LpL^p-integrability, dimensions of supports of fourier transforms and applications

It is proved that there does not exist any non zero function in $L^p(\R^n)$ with $1\leq p\leq 2n/\alpha$ if its Fourier transform is supported by a set of finite packing $\alpha$-measure where $0<\alpha<n$. It is shown that the assertion fails for $p>2n/\alpha$. The result is applied to prove $L^p$ Wiener-Tauberian theorems for $\R^n$ and M(2)

LpL^p Fourier asymptotics, Hardy type inequality and fractal measures

Suppose $\mu$ is an $\alpha$-dimensional fractal measure for some $0<\alpha<n$. Inspired by the results proved by R. Strichartz in 1990, we discuss the $L^p$-asymptotics of the Fourier transform of $fd\mu$ by estimating bounds of $$\underset{L\rightarrow\infty}{\liminf}\ \frac{1}{L^k} \int_{|\xi|\leq L}\ |\widehat{fd\mu}(\xi)|^pd\xi,$$ for $f\in L^p(d\mu)$ and $2<p<2n/\alpha$. In a different direction, we prove a Hardy type inequality, that is, $$\int\frac{|f(x)|^p}{(\mu(E_x))^{2-p}}d\mu(x)\leq C\ \underset{L\rightarrow\infty}{\liminf} \frac{1}{L^{n-\alpha}} \int_{B_L(0)} |\widehat{fd\mu}(\xi)|^pd\xi$$ where $1\leq p\leq 2$ and $E_x=E\cap(-\infty,x_1]\times(-\infty,x_2]...(-\infty,x_n]$ for $x=(x_1,...x_n)\in\R^n$ generalizing the one dimensional results proved by Hudson and Leckband in 1992

A Java implementation of Coordination Rules as ECA Rules

This paper gives an insight in to the design and implementation of the coordination rules as ECA rules. The language specifications of the ECA rules were designed and the corresponding implementation of the same using JAVA as been partially done. The paper also hints about the future work in this area which deals with embedding this code in JXTA, thus enabling to form a P2P layer with JXTA as the back bone

Sharp weighted estimates for multi-frequency Calder\'on-Zygmund operators

In this paper we study weighted estimates for the multi-frequency $\omega-$Calder\'{o}n-Zygmund operators $T$ associated with the frequency set $\Theta=\{\xi_1,\xi_2,\dots,\xi_N\}$ and modulus of continuity $\omega$ satisfying the usual Dini condition. We use the modern method of domination by sparse operators and obtain bounds $\|T\|_{L^p(w)\rightarrow L^p(w)}\lesssim N^{|\frac{1}{r}-\frac{1}{2}|}[w]_{\mathbb{A}_{p/r}}^{max(1,\frac{1}{p-r})},~1\leq r<p<\infty,$ for the exponents of $N$ and $\mathbb{A}_{p/r}$ characteristic $[w]_{\mathbb{A}_{p/r}}$

Liouville numbers, Liouville sets and Liouville fields

Following earlier work by E.Maillet 100 years ago, we introduce the definition of a Liouville set, which extends the definition of a Liouville number. We also define a Liouville field, which is a field generated by a Liouville set. Any Liouville number belongs to a Liouville set S having the power of continuum and such that the union of S with the rational number field is a Liouville field.Comment: Proceedings of the American Mathematical Society, to appea

Liouville Numbers and Schanuel's Conjecture

In this paper, using an argument of P. Erdos, K. Alniacik and E. Saias, we extend earlier results on Liouville numbers, due to P. Erdos, G.J. Rieger, W. Schwarz, K. Alniacik, E. Saias, E.B. Burger. We also produce new results of algebraic independence related with Liouville numbers and Schanuel's Conjecture, in the framework of G delta-subsets.Comment: Archiv der Math., to appea

Hole dynamics in an antiferromagnet across a deconfined quantum critical point

We study the effects of a small density of holes, delta, on a square lattice antiferromagnet undergoing a continuous transition from a Neel state to a valence bond solid at a deconfined quantum critical point. We argue that at non-zero delta, it is likely that the critical point broadens into a non-Fermi liquid `holon metal' phase with fractionalized excitations. The holon metal phase is flanked on both sides by Fermi liquid states with Fermi surfaces enclosing the usual Luttinger area. However the electronic quasiparticles carry distinct quantum numbers in the two Fermi liquid phases, and consequently the limit of the ratio A_F/delta, as delta tends to zero (where A_F is the area of a hole pocket) has a factor of 2 discontinuity across the quantum critical point of the insulator. We demonstrate that the electronic spectrum at this transition is described by the `boundary' critical theory of an impurity coupled to a 2+1 dimensional conformal field theory. We compute the finite temperature quantum-critical electronic spectra and show that they resemble "Fermi arc" spectra seen in recent photoemission experiments on the pseudogap phase of the cuprates.Comment: 33 pages, 8 figures, Longer version of cond-mat/0611536, with additional results for electron spectrum at non-zero temperatur