Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients

Abstract

We study the representation of bilinear forms associated with Riesz fractional gradients by symmetric integral kernels. Motivated by [SS18, Open Problem 1.10], we give a complete description of the constant coefficient case. We prove that any symmetric representing kernel is uniquely determined and we derive its explicit formula in terms of the constant matrix $A$. This yields a sharp characterization of the matrices for which the kernel is either nonnegative or uniformly positive. In particular, [SS18, Open Problem 1.10] has a negative answer (even for constant matrices). We also identify a class of nonconstant antisymmetric divergence free perturbations that are invisible to the fractional bilinear form and hence leave the representing kernel unchanged. The general case of nonconstant matrices exhibits a substantially different behavior and will be the main subject of our forthcoming paper [DSV26].