## New characterizations and applications ofinhomogeneous Besov and Triebel–Lizorkin spaces onhomogeneous type spaces and fractals

### Volume 403 / 2002

#### Abstract

Let $d>0$ and $\theta\in (0,1]$. We consider homogeneous type spaces, $(X,\varrho,\mu)_{d,\theta}$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces ${B^s_{pq}(X)}$ and Triebel–Lizorkin spaces ${F^s_{pq}(X)}$ have the lifting properties for $|s|<\theta$. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional integrals and derivatives are independent of the choices of approximations to the identity, and obtain some Poincaré-type inequalities. We also establish frame decompositions of ${B^s_{pq}(X)}$ and ${F^s_{pq}(X)}$. Applying these, we obtain estimates of the entropy numbers of compact embeddings between ${B^s_{pq}(X)}$ or ${F^s_{pq}(X)}$ when $\mu(X)<\infty$. Parts of these results are new even when $(X,\varrho,\mu)_{d,\theta}$ is the $n$-dimensional Euclidean space, or a compact $d$-set, ${\mit\Gamma}$, in $\mathbb R^n$, which includes various kinds of fractals. We also establish some limiting embeddings between these spaces, and by considering spaces $L^p(\log L)_a(X)$, we then establish some limiting compact embeddings and obtain estimates of their entropy numbers when $\mu(X)<\infty.$ We also discuss the relationship between Haj/lasz–Sobolev spaces of order 1 and the spaces defined by our methods. Finally, we give some applications of the estimates of the entropy numbers to estimates of eigenvalues of some positive-definite self-adjoint operators related to quadratic forms.