On the gentle properties of anisotropic Besov spaces

In this paper, we prove that anisotropic homogeneous Besov spaces Ḃp,qs,u(Rd) are gentle spaces, for all parameters s,p,qs,p,q and all anisotropies u. Using the Littlewood–Paley decomposition, we study their completeness, separability, duality and homogeneity. We then define the notion of anisotropic orthonormal wavelet basis of L2(Rd)L2(Rd), and we show that the homogeneous version of Triebel families of anisotropic orthonormal wavelet bases associated to the tensor product of Lemarié–Meyer (resp. Daubechies) wavelets are particular examples. We characterize the Ḃp,qs,u(Rd) spaces using Lemarié–Meyer wavelets. In fact, we show that these bases will be either unconditional bases or unconditional ∗∗-weak bases of Ḃp,qs,u(Rd), depending on whether Ḃp,qs,u(Rd) is separable or not. By introducing an anisotropic version of the class of almost diagonal matrices related to anisotropic orthonormal wavelet bases, we prove that these spaces are stable under changes of anisotropic orthonormal wavelet bases. As a consequence, we extend the characterization of Ḃp,qs,u(Rd) using Daubechies wavelets.