Martingale transform and Square function: some weak and restricted weak sharp weighted estimates

Following the ideas of A. Lerner, F. Nazarov, S. Ombrosi from [12] we prove that there is a sequence of weights $w\in A_1^d$ such that $[w]^d_{A_1}\to \infty$,andmartingaletransforms$T$suchthatwithanabsolutepositive$c$ $\mathrm{\parallel }T:L^1(w)\to L^{1,\mathrm{\infty }}(w)\mathrm{\parallel }\ge c[w{]}_{A_1}^d\log [w{]}_{A_1}^d$. We also show the existence of the sequence of weights (now in $A_2$) such that $[w{]}_{A_2}^d\to \mathrm{\infty }$, and such that the following holds: $[w{]}_{A_2^d}\asymp \mathrm{\parallel }{M}^d{\mathrm{\parallel }}_{w^{-1}}^2$; $|S_{w}: L^{2} (w) \to L^2(w^{-1})| \ge c, |M^d|{w^{-1}}\sqrt{\log |M^d|{w^{-1}}}$;$|S_{w}: L^{2,1} (w) \to L^2(w^{-1})| \ge c, |M^d|{w^{-1}}\sqrt{\log |M^d|{w^{-1}}}; $ $|S: L^2(w)\to L^{2, \infty}(w)|=|S_{w^{-1}}: L^{2}(w^{-1}) \to L^{2,\infty}(w)|\le C, |M^d|{w^{-1}}\le C, ([w]^d{A_2})^{1/2}$. Finally, it is shown that for test functions of the form ${\chi }_I$ the weak norm $|S_w \chi_I|{L^{2, \infty}} \le C, [w]{A_2^d}^{1/2}|\chi_I|_w$.