results are motivated by and applicable to:
  • Constructing solutions to second-order degenerate parabolic equations with unbounded lower-order terms.
  • Establishing boundedness results for degenerate parabolic Riesz transforms.
  • Refining the understanding of the Kato problem in weighted spaces.

In summary, the paper successfully generalizes the classical Hardy-Littlewood-Sobolev inequality to the degenerate setting by identifying the precise weight conditions (Reverse Hölder) required for the heat semigroup to bridge the gap between $L^p$ and $L^q$ spaces, and by utilizing real interpolation to extend these bounds to Lorentz spaces and functional calculus.