Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order

Technical Summary: Regularity Estimates for Fractional Orthotropic p-Laplacians of Mixed Order

Problem Statement
This paper investigates the regularity of weak solutions to nonlocal equations of the form $Lu = f$ in a domain $Q = (-1, 1)^n$, where $L$ is a nonlinear integro-differential operator defined by:
$$Lu(x) = \text{PV} \int_{\mathbb{R}^n} |u(y) - u(x)|^{p-2}(u(y) - u(x)) \mu(x, dy)$$
for $p > 1$. The operator is characterized by a family of measures $(\mu(x, dy))_{x \in \mathbb{R}^n}$ that are anisotropic and singular. The primary reference measure is $\mu_{\text{axes}}$, which corresponds to the sum of one-dimensional fractional $p$-Laplacians with potentially different orders of differentiability $s_1, \dots, s_n \in (0, 1)$ along each coordinate axis. Unlike the standard fractional $p$-Laplacian, which is invariant under orthogonal transformations, the operator associated with $\mu_{\text{axes}}$ (and its local limit, the orthotropic $p$-Laplacian) lacks this rotational invariance.

The authors aim to establish robust regularity estimates—specifically a Sobolev-type inequality, a weak Harnack inequality, and local Hölder estimates—for a class of admissible measures that are locally comparable to $\mu_{\text{axes}}$. A central challenge is that the constants in these estimates must be independent of the specific orders $s_1, \dots, s_n$, depending only on a lower bound $s_0$ and the dimension $n$. This "robustness" is crucial for recovering regularity results for local operators (such as the orthotropic $p$-Laplacian) by taking the limit as the fractional orders approach 1.

Methodology
The authors employ a combination of nonlocal analysis techniques adapted to anisotropic geometries:

1. Anisotropic Geometry: Instead of standard Euclidean balls or cubes, the analysis utilizes a family of rectangles $M_r(x)$ defined by $M_r(x) = \prod_{k=1}^n (x_k - r^{s_{\max}/s_k}, x_k + r^{s_{\max}/s_k})$, where $s_{\max} = \max\{s_1, \dots, s_n\}$. These sets form a doubling metric measure space, allowing the use of maximal function theorems and John–Nirenberg inequalities adapted to this geometry.
2. Function Spaces: The study is conducted within nonlocal analogs of Sobolev spaces, denoted $V^{p,\mu}(\Omega|\mathbb{R}^n)$ and $H^{p,\mu}_{\Omega}(\mathbb{R}^n)$, which incorporate the specific integrability and fractional regularity dictated by the measure $\mu$.
3. Assumptions on Measures: The class of admissible measures $\mathcal{K}(p, s_0, \Lambda)$ is defined by three conditions:
   • Integrability and symmetry of the measure family.
   • Specific tail behavior relative to the anisotropic rectangles $E^k_r(x)$.
   • Local comparability of the associated energy functionals to the reference functional defined by $\mu_{\text{axes}}$.
4. Technical Tools:
   • Sobolev-type Inequality: The proof utilizes the Hardy–Littlewood maximal function and a sharp maximal function theorem adapted to anisotropic rectangles. A key technical lemma (Lemma 2.4) handles the discrete gradient estimates required for the anisotropic setting.
   • Moser Iteration: The weak Harnack inequality is derived using Moser's iteration technique. This requires establishing a discrete gradient estimate for the nonlinear term (Lemma 3.4), which serves as a nonlocal, nonlinear analog to the inequality $|\nabla v|^{p-2}\nabla v \cdot \nabla(-v^{-t}\tau^p) \geq c_1 |\nabla(v^{\frac{-t+p-1}{p}}\tau)|^p - c_2|\nabla \tau|^p v^{-t+p-1}$.
   • Scaling Arguments: The authors utilize a specific anisotropic scaling (diffeomorphism $\Psi$) to reduce general estimates to a normalized setting, facilitating the proof of Hölder continuity.

Key Contributions and Results

1. Robust Sobolev-type Inequality (Theorem 2.7): The authors prove a Sobolev embedding for the reference measure $\mu_{\text{axes}}$ where the constant depends only on $n, p, s_0$, and the Sobolev exponent $p^*$, but not on the individual $s_k$. This is extended to the general class of admissible measures via local comparability (Corollary 2.8).
2. Weak Harnack Inequality (Theorem 1.4): For non-negative weak supersolutions to $Lu = f$, the paper establishes a weak Harnack inequality. The result provides a lower bound for the infimum of the solution in a smaller domain in terms of its $L^{p_0}$ norm and the tail of the solution outside the domain. The constants are robust with respect to the differentiability orders.
3. Local Hölder Estimate (Theorem 1.5): Combining the weak Harnack inequality with the decay of oscillation, the authors prove that bounded weak solutions are locally Hölder continuous ($C^\alpha$). The Hölder exponent $\alpha$ and the constant $C$ depend on $n, p, s_0, q, \Lambda$, but are independent of the specific values of $s_1, \dots, s_n$.
4. Recovery of Local Results: A significant aspect of the work is that the robustness of the constants allows the authors to recover regularity estimates for the local orthotropic $p$-Laplacian (defined in equation 1.3) by taking the limit as $s_k \nearrow 1$.

Significance and Claims
The paper claims that its primary strength lies in the robustness of the derived estimates. By ensuring that constants depend only on a lower bound of the differentiability orders ($s_0$) rather than the orders themselves, the authors bridge the gap between nonlocal fractional operators and their local counterparts. This approach allows for the derivation of regularity results for the orthotropic $p$-Laplacian (a well-known operator in analysis) directly from the theory of nonlocal operators, without needing separate, specific proofs for the local case.

The authors note that while the Harnack inequality fails for singular equations of this type, the weak Harnack inequality is sufficient to prove the decay of oscillation and subsequent Hölder continuity. The work extends previous results by Chaker, Kim, and others (specifically [14, 15, 25]) from the linear case ($p=2$) and isotropic settings to the fully nonlinear ($p>1$) and anisotropic setting. The paper acknowledges that global boundedness of solutions is assumed, noting that proving global or local boundedness for anisotropic nonlocal equations remains an open question.

The results are situated within the broader context of the De Giorgi–Nash–Moser theory for nonlocal operators, extending these foundational ideas to operators with anisotropic and singular kernels that model complex physical phenomena, such as systems driven by Lévy processes with anisotropic jumps.