\(L^p\)-estimates for singular integrals with kernels belonging to certain block spaces. (English) Zbl 1037.42015

Al-Qassem, Hussain; Pan, Yibiao

Rev. Mat. Iberoam. 18, No. 3, 701-730 (2002).

Let \(\mathbb{R}^n\), \(n\geq 2\), be the \(n\)-dimensional Euclidean space and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Let \(\Omega\) be a homogeneous function of degree 0 satisfying \(\Omega\in L^1(S^{n-1})\) and \(\int_{S^{n-1}} \Omega(y')\, d\sigma(y')= 0\), where \(y'= y/| y|\in S^{n-1}\) for any \(y\neq 0\). For \(\gamma> 0\), let \(\Delta_\gamma\) denote the set of all measurable functions \(h\) on \(\mathbb{R}^+\) such that \[ \sup_{R> 0} R^{-1} \int_0^R | h(t)|^\gamma\, dt<\infty. \] In this paper the authors study a singular integral operator defined by \[ Tf(x)= \text{p.v. }\int_{\mathbb{R}^n} f(x-{\mathcal P}(u))K(u)\, du \] where \(K(x)=| x|^{-n} \Omega(x) h(| x|)\) and \({\mathcal P}= (P_1,\dots, P_m)\) is a mapping from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) with each \(P_j\) being a polynomial function.
Let \(B_q^{\mu,\nu} (S^{n-1})\) be the block spaces defined in the book by S. Z. Lu, M. H. Taibleson and G. Weiss [“Spaces generated by blocks”, Beijing Normal University Press, Beijing (1989), per bibl.]. The main theorem of this paper is the following
Theorem. If \(\Omega\) belongs to the block space \(B_q^{0,0} (S^{n-1})\), \(q>1\), \(h\in \Delta_\gamma\) for some \(\gamma> 1\), then for any \(p\) satisfying \(| 1/p- 1/2|\leq \min\{1/2, 1/\gamma'\}\), there exists a constant \(C\) independent of the coefficients of \({\mathcal P}\) such that \(\| Tf\|_{L^p(\mathbb{R}^m)}\leq C\| f\|_{L^p(\mathbb{R}^m)}\).
The authors also obtain a similar boundedness result for the associated maximal operator.

Reviewer: Dashan Fan (Milwaukee)

MSC:

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

Keywords:

\(L^p\)-boundedness; singular integrals; block spaces; maximal operators; oscillatory integrals

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