For $\alpha\geq0$, the bilinear Bochner–Riesz operator of order $\alpha$ in $\mathbb{R}^n$ is defined as

\begin{equation} \mathcal{B}^\alpha_R(f,g)(x)=\int_{\mathbb{R}^n\times \mathbb{R}^n} \left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)^\alpha_+ \hat{f}(\xi) \hat{g}(\eta) e^{2\pi ix\cdot(\xi+\eta)} d\xi d\eta. \end{equation}

For $\alpha>n-\frac{1}{2}$, $\mathcal{B}^{\alpha}_R$ maps $L^{p_1}(\mathbb{R}^n)\times L^{p_2}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ whenever $p_1,p_2\geq1$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. Thus, $\alpha=n-\frac{1}{2}$ is commonly referred as the critical index for the bilinear Bochner–Riesz operator. Recently, there have been some results on $L^p$-boundedness of the bilinear Bochner–Riesz operator $\mathcal{B}^{\alpha}_R$ when $\alpha\leq n-\frac{1}{2}$.

In this talk, we extend the bilinear Bochner–Riesz operator to convex domains in the plane and discuss some $L^p$-boundedness results.

The video of this talk is available on the IISc Math Department channel.