2 results
$L^{p}$ BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 99 / Issue 3 / December 2015
- Published online by Cambridge University Press:
- 17 August 2015, pp. 380-398
- Print publication:
- December 2015
-
- Article
-
- You have access
- Export citation
-
In an extrapolation argument, we prove certain
$L^{p}\,(1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions 
${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function 
$h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley 
$g_{{\it\lambda}}^{\ast }$-functions are given.
On a Class of Singular Integral Operators With Rough Kernels
-
- Journal:
- Canadian Mathematical Bulletin / Volume 49 / Issue 1 / 01 March 2006
- Published online by Cambridge University Press:
- 20 November 2018, pp. 3-10
- Print publication:
- 01 March 2006
-
- Article
-
- You have access
- Export citation
-
In this paper, we study the
${{L}^{p}}$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on ${{L}^{p}}$ provided that their kernels satisfy a size condition much weaker than that for the classical Calderón–Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.
