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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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