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Some weighted norm inequalities for the area integral. (English) Zbl 0598.34019
We derive the weighted norm inequality \[ (\int_{{\mathbb{R}}^ n}S(f)(x)^ pw(x)dx)^{1/p}\leq c(p,n)(\int_{{\mathbb{R}}^ n}| f(x)|^ pw^*(x)dx)^{1/p},\quad 1<p\leq 2, \] where S(f) is the classical Poisson area integral of f and \(w^*\) is the Hardy-Littlewood maximal function of w. We also show that the inequality fails for \(p>2\) and derive a replacement result for this case. Results are also considered for area integrals formed with approximations to the identity which are Schwartz functions \(\phi\). In this way, we extend a result of Chang, Wilson and Wolff for the case when \(p=2\) and \(\phi\) has compact support.

34L99 Ordinary differential operators
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