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

Abstract

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 $[w{]}_{A_1}^d\to \infty$, and martingale transforms $T$ such that with an absolute positive $c$ $\Vert T:L^1(w)\to L^{1,\infty }(w)\Vert \ge c[w{]}_{A_1}^d\log [w{]}_{A_1}^d$. We also show the existence of the sequence of weights (now in $A_2$) such that $[w{]}_{A_2}^d\to \infty$, and such that the following holds: $[w{]}_{A_2^d}\asymp \Vert M^d{\Vert }_{w^{-1}}^2$; $\Vert S_w:L^2(w)\to L^2(w^{-1})\Vert \ge c\Vert M^d{\Vert }_{w^{-1}}\sqrt{\log \Vert M^d{\Vert }_{w^{-1}}}$; $\Vert S_w:L^{2,1}(w)\to L^2(w^{-1})\Vert \ge c\Vert M^d{\Vert }_{w^{-1}}\sqrt{\log \Vert M^d{\Vert }_{w^{-1}}};$ $\Vert S:L^2(w)\to L^{2,\infty }(w)\Vert =\Vert S_{w^{-1}}:L^2(w^{-1})\to L^{2,\infty }(w)\Vert \le C\Vert M^d{\Vert }_{w^{-1}}\le C([w{]}_{A_2}^d{)}^{1/2}$. Finally, it is shown that for test functions of the form ${\chi }_I$ the weak norm $\Vert S_w{\chi }_I{\Vert }_{L^{2,\infty }}\le C[w{]}_{A_2^d}^{1/2}\Vert {\chi }_I{\Vert }_w$.

References in corpus (2)

• On the sharp upper bound related to the weak Muckenhoupt-Wheeden conjecture
• Improving Beckner's bound via Hermite functions

Cited by in corpus (1)

• Strong weighted and restricted weak weighted estimates of the square function