zbMATH — the first resource for mathematics

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 \({\mathcal P}\).

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