On the Bilinear Bochner–Riesz Problem at Critical Index

Abstract

In this paper, we study maximal and square functions associated with bilinear Bochner–Riesz means at the critical index. In particular, we prove that they satisfy weighted estimates from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)\) for bilinear weights \((w_1,w_2)\in A_{\textbf{P}}\) where \(p_1,p_2>1\) and \(\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}\). Also, we show that both the operators fail to satisfy weak-type estimates at the end-point\(\big (1,1,\frac{1}{2}\big )\).