#### APRG Seminar

##### Venue: Microsoft Teams (online)

Let $X_1,X_2, X_3$ be Banach spaces of measurable functions in $L^0(\mathbb R)$ and let $m(\xi,\eta)$ be a locally integrable function in $\mathbb R^2$. We say that $m\in \mathcal{BM}_{(X_1,X_2,X_3)}(\mathbb R)$ if $$B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i \langle\xi+\eta, x\rangle}d\xi d\eta,$$ defined for $f$ and $g$ with compactly supported Fourier transform, extends to a bounded bilinear operator from $X_1 \times X_2$ to $X_3$.

In this talk we present some properties of the class $\mathcal{BM}_{(X_1,X_2,X_3)}(\mathbb R)$ for general spaces which are invariant under translation, modulation and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus in the case $m(\xi,\eta)=M(\xi-\eta)$ and find conditions for these classes to contain non zero multipliers in terms of the Boyd indices for the spaces.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 18 Sep 2021