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Regularity for the fractional logarithmic $p$ -Laplacian

This paper establishes the Harnack inequality (with tail terms) and local Hölder regularity for the fractional logarithmic $p$ -Laplacian by adapting De Giorgi-Nash-Moser techniques, while also demonstrating the necessity of tail terms for the Harnack inequality and providing new results even in the linear case $p=2$ .

Original authors: Nirjan Biswas, Stuti Das, Abhrojyoti Sen

Published 2026-07-14

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CC BY 4.0

Original authors: Nirjan Biswas, Stuti Das, Abhrojyoti Sen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible web where every point talks to every other point. Usually, when we study how things smooth out or spread across this web, we use a tool called the fractional p-Laplacian. Think of this tool as a super-smoothie blender that mixes values from far away to make a solution nice and steady.

But what happens if we tweak the blender's settings just a tiny bit? What if we ask, "What if we change the order of the blending?" That's exactly what the authors of this paper did. They took that famous blender and asked it to differentiate itself with respect to its own settings. The result? A new, quirky machine called the fractional logarithmic p-Laplacian.

The Weird Kernel: A Kernel with a Mood Swing

The most exciting (and tricky) thing about this new machine is its "kernel"—the rulebook it uses to decide how much influence a distant point has on a nearby one.

In the old, standard blender, the rulebook was always friendly and positive; it always said, "Hey, that distant point matters, and it matters in a good way." But this new logarithmic machine has a mood swing.

Nearby: When points are close together, the rulebook gets super intense and singular (very loud and chaotic).
Far Away: When points get too far apart, the rulebook flips a switch and turns negative. It starts saying, "Actually, that distant point is pulling in the opposite direction!"

This sign-changing behavior is the paper's main character. It's like a friend who is super supportive when you're next to them but starts criticizing you from across the room. Because of this flip-flop, the math gets messy. You can't just ignore the distant friends anymore; you have to account for both the supportive ones and the critical ones separately.

The Big Discovery: You Can't Ignore the "Tail"

The authors set out to prove two big things:

Harnack Inequality: This is a fancy way of saying, "If a solution is positive in one spot, it can't suddenly drop to zero nearby without a good reason." It's a guarantee of smoothness.
Hölder Regularity: This means the solutions don't have jagged, jagged edges; they are nicely smooth curves.

They proved both of these! But here is the twist that makes this paper special: They proved that you cannot ignore the "tail" terms.

In the old math world, if a solution was positive everywhere in the universe, you could sometimes ignore what was happening far away. The authors showed that for this new logarithmic machine, that is impossible. Even if your solution is positive everywhere, the "negative mood" of the kernel from the distant parts of the universe still pulls on your local solution.

They explicitly ruled out the idea that you could get a "purely local" Harnack inequality (one that only looks at the immediate neighborhood) without including these tail terms. They even built a specific counter-example (a mathematical proof by construction) showing that if you try to drop the tail terms, the whole inequality breaks. It's like trying to predict the weather in your backyard by ignoring the storm system three states away; for this specific type of math, the storm always matters.

How They Did It: The De Giorgi-Nash-Moser Dance

To prove their results, the authors didn't invent a new dance; they used a classic, rigorous routine called the De Giorgi-Nash-Moser iteration. Think of this as a game of "zooming in."

They start with a big ball of uncertainty.
They use a special energy estimate (a way of measuring how much the solution is "wiggling") to show that the solution can't wiggle too much.
They zoom in to a smaller ball, prove the wiggles are smaller there, and repeat.

But because their kernel changes sign, they had to invent a new way to measure the "wiggles" from the outside world. Instead of one big "tail" number, they had to split it into two:

Tail+: Measuring the influence of the positive parts of the distant solution.
Tail-: Measuring the influence of the negative parts.

They proved that these two tails work together in a specific pairing to keep the solution under control. If you only look at one, the math falls apart.

The Confidence Level

The authors are 100% sure about their main findings. They didn't just simulate this on a computer or guess; they provided a rigorous mathematical proof.

They proved the Harnack inequality (with the necessary tail terms).
They proved the solutions are locally Hölder continuous (smooth).
They proved that removing the tail terms is impossible by constructing a specific counter-example.

The "Linear" Surprise

One cool side note: Even if you simplify the problem to the most basic, linear case (where $p=2$ , which is like the standard fractional Laplacian), these results are brand new. No one had figured out how to handle this specific "logarithmic derivative" operator before, even in the simple case.

The Bottom Line

This paper introduces a new mathematical operator that acts like a blender with a sign-changing rulebook. The authors proved that solutions to equations using this operator are smooth and follow a specific inequality, but only if you account for the distant "tails" of the solution. They showed that ignoring the distant world is a fatal mistake for this specific type of math, and they did it all with a solid, step-by-step proof that leaves no room for doubt.

Technical Summary: Regularity for the Fractional Logarithmic p-Laplacian

Problem Setup and Motivation
This work investigates the local regularity theory for weak solutions to the nonlocal equation:
$(-\Delta_p)^{s+\log} u = f \quad \text{in } \Omega,$
where $\Omega \subset \mathbb{R}^d$ is a bounded open set, $f \in L^q_{loc}(\Omega)$ with $q > 1$ , and the operator $(-\Delta_p)^{s+\log}$ is defined as the first-order derivative of the fractional $p$ -Laplacian $(-\Delta_p)^t$ with respect to the order $t$ , evaluated at $t=s \in (0,1)$ . Specifically:
$(-\Delta_p)^{s+\log} u(x) := \frac{d}{dt}(-\Delta_p)^t u(x) \Big|_{t=s}.$
The operator arises naturally in the study of the asymptotic behavior of fractional operators as the order varies. While the endpoint case $s=0$ (the logarithmic Laplacian) has been studied extensively (e.g., Chen-Weth, Dyda-Jarohs-Sk), the interior case $s \in (0,1)$ , particularly for the quasilinear $p \neq 2$ setting, represents a novel direction initiated by Chen-Chen-Hauer.

The kernel of this operator, $K_{s+\log}(r)$ , exhibits two critical features that distinguish it from the standard fractional $p$ -Laplacian:

Increased Singularity: Near the diagonal ( $r \to 0$ ), the kernel behaves like $r^{-d-sp} \log(1/r)$ , which is more singular than the standard $r^{-d-sp}$ .
Sign Change: At large scales, the kernel changes sign. Specifically, $K_{s+\log}(r) \leq 0$ for $r \geq e^{B_{d,s,p}/p}$ . This sign-changing nature complicates the analysis of nonlocal interactions, as the standard "tail" terms used in fractional theory no longer suffice to control the far-field contributions.

Methodology
The authors adapt the classical De Giorgi-Nash-Moser theory to this specific nonlocal setting. The strategy involves several non-trivial modifications to handle the logarithmic weight and the sign-changing kernel:

Decomposition of the Kernel and Tails:
Due to the sign change, a single nonlocal tail term is insufficient. The kernel is decomposed into a fractional part and two sign-definite logarithmic parts: $K_{s+\log} \sim r^{-d-sp} + K^+_{s+\log} - K^-_{s+\log}$ . Consequently, the authors introduce two distinct tail terms to capture the far-field effects:
- $Tail_+(u; x_0, R)$ : Captures interactions where the logarithmic weight is positive (intermediate scales).
- $Tail_-(u; x_0, R)$ : Captures interactions where the logarithmic weight is negative (far field).
  The analysis reveals a specific duality: for subsolutions, the positive part of the solution couples with the negative kernel part, and vice versa. This necessitates a pairing of tails in the estimates, specifically $Tail_+(u^-) + Tail_-(u^+)$ .
Energy Estimates and Iteration:
The proof relies on establishing a Caccioppoli inequality and a logarithmic energy estimate. A key technical hurdle is the logarithmic growth of the kernel, which introduces factors of $(1 - \log r)$ in local energy estimates. To manage this, the authors utilize a fractional logarithmic Poincaré inequality (Proposition 2.5) which provides a matching decay factor, allowing the De Giorgi iteration to proceed in a scale-invariant manner.
Handling the Source Term:
The presence of $f \in L^q_{loc}$ with $q > d/sp$ (scaling subcritical) requires careful handling of the integrability exponent. The authors introduce a parameter $\sigma$ to ensure strict inequalities necessary for the convergence of the iteration lemmas, preventing the breakdown that occurs in the critical case.

Key Contributions and Results

Harnack Inequality with Tails (Theorem 1.2):
The paper establishes a Harnack inequality for non-negative weak solutions. Unlike the fractional $p$ -Laplacian, where global non-negativity eliminates the tail term, the sign-changing kernel here ensures that tail terms persist even if $u \geq 0$ globally. The inequality takes the form:
$\text{ess sup}_{B_{r/2}} u \leq C \left( \text{ess inf}_{B_r} u + \text{Tail Terms} + \text{Source Terms} \right).$
Crucially, the tail terms involve the specific pairing $Tail_+(u^-) + Tail_-(u^+)$ , reflecting the sign-changing structure of the kernel. The authors prove that these tail terms cannot be removed in general, even for non-negative solutions, by constructing a counterexample where the solution is supported outside the domain of interest.
Local Hölder Regularity (Theorem 1.4):
The authors prove that weak solutions are locally Hölder continuous. The Hölder exponent $\alpha$ depends on the dimension $d$ , the order $s$ , the exponent $p$ , and the integrability $q$ of the source term $f$ . The estimate includes a modulus of continuity that decays with the radius $r$ , scaled by logarithmic factors and the tail terms.
Counterexample to Tail-Free Harnack:
In Section 7, the authors construct a specific example (for $p=2$ ) demonstrating that if a solution is supported outside the ball $B_R$ , the tail term in the Harnack inequality is unavoidable. This confirms that the "tail-free" Harnack inequality, known for the standard fractional Laplacian under global non-negativity, does not hold for the fractional logarithmic operator due to the kernel's sign change.

Significance and Claims
The paper claims that these results are new even in the linear case $p=2$ . The work extends the regularity theory to a class of operators that are "zero-order" in a specific asymptotic sense but possess a kernel more singular than the standard fractional Laplacian near the diagonal and sign-changing at infinity.

The authors emphasize that the presence of the logarithmic factor and the sign change fundamentally alter the structure of the regularity estimates, requiring the introduction of dual tail terms and specific logarithmic decay factors in the bounds. The results provide a systematic local regularity theory for this class of nonlocal equations, answering affirmatively the questions regarding the existence of Harnack inequalities and Hölder continuity for weak solutions. The work does not claim to establish boundary regularity or global continuity, noting that these require separate barrier arguments outside the current scope.