, Volume 5, Issue 3, pp 369-378
Date: 01 Jun 2010

Rough bilinear fractional integrals with variable kernels

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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 (ℝ n L s (S n−1) for some s ⩾ 1, and S n−1 denotes the unit sphere of ℝ n . 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].