A Lions' type formula for some reproducing kernel Hilbert spaces of fractional harmonic functions

Abstract: In \cite{Lions}, J. L. Lions considered a reproducing kernel Hilbert space (RKHS) of harmonic functions on a regular domain with Sobolev traces and obtained a formula that expresses the kernel of this space as an integral on the boundary of some derivatives of the Green function associated with the Laplace operator and the homogeneous Dirichlet boundary condition. This result was simplified and extended later by Englis, Lukkassen, Peetre, and Persson in \cite{ELPL} to more general elliptic systems of even orders. In particular, they emphasized that the resemblance between Lions' type formula and the Hadamard variational formula only appears when the operator is of order $2$. In this paper, we investigate some RKHS of $a$-harmonic functions, where $a$ in $(0,1)$ refers to a fractional exponent of the Laplace operator. For such fractional order pseudo-differential operators, the local nonhomogeneous Dirichlet problem can be addressed by means of some $a$-transmission Sobolev spaces, which were introduced by Hörmander in the sixties and recently developed by Grubb in a series of papers. We deduce from these works a fractional Poisson formula, which is applied to obtain a Lions' type formula. We observe, in particular, that despite the order of the operator not being $2$, this formula resembles the Hadamard variational formula that we prove in the companion paper \cite{sidy-franck_1}. As a complementary remark, we observe that for a family of RKHS associated with the steady Stokes system, a second order system, there is also a Lions' type formula for their two-point kernels, which turn out not to be similar to the corresponding Hadamard variation formula.

• The paper establishes that spaces of fractional harmonic functions form reproducing kernel Hilbert spaces for fractional orders a in (0,1) and trace indices s > -a - 1/2.
• It derives the kernel as a boundary integral of products of smoothed fractional Green-function traces, using Grubb’s transmission theory and a fractional Poisson formula.
• The result refutes the conjecture that Lions–Hadamard structural parallels occur only for second-order operators, while the Stokes example shows that boundary-trace structure remains decisive.

Background: Lions' formula for harmonic RKHS

The paper by Djitte and Sueur (2602.09669) extends a classical result of J.-L. Lions on reproducing kernel Hilbert spaces (RKHS) of harmonic functions to the fractional setting. Recall that a Hilbert space $H$ of functions on a bounded connected open set $\Omega \subset \mathbb{R}^N$ of class $C^\infty$ is an RKHS if every point evaluation $E_x(u) = u(x)$ is continuous; by Riesz' theorem each such functional is represented by an element $K_x$, and the two-point kernel is $K(x,y) = \langle K_x, K_y\rangle$.

Lions considered the space $\mathcal{H}_s(\Omega)$ of harmonic functions in $H^{s+1/2}(\Omega)$ whose Dirichlet trace lies in $H^s(\partial\Omega)$, with the inner product transported from $H^s(\partial\Omega)$ via the Poisson map. Writing $\Omega \subset \mathbb{R}^N$0 for the smoothed Laplace–Beltrami operator $\Omega \subset \mathbb{R}^N$1 on $\Omega \subset \mathbb{R}^N$2 and $\Omega \subset \mathbb{R}^N$3 for the Green function of $\Omega \subset \mathbb{R}^N$4 with homogeneous Dirichlet condition, Lions' formula reads

$\Omega \subset \mathbb{R}^N$5

Englis, Lukkassen, Peetre and Persson later gave a direct proof and extended the result to elliptic systems of even order. Crucially, they observed that Lions' formula resembles Hadamard's variational formula,

$\Omega \subset \mathbb{R}^N$6

only when the operator has order $\Omega \subset \mathbb{R}^N$7. They explicitly conjectured that this connection "seems to be an isolated phenomenon peculiar to the second order case." The present paper disproves that conjecture.

Fractional Laplacians and their Hadamard-type variation

For $\Omega \subset \mathbb{R}^N$8, the fractional Laplacian $\Omega \subset \mathbb{R}^N$9 is defined via the principal-value integral with normalization constant $C^\infty$0 matching the Fourier multiplier symbol $C^\infty$1, or equivalently as the generator of a symmetric stable process. Its Green function $C^\infty$2 solves $C^\infty$3 in $C^\infty$4 with $C^\infty$5 in $C^\infty$6 — an exterior condition reflecting the nonlocality of the operator.

In the companion paper (Djitte et al., 6 Feb 2026), the authors established a Hadamard-type variational formula for $C^\infty$7: under normal perturbations of the boundary with velocity $C^\infty$8,

$C^\infty$9

where $E_x(u) = u(x)$0 is the fractional trace weighted by the distance function $E_x(u) = u(x)$1 to the boundary. By Grubb's boundary regularity results, $E_x(u) = u(x)$2 is $E_x(u) = u(x)$3 on $E_x(u) = u(x)$4, so the integrand is well defined. This formula is the structural template against which the new kernel formula should be compared.

The $E_x(u) = u(x)$5-transmission framework

The analytic backbone of the paper is Hörmander's theory of pseudodifferential operators satisfying the $E_x(u) = u(x)$6-transmission property, developed extensively by Grubb. The fractional Laplacian satisfies this property with $E_x(u) = u(x)$7, which permits solvability of nonhomogeneous local Dirichlet problems in the $E_x(u) = u(x)$8-transmission Sobolev spaces $E_x(u) = u(x)$9, defined via the order-reducing operators $K_x$0 in the half-space and by localization for general domains.

Two facts from Grubb's work are essential. First, the fractional trace map $K_x$1 extends to a surjective continuous map from $K_x$2 onto $K_x$3 with kernel $K_x$4. Second, for any $K_x$5 there exists a unique solution $K_x$6 of

$K_x$7

with maximal regularity permitted by the trace theorem. These ingredients define the solution operator $K_x$8 as an isomorphism between $K_x$9 and the space of $K(x,y) = \langle K_x, K_y\rangle$0-harmonic functions, where

$K(x,y) = \langle K_x, K_y\rangle$1

The fractional Poisson formula

A key intermediate result is a fractional Poisson representation: for $K(x,y) = \langle K_x, K_y\rangle$2 (the space of $K(x,y) = \langle K_x, K_y\rangle$3-harmonic functions in $K(x,y) = \langle K_x, K_y\rangle$4 supported in $K(x,y) = \langle K_x, K_y\rangle$5, equipped with the scalar product of $K(x,y) = \langle K_x, K_y\rangle$6 applied to $K(x,y) = \langle K_x, K_y\rangle$7),

$K(x,y) = \langle K_x, K_y\rangle$8

The proof is instructive: it approximates the trace by smooth data, applies Grubb's exact Green formula to the pair $K(x,y) = \langle K_x, K_y\rangle$9 where $\mathcal{H}_s(\Omega)$0 is a mollification of the Green function at $\mathcal{H}_s(\Omega)$1, and passes to the limit using dominated convergence together with the standard estimate $\mathcal{H}_s(\Omega)$2. Interior elliptic regularity then identifies pointwise limits. This regularization argument handles the interior singularity of the Green function, which is the main technical obstruction beyond the classical case.

Main result: the fractional Lions formula

The central theorem states that for all $\mathcal{H}_s(\Omega)$3 and $\mathcal{H}_s(\Omega)$4, the space $\mathcal{H}_s(\Omega)$5 is an RKHS whose two-point kernel is

$\mathcal{H}_s(\Omega)$6

The proof follows Lions' strategy: continuity of evaluations follows from the Poisson formula and Cauchy–Schwarz; Riesz representation plus identification with the Poisson formula yields $\mathcal{H}_s(\Omega)$7 almost everywhere; taking inner products gives the kernel.

This result directly contradicts the conjecture of Englis–Lukkassen–Peetre–Persson. Their formula for second-order operators mirrors the Hadamard variational formula only because the operator order equals $\mathcal{H}_s(\Omega)$8; yet here, for operators of fractional order $\mathcal{H}_s(\Omega)$9, the kernel formula is structurally identical to the fractional Hadamard formula of the companion paper — both are boundary integrals of products of $H^{s+1/2}(\Omega)$0-smoothed fractional traces of the Green function, differing only by the constant prefactor $H^{s+1/2}(\Omega)$1 versus $H^{s+1/2}(\Omega)$2 and the smoothing power. The resemblance between reproducing kernels and shape derivatives is therefore not peculiar to order $H^{s+1/2}(\Omega)$3.

Two consistency checks support the result. As $H^{s+1/2}(\Omega)$4, one recovers the classical formula in the sense that $H^{s+1/2}(\Omega)$5, reflecting the shift by $H^{s+1/2}(\Omega)$6 in the boundary Sobolev index between the fractional and classical parametrizations. Moreover, since the boundary behavior of $H^{s+1/2}(\Omega)$7 is well understood, the formula provides a tool for analyzing the boundary behavior of the reproducing kernel itself.

The steady Stokes counterpoint

As a complementary observation, the authors show that the analogy can fail even at order $H^{s+1/2}(\Omega)$8 when the operator is a system. For $H^{s+1/2}(\Omega)$9, consider the RKHS $H^s(\partial\Omega)$0 of traces on $H^s(\partial\Omega)$1 of solutions to the steady Stokes problem $H^s(\partial\Omega)$2, $H^s(\partial\Omega)$3, endowed with the $H^s(\partial\Omega)$4 scalar product. Using the classical Poisson formula involving the stress tensor $H^s(\partial\Omega)$5 of the Stokes Green function $H^s(\partial\Omega)$6, the same reasoning yields a Lions-type formula:

$H^s(\partial\Omega)$7

By contrast, the known Hadamard variational formula for the Stokes Green function (due to Simon, Fujiwara–Ozawa, Kozono–Ushikoshi) involves the Neumann traces $H^s(\partial\Omega)$8 rather than the Dirichlet traces of the traction $H^s(\partial\Omega)$9. The two formulas are thus not structurally similar. This shows that the kernel–variation correspondence depends not merely on the order of the operator but on the specific structure of the boundary operator appearing in the Green identity — the traction trace for Stokes versus the conormal derivative for the Laplacian, and the fractional normal trace for $H^s(\partial\Omega)$0.

Limitations and open questions

Several restrictions frame the applicability of the results. The domain is assumed $H^s(\partial\Omega)$1 throughout, and the fractional exponent is confined to $H^s(\partial\Omega)$2; extension to $H^s(\partial\Omega)$3 would require handling higher-order transmission conditions and additional boundary operators. The range $H^s(\partial\Omega)$4 is dictated by the trace theorem, so kernels for rougher trace spaces are not covered. The Hadamard comparison relies on the companion paper's variational formula, which was proved for more general perturbations but within its own hypotheses. Finally, the Stokes section is presented as a remark without full proofs, and the precise mechanism determining when a Lions-type kernel resembles the corresponding Hadamard formula — apparently governed by which trace operator appears in the exact Green formula — is left unformalized. Identifying that mechanism for general elliptic boundary value problems remains an open question raised implicitly by the contrast between the fractional Laplacian and the Stokes system.

Conclusion

The paper establishes a Lions-type integral formula for the reproducing kernels of Hilbert spaces of $H^s(\partial\Omega)$5-harmonic functions built on Grubb's $H^s(\partial\Omega)$6-transmission Sobolev theory, and shows that this formula parallels the Hadamard variational formula for the fractional Green function. In doing so it refutes the long-standing conjecture that the kernel–variation resemblance is exclusive to second-order operators, while the Stokes example demonstrates that the correspondence is sensitive to the boundary operator structure rather than to operator order alone.