On a Hardy and Littlewood imbedding theorem (Q1076269)

The Hardy-Littlewood imbedding theorem, cited by the authors, is the following: \[ \{\int^{1}_{0}[(1-\rho)^{1/p- 1/r}(\int^{2\pi}_{0}| f(\rho e^{i\theta})| \quad^ rd\theta)^{1/r}]^ q(1-\rho)^{-1}d\rho \}^{1/q}\leq C\| f\|_{H^ p}, \] where \(0<p\leq r\leq \infty\), \(p\leq q\leq \infty\). The authors give general versions of this theorem in the following setting. Let (X,\(\mu)\), (T,\(\nu)\) be positive measure spaces. Assume that to each (x,t)\(\in X\times T\) one associates a \(\mu\)-measurable set B(x,t)\(\subset X\) so that \({\mathcal B}=\{B(x,t)\}\) satisfies (i) \(x\in B(x,t)\), \(t\in T\); (ii) if \(y\in B(x,t)\), then \(x\in B(y,t)\); and (iii) \(0<\mu (P(x,t))<\infty\). Now for functions f on \(X\times T\) one sets \(M_{{\mathcal B}}f(x)=\sup_{t\in T} \sup_{y\in B(x,t)}| f(y,t)|.\) Using this maximal function, they give general versions of the above theorem, one of which contains the above theorem as an easy corollary. Their proofs are simple. Several applications are given to the real Hardy spaces of Fefferman-Stein, Calderón-Torchinsky, the complex Hardy spaces on smooth domains in \({\mathbb{C}}^ n\), and the Hardy spaces in the poly half-space.