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On some sharp weighted norm inequalities. (English) Zbl 1140.42005
Let $$M$$ be the Hardy-Littlewood maximal operator and $$\omega$$ be a weight function, i.e. a locally integrable non-negative function on $$\mathbb R^n$$. It is proved that $\| M(Tf)\| _{L^p_{\omega}} \leq c\| \omega\| _{A_p}^{\alpha_p}\| Mf\| _{L^p_{\omega}}$ for any Calderón-Zygmund operator $$T$$ with a constant $$c$$ depending only on $$1 < p < \infty$$ and the dimension $$n$$. Here $$\| \omega\| _{A_p}$$ denotes the Muckenhoupt $$A_p$$ characteristic of $$\omega$$, and it is proved that the exponent $$\alpha_p = \max\{1,1/(p-1)\}$$ is sharp. Moreover, if $$Tf$$ is either the area integral $$S(f)$$ or the Littlewood-Paley function $$g_{\mu}^*(f)$$, $$\mu > 3$$, then the analogous norm estimate holds with exponent $$\alpha_p = \max\{1/2,1/(p-1)\}$$, and this exponent is again sharp. The proof is based on estimates for local sharp maximal functions.

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