Variational Estimates for Bilinear Ergodic Averages Along Sublinear Sequences

We prove long variational estimates for the bilinear ergodic averages \[ A_{N;X}(f,g)(x) = \frac{1}{N} \sum_{n=1}^N f(T^{\lfloor \sqrt{n} \rfloor}x) g(T^nx) \] on an arbitrary measure preserving system $(X,\mu,T)$ for the full expected range, i.e. whenever $f \in L^{p_1}(X)$ and $g \in L^{p_2}(X)$ with $1<p_1,p_2<\infty$. In particular, if $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ we show that the long $r$-variation of $A_{N;X}$ maps $L^{p_1}(X) \times L^{p_2}(X)$ into $L^p(X)$ for any $p>\frac{1}{2}$, which is sharp up to the endpoint. If $p \geq 1$ we obtain long variational estimates for the full expected range $r>2$ and if $p<1$ we obtain a range of $r>2+\varepsilon_{p_1,p_2}$ where $\varepsilon_{p_1,p_2}>0$ depends only on $p_1$ and $p_2$. As a consequence, we obtain bilinear maximal estimates \[ \left\| \sup_{N \in \mathbb{N}} |A_{N;X}(f,g)| \right\|_{L^p(X)} \leq C_{p_1,p_2} \|f\|_{L^{p_1}(X)} \|g\|_{L^{p_2}(X)} \] for any $1<p_1,p_2 \leq \infty$.