APRG Seminar

Title: Bilinear spherical maximal functions of product type
Speaker: Kalachand Shuin (IISER Bhopal)
Date: 30 September 2020
Time: 4 pm
Venue: Microsoft Teams (online)

The spherical averages often make their appearance in partial differential equations. For instance, the solution of the wave equation \begin{equation} u_{tt}=\Delta u,\ \ u(x,0)=0,\ \ u_{t}(x,0)=f(x),\ \ in\ \ \mathbb{R}^{3}\ \ is \end{equation} \begin{equation} u(x,t)=\frac{t}{4\pi}\int_{\mathbb{S}^{2}}f(x-ty)d\sigma(y), \end{equation}

where $d\sigma$ is the rotation invariant, normalized surface measure on the sphere $\mathbb{S}^{2}$. In [Proc. Natl. Acad. Sci. USA (1976)], Stein proved the following result:

Theorem. Let $n \geq 3$. Then \begin{equation}\Vert \sup_{t>0} \int_{\mathbb{S}^{n-1}}f(x-ty)d\sigma(y) \Vert_{L^{p}(\mathbb{R}^{n})} \leq C_{p}\Vert f\Vert_{L^{p}(\mathbb{R}^{n})} \end{equation} if, and only if $\frac{n}{n-1}<p\leq\infty$.

The above result was extended to dimension $n=2$, by Bourgain in [J. d’Anal. Math. (1986)]. Later, in [J. d’Anal. Math. (2019)] Lacey proved sparse domination for both lacunary and full spherical maximal functions.

In this talk, I shall talk about the bilinear spherical maximal functions of product type, which is defined in the spirit of bilinear Hardy–Littlewood maximal function. The lacunary and full bilinear spherical maximal functions are defined by \begin{equation} \mathcal{M}_{lac}(f_1,f_2)(x):= \sup_{j\in\mathbb{Z}} \prod_{i=1,2} \int_{\mathbb{S}^{n-1}} f_i(x-2^{j}y_i)d\sigma(y_i), \end{equation} \begin{equation} \mathcal{M}_{full}(f_1,f_2)(x):= \sup_{r>0} \prod_{i=1,2} \int_{\mathbb{S}^{n-1}} f_i(x-ry_i)d\sigma(y_i), \end{equation} where $f_{1},f_{2}\in\mathcal{S}(\mathbb{R}^{n})$, the Schwartz class. We have investigated the sparse domination and weighted boundedness of both the operators $\mathcal{M}_{lac}$ and $\mathcal{M}_{full}$ with respect to the bilinear Muckenhoupt weights $A_{\vec{p},\vec{r}}$. (Joint with Saurabh Shrivastava and Luz Roncal.)


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Last updated: 04 Dec 2020