Estimates in Beurling--Helson type theorems. Multidimensional case

We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p(\mathbb T^m)}$ as $|\lambda|\rightarrow \infty, ~\lambda\in\mathbb R,$ for $C^1$ -smooth real functions $\varphi$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogues for the spaces $A_p(\mathbb R^m)$

On the Fourier transform of the characteristic functions of domains with C1C^1 -smooth boundary

We consider domains $D\subseteq\mathbb R^n$ with $C^1$ -smooth boundary and study the following question: when the Fourier transform $\hat{1_D}$ of the characteristic function $1_D$ belongs to $L^p(\mathbb R^n)$?Comment: added two references; added footnotes on pages 6 and 1

Directed Polymer -- Directed Percolation Transition

We study the relation between the directed polymer and the directed percolation models, for the case of a disordered energy landscape where the energies are taken from bimodal distribution. We find that at the critical concentration of the directed percolation, the directed polymer undergoes a transition from the directed polymer universality class to the directed percolation universality class. We also find that directed percolation clusters affect the characterisrics of the directed polymer below the critical concentration.Comment: LaTeX 2e; 12 pages, 5 figures; in press, will be published in Europhys. Let

Classical and relativistic dynamics of supersolids: variational principle

We present a phenomenological Lagrangian and Poisson brackets for obtaining nondissipative hydrodynamic theory of supersolids. A Lagrangian is constructed on the basis of unification of the principles of non-equilibrium thermodynamics and classical field theory. The Poisson brackets, governing the dynamics of supersolids, are uniquely determined by the invariance requirement of the kinematic part of the found Lagrangian. The generalization of Lagrangian is discussed to include the dynamics of vortices. The obtained equations of motion do not account for any dynamic symmetry associated with Galilean or Lorentz invariance. They can be reduced to the original Andreev-Lifshitz equations if to require Galilean invariance. We also present a relativistic-invariant supersolid hydrodynamics, which might be useful in astrophysical applications.Comment: 22 pages, changed title and content, added reference