In this talk, the speaker considers two types of maximal operators given by Fourier multipliers, and suggests criteria for Fourier multipliers so that the associated maximal operators are bounded on $L^p$ for each $p$. In other words, we first consider maximal operators given by taking supremum over $t > 0$ where $t$ is a dilation factor of Fourier multipliers, $m(t\xi)$. The condition for $m$ may be understood as a vector-valued analogue of the Hörmander–Mikhlin multiplier theorem. For the second type of maximal operators, we take the supremum over $t \in E$ with $0 \leq \dim(E) < 1$. Together with the dimension of $E$, the condition for $m$ associated with the first maximal operators is still valid for the second maximal operators.

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