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Carleson conditions for the Poisson integral. (English) Zbl 0748.42009
The authors study the weighted norm inequality $\left(\iint_{\mathbb{R}_ +^{n+1}}| Pf(x,t)|^ qw(x,t)dx dt\right)^{1/q}\leq c\left(\int_{\mathbb{R}^ n}| f(x)|^ pv(x)dx\right)^{1/p}$ with a constant $$c$$ independent of $$f$$, where $$Pf$$ is the Poisson integral of $$f$$ and $$1<p\leq q<\infty$$. They obtain necessary and sufficient conditions for this inequality to be true. These conditions simplify the previous ones obtained by the same authors. The case $$p=q$$ which brings additional difficulties is considered separately.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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