Change of variables for weighted Hardy spaces on a domain (Q842317)

The author introduces a generalized version of weighted Hardy spaces \(H^p(\Omega, T, \lambda)\) on domains \(\Omega\) in \(\mathbb R^n\), where \(\lambda\) is a measure on \(\Omega\) satisfying a doubling condition, and \(T\) is a function on \(\Omega\) satisfying (i) \(0<T(x)\leq d(X,\Omega^c)\) and (ii) \(|T(x)-T(y)|\leq |x-y|\). The spaces are real-variable Hardy spaces defined in terms of the \(L^p(\Omega,\lambda)\) norm of a certain maximal function. The spaces \(H^p(\Omega, T,\lambda)\) include as special cases the local Hardy space \(h^p(\mathbb R^n)\) of \textit{D. Goldberg} [Duke Math. J. 46, 27--42 (1979; Zbl 0409.46060)], and the space \(H^p(\Omega) \) introduced by the author [Stud. Math. 96, No. 3, 205--228 (1990; Zbl 0716.42017)]. In the paper it is proved that the spaces are transformed to the same kind of spaces by certain smooth changes of variable. The paper also gives an atomic decomposition theorem for functions in \(H^p(\Omega, T, \lambda)\).