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A note on Triebel-Lizorkin spaces. (English) Zbl 0698.42008
Approximation and function spaces, Proc. 27th Semest., Warsaw/Pol. 1986, Banach Cent. Publ. 22, 391-400 (1989).
[For the entire collection see Zbl 0681.00013.]
The author gives two characterizations of anisotropic Triebel-Lizorkin spaces, (A) one in terms of differences, (B) the other in terms of mean oscillation, which are generalizations and improvements of the usual (isotropic) cases. His result in the isotropic case of (B) is as follows. For \(r\geq 1\), k integer, a ball \(B=B(x,t)\) with center x and radius t, and \(f\in L^ 1_{loc}({\mathbb{R}}^ n)\), let \[ osc^ k_ r(f,B)=\inf (| B|^{-1}\int_{B}| f(y)-P(y)|^ r dy)^{1/r}, \] where the infimum is taken over all polynomials P of degree \(\leq k\). Suppose \(0<p<\infty\), \(0<q\leq \infty\), \(r\geq 1\), \(k>\epsilon >\max \{0,n(1/p-1/r),n(1/q-1/r)\}\). Then \(f\equiv L^ 1_{loc}({\mathbb{R}}^ n)\) belongs to the homogeneous Triebel-Lizorkin space \(\dot F{}^{\epsilon}_{pq}\) if and only if \[ (\int^{\infty}_{0}[osc_ r^{k-1}(f,B(x,t))]^ q dt/t^{1+\alpha q})^{1/q}<\infty. \] He also characterizes anisotropic Triebel-Lizorkin spaces on domains, using his results on \({\mathbb{R}}^ n\).
Reviewer: K.Yabuta

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces