Frontiers of Mathematics in China

, Volume 5, Issue 3, pp 369–378

Rough bilinear fractional integrals with variable kernels

Research Article

DOI: 10.1007/s11464-010-0061-1

Cite this article as:
Chen, J. & Fan, D. Front. Math. China (2010) 5: 369. doi:10.1007/s11464-010-0061-1

Abstract

We study the rough bilinear fractional integral
$$ \tilde B_{\Omega ,\alpha } (f,g)(x) = \int_{\mathbb{R}^n } {f(x + y)g(x - y)\frac{{\Omega (x,y')}} {{\left| y \right|^{n - \alpha } }}dy} , $$
, where 0 < a < n, Ω is homogeneous of degree zero on the y variable and satisfies Ω ∈ L(ℝnLs(Sn−1) for some s ⩾ 1, and Sn−1 denotes the unit sphere of ℝn. By assuming size conditions on Ω, we obtain several boundedness properties of \( \tilde B_{\Omega ,\alpha } (f,g) \):
$$ \tilde B_{\Omega ,\alpha } :L^{p_1 } \times L^{p_2 } \to L^p , $$
where
$$ \frac{1} {p} = \frac{1} {{p_1 }} + \frac{1} {{p_2 }}\frac{\alpha } {n}. $$
Our result extends a main theorem of Y. Ding and C. Lin [Math. Nachr., 2002, 246–247: 47–52].

Keywords

Bilinear operator multilinear fractional integral variable kernel 

MSC

42B25 42B20 42B99 

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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