Variational Inequalities for Bilinear Averaging Operators over Convex Bodies

TL;DR

This paper establishes $q$-variation inequalities for bilinear averaging operators over convex bodies, extending the understanding of their boundedness in various function spaces and applying to discrete, ergodic, and square function contexts.

Contribution

It proves new $q$-variation inequalities for bilinear averages over convex bodies, including applications to discrete, ergodic, and square function operators.

Findings

Boundedness of $V_q$ in $L^p$ spaces for bilinear averages

Extension to $L^{p, olinebreak ext{,} olinebreak ext{ } ext{infinity}}$ and BMO spaces

Applicability to discrete, ergodic, and square function operators

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 $\Bbb R^{2d}$. We prove that $$\|V_q(\mathbf{A}_t^G(f_1,f_2): t>0) \|_{L^p} \lesssim \|f_1\|_{L^{p_1}} \|f_2\|_{L^{p_2}},$$ 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.