A spectral multiplier theorem for fractional Hardy operators

We consider the fractional Laplacian with added critical or subcritical Hardy potential, called Hardy operator. We prove a Hörmander spectral multiplier theorem, i.e., the boundedness of sufficiently smooth multipliers of these Hardy operators on Lebesgue spaces. The range of admissible Lebesgue exponents is optimal, but the smoothness assumption is probably not. The proof combines the Helffer–Sjöstrand formula with a quantitative Jaffard lemma for operator-valued matrices.