Anisotropic Spaces I. (Interpolation of Abstract Spaces and Function Spaces)

Wavelet Characterizations for Anisotropic Besov Spaces

“…Thus, a standard reiteration theorem provides (38) for 1 ≤ p < ∞. Moreover, we note that in [28] one also finds the identity…”

“…We shall relate our definition for anisotropic Besov spaces to that via differences used in [29]; i.e., we shall argue that at least for the crucial range of parameters both definitions lead to the same function spaces with equivalent quasi-norms, which at least in the case of p ≥ 1 is well-known; see [26]. Hence, our final characterization results apply also to the anisotropic Besov spaces investigated in [28,29].…”

“…We remark that at least for 1 ≤ p < ∞ (38) may be concluded from results in [28,29] as follows: For¯ ∈ N 2 0 anisotropic Sobolev spaces are defined by…”

“…Therefore, the following identity (35) leads to a generalization of known interpolation results. Moreover, since for 1 ≤ p < ∞ the interpolation spaces are already known (see [2,28,29]), Eq. (32) gives directly (35); i.e., nothing new has to be proved.…”

Anisotropic Spaces. II. (Equivalent Norms for Abstract Spaces, Function Spaces with Weights of SOBOLEV‐BESOV type)

“…In [7] was given a description of the interpolation theory of these spaces, and in the case that all quotients 2 are rationalnumbersadescription of equivalent norms in Bi too. I n this paper Theorem 2 (Subsection 1.4) will show that these additional assumptions are not necessary to describe equivalent norms in BI for 1 sqz-.…”