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Title: Localisation theorems for Bochner-Riesz means.
Speaker: Jotsaroop Kaur (University of Milano, Italy)
Date: 21 August 2015
Time: 4 pm
Venue: LH 1

Define $S_R^\alpha f := \int_{\mathbb{R}^d} (1-\frac{|\xi|^2}{R^2})_+^\alpha$ $\hat{f} (\xi) e^{i2\pi (x.\xi)} d\xi$, the Bochner Riesz means of order $\alpha \geq 0$. Let $f\in L^2(\mathbb{R}^d)$ and $f \neq 0$ in an open, bounded set $B.$ It is known that $S_R^\alpha f$ goes to 0 a.e. in $B$ as $R\rightarrow\infty.$ We study the pointwise convergence of Bochner Riesz means $S_{R}^\alpha f, \alpha>0$ as $R \rightarrow \infty$ on sets of positive Hausdorff measure in $\mathbb{R}^d$ by making use of the decay of the spherical means of Fourier Transform of fractal measures. We get an improvement in the range of the Hausdorff dimension of the sets on which it converges. When $0<\alpha<\frac{d-1}{4},$ we get the best possible result in $\mathbb{R}^2$ and in higher dimensions we improve the result by L.Colzani, G. Gigante and A. Vargas. Steins interpolation theorem also gives us the corresponding result for $f\in L^p(\mathbb{R}^d), 1<p<2.$

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Last updated: 18 May 2024