Anisotropic Function Space

Isotropic classical Besov Bp,qs(Rn)and Triebel-Lizorkin Fp,qs(Rn)space with 0<p≤∞,0<q≤∞have been well studied.Numbers of Known results contain Holder-Zygmund space C(Rn)=B∞,∞s(Rn), s∈R,fractional Sobolev spaces Hps(Rn)=Fp.2s(Rn),s∈R,0<p<∞,in par-ticular,F∞,20(Rn)=Lp(Rn),0<p<∞.H.Triebel also develops a systematic treatment of the theory of Bp,qs(Rn)and Fp,qs(Rn)space.To establish and perfect the anisotropic function space,the primary task is to distinguish isotropic and anisotropic function space. Therefore,we first give the visual description of two function spaces. Classic isotropic function space structure depends on binary decomposition, precisely, is dependent on standard norm‖·‖,a list of standard norm balls{2j{‖x‖≤1}}j∈{0}UN and positive sequence{2k}k∈N. Standard norm‖·‖guaranteed that{Φk}0∞have close support sets,the balls{2j{‖x‖≤1}}j∈{0)UN specify the support set of {Φk}0∞,and positive sequence {2k}k∈N plays a large role in proof of∑Φk=1. Because of the one-to-one relationship between the balls of{2j{‖x‖≤1}}j∈N and matrix column of{Aj=2jI}j∈N,we use matrix replace ball.Therefore,roughly, index group column of{(‖·‖,Ak,2k)}k∈N decided the isotropic function space. Index group column method can establish more such function spaces.We will in the third chapter present the specific content.Let∨k∈nN,Ak be the diagonal matrixes rather than the times of the unit matrix.∑aii=n,aii>0. Let quasi norm p(x)satisfies that∨x∈Rn,p(Akx)=2kp(x).In fact,the so-called anisotropic function space means the function space that{(p(x),Ak,2k)}k∈N induced.Obviously,index group columns that isotropic and anisotropic function space depend on differ.In geometrical speaking,isotropic function space makes"stan-dard norm unit sphere"(more accurately are tight sets)uniformly into a list of "balls". However:anisotropic one makes"quasi norm unit sphere" into a list of "quasi norm balls" in the way that all directions inhomogeneous expand.At the same time, anisotropic function spaces have been developed to be a complete theory.The aim of this paper is threefold:1. We look back to the process about how Besov and Triebel-Lizorkin space is built.2. we try to introduce the theory of anisotropic function spaces and enrich our knowledge on this new function space3. At last, we explore some general anisotropic function spaces.