Hardy spaces and the \(Tb\) Theorem. (English) Zbl 1059.42015
Summary: It is well known that Calderón-Zygmund operators \(T\) are bounded on \(H^p\) for \(\frac{n}{n+1}<p\leq 1\) provided \(T^*(1)= 0\). In this article, it is shown that if \(T^*(b)=0\), where \(b\) is a para-accretive function, \(T\) is bounded from the classical Hardy space \(H^p\) to a new Hardy space \(H^p_b\). To develop an \(H^p_b\) theory, a discrete Calderón-type reproducing formula and Plancherel-Pólya-type inequalities associated to a para-accretive function are established. Moreover, the result of G. David, J. L. Journé and S. Semmes [Rev. Mat. Iberoam. 1, No. 4, 1–56 (1985; Zbl 0604.42014)] about the \(L^p\), \(1<p<\infty\), boundedness of the Littlewood-Paley \(g\) function associated to a para-accretive function is generalized to the case of \(p\leq 1\). A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.
Keywords:
Calderón-Zygmund operator; discrete Calderón formula; Hardy space; Plancherel-Pólya inequality; Littlewood-Paley \(g\) function \(Tb\) theroemCitations:
Zbl 0604.42014References:
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