Mathematics Colloquia and Seminars

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We consider Hardy spaces $H^p\left(T_\Gamma\right)$ in tube domains over open cones ($T_\Gamma\subset\mathbb{C}^n$).

When $p\ge 1$, these spaces have properties very similar to those of Lebesgue $L^p\left(\mathbb{R}^n\right)$ spaces. When $p<1$, the situation is dramatically different. These spaces are not even normed (just pre-normed)/ However, they have very interesting properties related to the Fourier transform. These properties make those spaces much nicer than their "brothers" with $p>1$. And it is possible to obtain general results (for any $p$) from those for $p\le 1$, which can be obtained more easily.

I am going to give a flavor of this idea showing how Fourier multipliers can be used. They can successfully be applied to various PDE problems (especially, with elliptic operators), to Approximation Theory problems, and for obtaining various inequalities. In particular, we will see how to derive Bernstein and Nikolski\u{\i} type inequalities for entire functions of exponential type $K$ belonging to $H^p\left(T_\Gamma\right)$.

Another result (joint work with Dr. Xin Li) for Hardy spaces $H^p\left(T_\Gamma\right)$ with $p\in\left(0,1\right]$ is the Poisson summation formula:

$$\sum_{m\in\Lambda}f\left(z+m\right) = \sum_{m\in\Lambda}\widehat{f}\left(m\right)e^{2\pi i\left(z,m\right)},\quad\forall z\in T_\Gamma.$$

The formula holds without any additional assumptions. Moreover, the series in both sides of this formula are analytic functions in $T_\Gamma$.