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Rough Marcinkiewics integral operators. (English) Zbl 0995.42013
Let $$\mathbb{R}^n$$, $$n\geq 2$$, be the $$n$$-dimensional Euclidean space and $$S^{n-1}$$ be the unit sphere in $$\mathbb{R}^n$$. Consider the Marcinkiewicz integral $\mu(f)(x)= \Biggl(\int_{\mathbb{R}}|F_t(x)|^2 2^{-2t} dt\Biggr)^{1/2},$ where $F_t(x)= \int_{|x-y|< 2^t} f(y)|x-y|^{-n+1} \Omega(x- y) dy,$ and $$\Omega\in L^1(S^{n-1})$$ is a homogeneous function of degree zero that satisfies $$\int_{S^{n-1}}\Omega= 0$$.
If $$\Omega$$ satisfies, for $$\alpha\geq 0$$, $\sup_{\xi'\in S^{n-1}} \int_{S^{n-1}}|\Omega(y')|(\log|\langle y',\xi'\rangle|^{- 1})^{1+ \alpha} dy'< \infty$ then one says that $$\Omega\in V_\alpha(n)$$. This class $$V_\alpha(n)$$ was initially introduced in [L. Grafakos and A. Stefanov, Indiana Univ. Math. J. 47, No. 2, 455-469 (1998; Zbl 0913.42014)]. It is known [J. Chen, D. Fan and Y. Pan, Math. Nachr. 227, 33-42 (2001; Zbl 0994.42012)] that if $$\Omega\in V_\alpha(n)$$ then $$\mu(f)$$ is bounded on $$L^p(\mathbb{R}^n)$$ provided $$(2\alpha+ 2)/(2\alpha+1)< p< 2\alpha+2$$.
In this reviewed paper, the authors study a more general operator $M_{\mathcal P}(f)= \Biggl(\int_{\mathbb{R}} |F_{{\mathcal P},t}(x)|^2 2^{-2t} dt\Biggr)^{1/2},$ where $F_{{\mathcal P},t}(x)= \int_{|y|< 2^t} f(x- {\mathcal P}(y)) \Omega(y')|y|^{-n-1} dy$ and $${\mathcal P}= (P_1,\dots, P_d):\mathbb{R}^n\to \mathbb{R}^d$$ is a polynomial mapping, $$d\geq 1$$, $$n\geq 2$$.
To prove the $$L^p$$-boundedness of $$M_{\mathcal P}$$, the condition $$\Omega\in V_\alpha(n)$$ is not sufficient. The authors assume that $$\Omega\in W_\alpha(n)$$, where $$W_\alpha(n)$$ is a subclass of $$V_\alpha(n)$$, that was introduced in a previous paper by A. Al-Salman and Y. Pan in order to study singular integrals. The following theorem is proved. Theorem. Let $$\alpha> 0$$, and $$\Omega\in W_\alpha(n)$$. The operator $$M_{\mathcal P}$$ is bounded on $$L^p(\mathbb{R}^n)$$ for $$(2\alpha+ 2)/(2\alpha+1)< p< 2\alpha+ 2$$. The bound of $$M_{\mathcal P}$$ is independent of the coefficients of $${\mathcal P}$$.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B15 Multipliers for harmonic analysis in several variables 42B25 Maximal functions, Littlewood-Paley theory
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