Delhi’s VAT Department- Mixed Results and Lessons for GST

The Value Added Tax (VAT) system is a system of indirect taxation that replaced the previous sales tax regime in India. Like its predecessor, VAT is implemented at the state level and applies to all goods traded within the state. The Delhi Value Added Tax Act was passed on 2004 and there were follow-up Rules in 2005. The new consumption tax system was finally put in force from 1 April 2005 in Delhi, along with 20 other states. Delhi VAT Act replaces the old Delhi Sales Tax Act, Delhi Sales Tax on Works Contract Act, Delhi Sales Tax on Right to use goods Act and Delhi Sales Tax on entry of motor vehicles. The Department of Trade and Taxes is the state department in charge of all matters related to VAT administration.VAT; GST; Delhi; India

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 regularity of solutions to Poisson's equation

In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In particular, our results show that there is no qualitative difference in the regularity of these solutions in the plane and in higher dimensions

On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.Comment: 21 page

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

The lattice point counting problem on the Heisenberg groups

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{\alpha,A}((z,t)) = \left(|z|^\alpha + A |t|^{\alpha/2}\right)^{1/\alpha}$, for $\alpha \ge 2$ and $A>0$. This natural family includes the canonical Cygan-Kor\'anyi norm, corresponding to $\alpha =4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $\alpha=2$ is the best possible, in all dimensions