Rough bilinear fractional integrals with variable kernels

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^∞(ℝⁿ)× L^s(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

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