Elementary Brezis-Browder type results and Representation formulae for s-harmonic functions

1. Problem Statement and Context

The paper addresses two primary challenges in the analysis of fractional partial differential equations (PDEs) on $\mathbb{R}^d$:

1. Representation of Solutions: Determining the precise structure of distributional solutions to the fractional Poisson equation $(-\Delta)^s u = T$ and characterizing $s$-harmonic functions (solutions to $(-\Delta)^s u = 0$). Specifically, the author seeks to establish when solutions can be represented as Riesz potentials plus a polynomial.
2. Brezis–Browder Type Results: Extending classical results by Brezis and Browder (originally for integer-order Sobolev spaces) to fractional Sobolev spaces. The core question is: Under what conditions does the duality pairing $\langle T, u \rangle$ between a distribution $T$ and a function $u$ equal the integral $\int_{\mathbb{R}^d} T(x)u(x) dx$, given that $Tu$ is bounded from below by an integrable function?

The work is motivated by the non-local nature of the fractional Laplacian $(-\Delta)^s$, which complicates the definition of distributional solutions and the behavior of functions at infinity compared to the classical local case ($s=1$).

2. Methodology

The author employs a combination of harmonic analysis, functional analysis, and approximation techniques:

• Quantitative Estimates on Balls: To analyze $s$-harmonic functions, the author derives quantitative interior regularity estimates on balls $B(0, R)$. This involves constructing a specific kernel representation using a regularized Poisson kernel and the operator $L_{m,s}$ (a finite difference approximation of the fractional Laplacian).
• Liouville-Type Arguments: The proof that $s$-harmonic functions are polynomials relies on sending the radius $R \to \infty$ in the derived estimates. This forces derivatives of order $|n| \geq 2s$ to vanish, implying polynomial growth.
• Approximation Sequences:
  • For the general fractional Sobolev spaces $\dot{H}^{s,p}$ (defined via Fourier multipliers), the author uses a truncation strategy involving smooth cut-off functions $\phi_\lambda(x) = \phi(\lambda^{-1}x)$.
  • For the low-order regime ($0 < s < 1$) using Gagliardo–Slobodeckiĭ seminorms, the author constructs bounded approximating sequences $(u_n)$ that converge to $u$ while preserving non-negativity and pointwise bounds ($0 \leq u_n \leq u$).
• Duality and Fatou's Lemma: To prove the Brezis–Browder equality, the author uses the approximation sequences to pass to the limit in the duality pairing, utilizing Fatou's Lemma and the Dominated Convergence Theorem under the assumption $Tu \geq -|f|$.

3. Key Contributions and Results

A. Representation Formulae for $s$-Harmonic Functions and Poisson Equation

• Theorem 1.1 (Liouville Theorem for Fractional Laplacian):
  • Result: If $u \in L^1_{2s}(\mathbb{R}^d)$ (a space of functions with specific polynomial decay at infinity) solves $(-\Delta)^s u = 0$ in the distributional sense, then $u$ is a polynomial of degree strictly less than $2s$.
  • Significance: This extends previous results known for $s \in (0,1)$ to the range $s \in (0, d/2)$ for $d \geq 1$. It establishes that the only "global" $s$-harmonic functions with controlled growth are polynomials.
• Theorem 1.2 (Representation of Solutions):
  • Result: For the equation $(-\Delta)^s u = T$ with $T \in \dot{H}^{-s}(\mathbb{R}^d) \cap L^1_{loc}(\mathbb{R}^d)$, a solution $u$ satisfying a specific growth condition at infinity (liminf of the average of $|u-l|$ is finite) is unique and given by:
    $$u(x) = A_{2s} \int_{\mathbb{R}^d} \frac{T(y)}{|x-y|^{d-2s}} dy + l$$
    where $l \in \mathbb{R}$ is a constant.
  • Significance: This provides a rigorous representation formula for distributional solutions, generalizing results for polyharmonic operators to the fractional setting. It confirms that the Riesz potential is the unique solution within the class of functions with controlled growth.

B. Brezis–Browder Type Results for Fractional Spaces

The paper establishes conditions under which the duality pairing equals the integral of the product:
$$\langle T, u \rangle = \int_{\mathbb{R}^d} T(x)u(x) dx$$

• Theorem 1.3 (General Case, $s > 0$):

  • Setting: Homogeneous Sobolev spaces $\dot{H}^{s,p}(\mathbb{R}^d)$ and their duals.
  • Condition: Requires $T \in \dot{H}^{-s,p'}(\mathbb{R}^d) \cap L^{\bar{q}}_{loc}(\mathbb{R}^d)$, where $\bar{q}$ depends on $d, s, p$. Specifically, if $s \notin \mathbb{N}$ and $1 < p < d/s$, $T$ must have higher local integrability ($L^{\bar{q}}$).
  • Result: If $Tu \geq -|f|$ for some $f \in L^1$, then $Tu \in L^1$ and the integral representation holds.
  • Significance: This extends the classical Brezis–Browder theorem to fractional orders. The requirement for extra local integrability of $T$ is necessary because the standard truncation sequences for fractional spaces (unlike the integer case) do not naturally remain in $L^\infty$.

• Theorem 4.2 (Low Order Regime, $0 < s < 1$):

  • Setting: Fractional Sobolev spaces defined via the Gagliardo–Slobodeckiĭ seminorm ($\dot{W}^{s,p}$).
  • Result: For $0 < s < 1$, the Brezis–Browder equality holds without requiring extra local integrability on $T$ (beyond $T \in Y' \cap L^1_{loc}$).
  • Significance: This is a sharp improvement over Theorem 1.3 for the range $0 < s < 1$. The author achieves this by constructing a specific bounded approximating sequence $(u_n)$ that converges strongly in the fractional norm, bypassing the need for the extra integrability condition required in the general case.

4. Significance and Applications

• Theoretical Foundation: The paper bridges the gap between classical integer-order potential theory and fractional calculus, providing rigorous justification for representation formulas that are often used heuristically in physics and applied mathematics.
• Liouville Theorems: The characterization of $s$-harmonic functions as polynomials is crucial for uniqueness results in boundary value problems and for understanding the asymptotic behavior of solutions.
• Non-local Analysis: The results clarify the specific integrability requirements needed to handle the non-local operator $(-\Delta)^s$ in distributional settings, distinguishing between the behavior of integer-order ($s \in \mathbb{N}$) and fractional-order ($s \notin \mathbb{N}$) operators.
• Applications: The author notes that these results are applicable to the study of Thomas–Fermi type integral equations and other non-local models arising in plasma physics and quantum mechanics.

In summary, Damiano Greco provides a comprehensive analysis of the structural properties of solutions to fractional Poisson equations and establishes robust duality results for fractional Sobolev spaces, refining the conditions under which distributional pairings can be computed as standard integrals.