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Seminários do IMPA

Análise / Equações Diferenciais Parciais

Título
Fourier Multipliers, Local Smoothing and Regularising Properties of Fractional Maximal Functions
Expositor
João Pedro Ramos

Universitat Bonn - Germany
Data
Terça-feira, 9 de janeiro de 2018, 15:30
Local
Sala 232
Resumo

On their 2003 paper, Kinnunen and Saksman proved that the fractional maximal function, defined by
\[
M_{\alpha} f(x) = \sup_{r\ge 0} r^{\alpha} \cdot \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|\, dy,
\]
actually regularises functions: if $f \in L^p,$ with $1 < p < n$, and $1 \le \alpha \le n/p$, then $M_{\alpha} f$ has a weak derivative, and
\[
\|\nabla M_{\alpha} f \|_{q} \le C_{p, \alpha,n} \|f\|_p,
\]
where $\frac{\alpha-1}{n} = p^{-1} - q^{-1}$.  Our aim in this talk is to propose a new approach for proving similar estimates for suitable spherical, lacunary and smooth versions of the fractional maximal function, making use of maximal Fourier multiplier operators that majorise them.

We will use several classical results and, time allowing, show how the recently developped decoupling techniques - and its consequences, such as local smoothing estimates - can be put into play to get results.