Variational Inequalities for Bilinear Averaging Operators over Convex Bodies

Abstract

We study $q$-variation inequality for bilinear averaging operators over convex bodies $(G_t{)}_{t>0}$ defined by \begin{align*} \mathbf{A}_t^G(f_1,f_2)(x) & =\frac{1}{|G_t|}\int_{G_t} f_1(x+y_1)f_2(x+y_2)\, dy_1\, dy_2, \quad x\in \Bbb R^d. \end{align*} where $G_t$ are the dilates of a convex body $G$ in ${\mathbb{R}}^{2d}$. We prove that $$\Vert V_q({\mathbf{A}}_t^G(f_1,f_2):t>0){\Vert }_{L^p}\lesssim \Vert f_1{\Vert }_{L^{p_1}}\Vert f_2{\Vert }_{L^{p_2}},$$ for $2<q<\infty$, $1<p_1,p_2\le \infty$, $1/2<p<\infty$ with $1/p=1/p_1+1/p_2$. The target space $L^p$ should be replaced by $L^{p,\infty }$ for $p_1=1$ and/or $p_2=1$, and by dyadic BMO when $p_1=p_2=\infty$. As applications, we obtain variational inequalities for bilinear discrete averaging operators, bilinear averaging operators of Demeter-Tao-Thiele, and ergodic bilinear averaging operators. As a byproduct, we also obtain the same mapping properties for a new class of bilinear square functions involving conditional expectation, which are of independent interest.

37 pages