Variational Estimates for Bilinear Ergodic Averages Along Sublinear Sequences

We prove long variational estimates for the bilinear ergodic averages $$A_{N;X}(f,g)(x)=\frac{1}{N}\sum _{n=1}^{N}f(T^{\lfloor \sqrt{n}\rfloor }x)g(T^{n}x)$$ 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 $1<p_1,p_2<\mathrm{\infty }$. 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 $p\ge 1$ we obtain long variational estimates for the full expected range $r>2$ and if $p<1$ we obtain a range of $r>2+{\epsilon }_{p_1,p_2}$ where ${\epsilon }_{p_1,p_2}>0$ depends only on $p_1$ and $p_2$. As a consequence, we obtain bilinear maximal estimates $${\parallel \underset{N\in \mathbb{N}}{\sup }\mathrm{\mid }{A}_{N;X}(f,g)\mathrm{\mid }\parallel }_{L^p(X)}\le C_{p_1,p_2}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{p_1}(X)}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{p_2}(X)}$$ for any $1<p_1,p_2\le \mathrm{\infty }$.