Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type (Q1602281)

Let \((X, d, \mu)\) be a space of homogeneous type, \(\roman{i.e.}\), \(X\) is a topological space, defined by a pseudometric \(d\), 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 \(\varphi(t)\) defined on \((0, \infty)\) is said to be of lower (upper) type \(\alpha \geq 0\) if \(\varphi(st)\leq(\geq) C_1s^\alpha \varphi(t)\) for \(0<s\leq 1\) and \(t>0\). \quad 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 the operators mentioned above. The authors use inhomogeneous approximations to the identity \(\{S_k\}_{k\in \mathbb N}\), defined by \textit{Y. Han} [J. Geometric Anal. 7, 259-284 (1997; Zbl 0915.42010)], and give a Littlewood-Paley theorem for a Littlewood-Paley \(g\)-function associated with this approximation to the identity. Based on it, they give inhomogeneous discrete Calderón-type reproducing formulas.