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\(T1\) theorems on generalized Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. (English) Zbl 0995.42011

Let \((X, d, \mu)\) be a space of homogeneous type, \(\roman{i.e.}\), \(X\) is a topological space, defined by a pseudo metric \(d\), and \(\mu\) is a positive Borel measure on \(X\), such that \(d(x,y)=d(y,x)\geq 0\) and \(=0\) iff \(x=y\), \(d(x,z)\leq K(d(x,y)+d(y,z))\), and \(\mu(B(x,2t))\leq A \mu(B(x,t))\). Here, \(B(x,t)=\{y\in X; d(x,y)<t\}\). The authors treat the space \((X, d, \mu)\) of homogeneous type, satisfying \(|d(x,y)-d(x',y)|\leq Cr^{1-\theta}d(x,x')^\theta\) if \(d(x,y), d(x',y)<r\), and \(A_1r\leq \mu(B_d(x,r))\leq A_2r\) for every \(x\in X\) and \(r>0\). A nonnegative function \(\phi(t)\) defined on \((0, \infty)\) is said to be of lower (upper) type \(\alpha \geq 0\) if \(\phi(st)\leq(\geq) C_1s^\alpha \phi(t)\) for \(0<s\leq 1\) and \(t>0\).
In the context of spaces of homogeneous type, G. David, J.L. Journé and S. Semmes showed how to construct an appropriate family of operators \(\{D_k\}_{k\in \mathbb Z}\) whose kernels satisfy certain size, smoothness and moment conditions and \(\sum_{k\in \mathbb Z}D_k=I\) on \(L^2\). Y. Han and E. Sawyer introduced a class of distributions on spaces of homogeneous type and established a Calderón-type reproducing formula for this class, associated with the family of operators in the above. Using this formula, they defined the homogeneous Besov spaces \(\dot B_p^{\alpha, q}\), \(1\leq p, q<\infty\), and the homogeneous Triebel-Lizorkin spaces \(\dot F_p^{\alpha, q}\), \(1< p, q<\infty\), like as in \(\mathbb R^n\). The authors consider more general functions \(\psi(t)\) than the measure function \(t^\alpha\), i.e., \(\psi(t)\) is of the form \(\psi(t)=\varphi_1(t)/\varphi_2(t)\), where \(\varphi_1\), \(\varphi_2\) are almost increasing functions of upper type. They define the homogeneous Besov spaces \(\dot B_p^{\psi, q}\), \(1\leq p, q<\infty\), and the homogeneous Triebel-Lizorkin spaces \(\dot F_p^{\psi, q}\), \(1< p, q<\infty\), on spaces of homogeneous type. They also treat \(T1\) theorems of boundedness of generalized Calderón-Zygmund singular integral operators on these spaces for kernels satisfying integral conditions of size and smoothness, which generalize the work on \(\mathbb R^n\) by Y.-S. Han and S. Hofmann [“\(T1\) theorems for Besov and Triebel-Lizorkin spaces”, Trans. Am. Math. Soc. 337, No. 2, 839-853 (1993; Zbl 0779.42010)].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
43A85 Harmonic analysis on homogeneous spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Citations:

Zbl 0779.42010
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