Variations on a theorem of Beurling

We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $0 < a < 1.$ We show that such functions are entire vectors for the Schr\"{o}dinger representations of the Heisenberg group. If an eigenfunction $f$ of the Fourier transform satisfies the above condition we show that the Hermite coefficients of $f$ have certain exponential decay which depends on $a$.Comment: 21 page

On the Hermite expansions of functions from Hardy class

Considering functions $f$ on $\R^n$ for which both $f$ and $\hat{f}$ are bounded by the Gaussian $e^{-{1/2}a|x|^2}, 0 < a < 1$ we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for $O(n)-$finite functions thus extending the one dimensional result of Vemuri.Comment: 22 page

On the structure of analytic vectors for the schrodinger representation

This article deals with the structure of analytic and entire vectors for the Schr\"{o}dinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.Comment: 19 page

On Hermite pseudo-multipliers

In this article we deal with a variation of a theorem of Mauceri concerning the $L^p$ boundedness of operators $M$ which are known to be bounded on $L^2.$ We obtain sufficient conditions on the kernel of the operaor $M$ so that it satisfies weighted $L^p$ estimates. As an application we prove $L^p$ boundedness of Hermite pseudo-multipliers.Comment: 28 page

Fourier multipliers and pseudo-differential operators on Fock-Sobolev spaces

Any bounded linear operator $T$ on $L^2(\mathbb{R}^n)$ gives rise to the operator $S= B \circ T \circ B^\ast$ on the Fock space \mathcal{F}(\C^n) where $B$ is the Bargmann transform. In this article we identify those $S$ which correspond to Fourier multipliers and pseudo-differential operators on $L^2(\mathbb{R}^n)$ and study their boundedness on the Fock-Sobolev spaces \mathcal{F}^{s,2}(\C^n).Comment: 18 page