The author considers operators whose bilinear form has a sparse domination, that is, for all bounded and compactly supported functions $f_1,f_2$ there are $\Cal{K}<\infty$ sparse sets such that the following holds: $$ \|\langle Tf_1,f_2\rangle\|\lesssim \sum_{k=1}^{\Cal{K}} \sum_{Q\in \Cal{S}_k} |Q| \langle f_1\rangle_Q\langle f_2\rangle_Q,\tag1 $$ where $\langle f\rangle_Q=\frac1{|Q|}\int_Q |f|$ and $\Cal{S}_k$ are sparse sets. In this paper, the author studies endpoint estimates for those operators. \par To state the main results, the author gives the following definitions: Let $\varepsilon\: [1,\infty]\to [1,\infty]$ be an increasing function with $K_{\varepsilon}\coloneq \sum_{k=-1}^{\infty} \varepsilon(2^{2^k}) ^{-1}<\infty$. For a cube $Q$ and a weight $w(x)\geq 0$ define $$ \rho_{w}(Q)\coloneq \frac1{w(Q)}\int_Q M({1}_Qw)(x)dx, $$ where $M$ is the usual Hardy-Littlewood maximal function and $w(Q)\coloneq \int_Q w(y)dy$. For a collection $\Cal{S}$ of cubes define the following maximal function: $$ M_{\varepsilon}w\coloneq \sup_{Q\in \Cal{S}}{1}_Q\langle w \rangle_Q \log(\rho_w(Q))\varepsilon(\rho_{\varepsilon}(Q)). $$ \par The first result was proved by T.~P. Hytonen and C. Pérez~Moreno in [Anal. PDE {\bf 6} (2013), no.~4, 777--818 (Corollary 14); MR3092729], that is: \par Theorem 1. Let $T$ be an operator that has a sparse bilinear domination as in (1) and let $w$ be an $A_1$ weight. Then we have the following quantitative estimate: $$ \|T: L^1{(w)}\to L^{1,\infty}(w)\|\lesssim [w]_{A_1}\log(e+[w]_{A_{\infty}}), $$ where $$ [w]_{A_1}\coloneq \sup_{Q \text{ a cube }}\frac{M({1}_Qw)(x)}{w(x)} {\text{ and }} [w]_{A_{\infty}}\coloneq \sup_{Q \text{ a cube }}\frac1{w(Q)}\int_Q M({1}_Qw)(x)dx. $$ \par The main, new theorem is: \par Theorem 2. Let $T$ be an operator that has a sparse bilinear domination as in (1) and let $\varepsilon$ be a function as above. Then for any weight $w(x)\geq 0$ we have $$ \|T: L^1{(M_{\varepsilon}w)}\to L^{1,\infty}(w)\|\lesssim K_{\varepsilon}. $$