Kernel representations of fractional gradient bilinear forms. Part II. Variable coefficients

Introduction

In the classical theory of elliptic equations, divergence form operators of the type

where $A:\mathbb{R}^{N}\to\mathbb{R}^{N\times N}$ is a matrix (or, more precisely, a matrix-valued function), provide a natural framework for describing anisotropy and spatial inhomogeneity. A fractional counterpart of this construction can be obtained by replacing the classical gradient with the Riesz fractional gradient. For $s\in(0,1)$ and $u\in C^\infty_c(\mathbb{R}^{N})$, the latter is defined by

where

(we refer to [1 Section 1.5] and the references therein for further background on fractional gradients).

The Riesz fractional gradient has been extensively studied as a vector-valued counterpart of the classical gradient, particularly in connection with fractional divergence operators and weak formulations. This framework has proved useful in the study of functions of bounded variation, the development of fractional Sobolev, Hardy, Trudinger and Morrey inequalities, and the analysis of $\Gamma$-convergence (see e.g. [SS15, SSS15, SSVS17, SSS18, SS18, CS19, Sil20, BCMC20, BCMC21]). In a related direction, [2] constructs nonlocal operators through suitable interaction kernels, recovering classical elliptic operators in divergence form in the limit $s\nearrow1$. This provides another connection between nonlocal kernels and elliptic operators with matrix-valued coefficients.

Despite these developments, the representation of energies involving variable matrix fields through suitable interaction kernels remains a delicate question. Indeed, this construction leads naturally to energies of the form

whose positivity and coercivity follow from the uniform ellipticity of $A$. However, the uniform ellipticity alone does not guarantee, in general, that this energy can be represented in terms of pairwise differences $u(x)-u(y)$ through a nonnegative symmetric kernel.

For the fractional Laplacian, such interactions are described by a positive symmetric kernel proportional to $\lvert x-y\rvert^{-N-2s}$ through the constant

This representation plays a fundamental role in the analysis of nonlocal equations, as it makes the interactions explicit and allows the use of techniques that rely on the positivity and decay of the kernel. For general matrices with variable coefficients, it is not clear whether an analogous representation exists. Indeed, the positivity of the energy does not, by itself, guarantee positivity of the underlying interaction kernel.

The present paper is the second part of our investigation of this question. In [DSV26], we considered the case of constant matrices, deriving an explicit formula for the associated kernel and studying how its sign depends on the constant matrix $A$. Here we turn to the variable coefficients case, which introduces new difficulties, both in identifying the kernel and in understanding its behavior near the diagonal and at infinity. Our starting point is the following question, raised in [SS18, Open Problem 1.10]:

Can one find a nonnegative kernel $k_A$ such that

and, for some $\alpha\in(0,2)$, $\eta>0$, $0<\lambda\leq\Lambda<+\infty$, and $M>0$,

and

such that

for every $u,v\in C^\infty_c(\mathbb{R}^N)$?

We show that the answer to this question is negative in general. Our analysis also provides a detailed description of the properties of representing kernels, starting with the restrictions on their singularity near the diagonal and their decay at infinity.

Singularity and decay of the kernel

Here we provide necessary conditions that can be obtained without an explicit representation of the kernel $k_A(x,y)$. In particular, we determine the only possible order of its singularity at the diagonal and we derive restrictions on its decay at infinity. These results show that the order $2s$ plays a distinguished role, both in the local behavior of the kernel and in its decay at infinity.

Throughout the paper we will use the notation

To begin with, we show that the singularity exponent in (1.5) is necessarily determined by the order of the fractional gradient. More precisely, we have the following:

We stress that the conclusion $\alpha=2s$ in Theorem 1.2 follows under assumptions weaker than those required in Open problem 1.1. Indeed, our proof neither uses the symmetry condition (1.4), nor requires the kernel to be nonnegative on the whole of $\mathbb{R}^{2N}\setminus\Delta$. The positivity of the kernel is only needed near the diagonal, and it is already contained in the lower bound in (1.5).

We next prove that the exponent $\eta$ appearing in (1.6) cannot be taken strictly larger than $2s$.

In addition, assume (1.9).

Then, $0 < \eta\le2s$.

We point out that, as in Theorem 1.2, also in Theorem 1.3 neither the symmetry assumption (1.4) nor the global nonnegativity of the kernel is needed. Moreover, the assumptions required in Theorem 1.3 are weaker than those used in Theorem 1.2. Indeed, in Theorem 1.2, the lower bound in (1.5) is essential, since it provides a positive contribution which prevents cancellations. In contrast, Theorem 1.3 only requires an upper bound on the kernel, since the lower bound of the relevant energy is provided by the uniform ellipticity of the matrix $A(x)$.

Theorem 1.3 shows that, if the kernel at infinity is controlled from above by a quantity proportional to $|x-y|^{-N-\eta}$, then necessarily $\eta\le2s$. We will also prove the complementary statement, namely if, for $|x-y|>1$, the kernel is bounded from below by a positive multiple of $|x-y|^{-N-\gamma}$, then necessarily $\gamma\ge2s$ (see the forthcoming Theorem 2.2). Accordingly, if the kernel is bounded from above and below by positive multiples of the same power of $|x-y|$, then that exponent must be exactly $2s$ (see the forthcoming Corollaries 2.3 and 2.4).

An explicit representation formula

The results discussed above provide necessary information on the singularity and the decay of any kernel representing the quadratic form (1.9), but do not give an explicit expression for this kernel. Thus, we next turn to the identification of the kernel in terms of the matrix $A(x)$.

We first work only assuming that $A$ is bounded, measurable and symmetric. Thus, given $\varepsilon>0$, we define the truncated kernel

We will show that, if a symmetric kernel $k_A(x,y)$ represents the bilinear form associated with $A(x)$, then the truncated kernels $k_{A,\varepsilon}(x,y)$ recover $k_A(x,y)$ at the level of the corresponding bilinear forms (see the forthcoming Theorem 3.1).

We stress that, without imposing any continuity assumption on $A(x)$, the kernel is identified in a weak sense through the family $k_{A,\varepsilon}(x,y)$. However, this result does not yet provide a pointwise formula for the kernel $k_A(x,y)$. To obtain such a representation, we impose an additional local regularity assumption on $A(x)$. For every compact set $K\subset\mathbb{R}^{N}$, we set

and we assume that $A$ satisfies the uniform fractional Dini condition