Pointwise multipliers of Besov spaces on Carnot–Carathéodory spaces

Let ( X , d , μ ) be a Carnot–Carathéodory space, namely, X is a smooth manifold, d is a control, or Carnot–Carathéodory, metric induced by a collection of vector fields of finite type. μ is a nonnegative Borel regular measure on X satisfying that there exists constant C 0 ∈ [ 1 , ∞ ) such that for all x ∈ X and 0 < r < diam X , μ ( B ( x , 2 r ) ) : = μ ( { y ∈ X : d ( x , y ) < 2 r } ) ≤ C 0 μ ( B ( x , r ) ) < ∞ ( doubling property ) . Using the discrete Calderón reproducing formula and the Plancherel–Pôlya characterization of the inhomogeneous Besov spaces developed by Han et al. [12], and Han et al. (2008) [10], pointwise multipliers of inhomogeneous Besov spaces are obtained.