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Tytuł książki

New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals

Seria
Rozprawy Matematyczne tom/nr w serii: 403 wydano: 2002
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Warianty tytułu
Abstrakty
EN
Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, $(X,ϱ,μ)_{d,θ}$, 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| < θ. 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 μ(X) < ∞. Parts of these results are new even when $(X,ϱ,μ)_{d,θ}$ is the n-dimensional Euclidean space, or a compact d-set, Γ, in ℝⁿ, 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 μ(X) < ∞. We also discuss the relationship between Hajłasz-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.
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Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 403
Liczba stron
102
Liczba rozdzia³ów
Opis fizyczny
Daty
wydano
2002
Twórcy
  • Department of Mathematics, Auburn University, Auburn, AL 36849-5310, U.S.A.
autor
  • Department of Mathematics, Beijing Normal University, Beijing 100875, People's Republic of China
Bibliografia
Języki publikacji
EN
Uwagi
Identyfikator YADDA
bwmeta1.element.bwnjournal-rm-doi-10_4064-dm403-0-1
Identyfikatory
DOI
10.4064/dm403-0-1
Kolekcja
DML-PL
Zawartość książki

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