|DATE||June 27 (Mon), 2022|
|TITLE||Bochner-Riesz mean for the Hermite expansion|
Bochner-Riesz mean is a summation method often tested when a given series is not convergent. It tends to converge better as the summation index $\delta$ grows. The question of whether the Bochner-Riesz means of the Fourier series of a given function is convergent in the $L^p$ spaces is now called Bochner-Riesz conjecture, a long-standing open problem in harmonic analysis. We study this problem by replacing the Fourier series with the Hermite-Fourier expansion. Similar to the Fourier series, the Hermite expansion is not convergent in $L^p$ for most $p$. First, we give a new necessary condition on $p$ and $\delta$ that does not appear in the classical Bochner-Riesz conjecture. This is due to the original phenomenon observed only in the Hermite expansion. Secondly, considering the problem in a local setting, we extend the previously known range of $p$ and the summability index $\delta$. Especially, in two dimensions, we obtain the optimal range of $p$, $\delta$ in which the local convergence in $L^p$ holds.