Automata in Groups and Dynamics and Induced Systems of PDE in Tropical Geometry

In this paper we develop a dynamical scaling limit from rational dynamics to automata in tropical geometry. We compare these dynamics and induce uniform estimates of their orbits. We apply these estimates to introduce a comparison analysis of theory of automata groups in geometric group theory with analysis of rational dynamics and some hyperbolic PDE systems. Frameworks of characteristic properties of automata groups are inherited to the corresponding rational or PDE dynamics. As an application we study the Burnside problem in group theory and translate the property as the infinite quasi-recursiveness in rational dynamics.     

Elements of functional calculus andL 2 regularity for some classes of fourier integral operators

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), where T are Fourier integral operators with one-sided right (respectively, left) singularities; this idea has its roots in the work of Greenleaf and Seeger. Such a result allows us to reduce the L² regularity problem for operators in n dimensions with one-sided singularities to the L² regularity problem for operators with two-sided singularities in n − 1 dimensions. As a consequence we deduce almost sharp L²-Sobolev estimates for operators in three-dimensions; an interesting special case is provided by certain restricted X-ray transforms associated to line complexes which are not well curved. We also provide a proof of almost-sharpness by looking at a restricted X-ray transform associated to the line complex generated by the curve t → (t, t^k). Appropriate notions of singularity, strong singularity, and type are also developed.     

Discretization and Transference of Bisublinear Maximal Operators

We develop a general condition for automatically discretizing strong type bisublinear maximal estimates that arise in the context of the real line. In particular, this method applies directly to Michael Lacey’s strong type boundedness results for the bisublinear maximal Hilbert transform and for the bisublinear Hardy-Littlewood maximal operator, furnishing the counterpart of each of these two results (without changes to the range of exponents) for the sequence spaces We then take up some transference applications of discretized maximal bisublinear operators to maximal estimates and almost everywhere convergence in Lebesgue spaces of abstract measures. We also broaden the scope of such applications, which are based on transference from by developing general methods for transplanting bisublinear maximal estimates from arbitrary locally compact abelian groups.     

Nonlinear elliptic equations with BMO coefficients in Reifenberg domains

We develop a unifying method to obtain the interior and boundary estimates for the weak solution of a nonlinear elliptic partial differential equation of p-Laplacian type with BMO coefficients in a δ-Reifenberg flat domain. Our results greatly improve the known results for such equations.     

Estimations du Noyau de la Chaleur sur les Espaces Homogènes

We give upper and lower gaussian estimates for the heat kernel for canonical sub-laplacians on homogeneous spaces of Lie groups. We prove that homogeneous spaces of Lie groups of polynomial growth have the doubling property. We deduce Poincaré inequalities on these spaces.

     

Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks

We develop a number of statistical aspects of symmetric groups (mostly dealing with the distribution of cycles in various subsets of Sn), asymptotic properties of (ordinary) characters of symmetric groups, and estimates for the multiplicities of root number functions of these groups. As main applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, a finiteness result for the number of Fuchsian presentations of such a group (resolving a long-standing problem of Roger Lyndon), as well as a proof of a well-known conjecture of Roichman concerning the mixing time of random walks on symmetric groups.     

The LIL for <Emphasis Type="Italic">U</Emphasis>-Statistics in Hilbert Spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest. R. Adamczak’s research partially supported by MEiN Grant 2 PO3A 019 30. R. Latała’s research partially supported by MEiN Grant 1 PO3A 012 29.     

On Lebesgue-type inequalities for greedy approximation

We study the efficiency of greedy algorithms with regard to redundant dictionaries in Hilbert spaces. We obtain upper estimates for the errors of the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm in terms of the best m-term approximations. We call such estimates the Lebesgue-type inequalities. We prove the Lebesgue-type inequalities for dictionaries with special structure. We assume that the dictionary has a property of mutual incoherence (the coherence parameter of the dictionary is small). We develop a new technique that, in particular, allowed us to get rid of an extra factor m^1/2 in the Lebesgue-type inequality for the Orthogonal Greedy Algorithm.     

Two-parameter estimates for joint spectral projections on complex spheres

We prove sharp two-parameter estimates for the L ^p -L ² norm, 1 ≤ p ≤ 2, of the joint spectral projectors associated to the Laplace–Beltrami operator and to the Kohn Laplacian on the unit sphere S ^2n-1 in . Then, by using the notion of contraction of Lie groups, we deduce the estimates recently obtained by H. Koch and F. Ricci for joint spectral projections on the reduced Heisenberg group h ¹.      

Analyticity estimates for the Navier–Stokes equations

We study spatial analyticity properties of solutions of the three-dimensional Navier–Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier–Stokes equations with data in Hr, r?1/2, and prove a stability result for the analyticity radius.     

Strichartz Estimates for the Schrödinger Equation on Polygonal Domains

We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.     

Asymptotic analysis for linear difference equations

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

     

A discontinuous mixed covolume method for elliptic problems

We develop a discontinuous mixed covolume method for elliptic problems on triangular meshes. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-order L²-error estimates are derived for the approximations of both velocity and pressure.     

Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in ${{\mathbb R}^n}$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L² estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.     

Sharp regularity results on second derivatives of solutions to the Monge‐Ampère equation with VMO type data

We establish interior estimates for L^p‐norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge‐Ampère equation det D²u = f(x) with zero boundary value, where f(x) is strictly positive, bounded, and satisfies a VMO‐type condition. These estimates develop the regularity theory of the Monge‐Ampère equation in VMO‐type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W^{2, p}‐estimates. © 2008 Wiley Periodicals, Inc.     

Error estimates of proper orthogonal decomposition eigenvectors and Galerkin projection for a general dynamical system arising in fluid models

We study estimates for proper orthogonal decomposition eigenvectors and eigenvalues as well as error estimates between the exact solution of a 2D Navier–Stokes model and the numerical approach when the proper orthogonal decomposition method is considered. These estimates are also extended when bifurcation diagram are calculated using the so called p-POD or SPOD methods with a new cut-off criterion to minimize noisy modes produced by the p-POD method.     

Classes of singular integral operators along variable lines

We prove estimates for classes of singular integral operators along variable lines in the plane, for which the usual assumption of nondegenerate rotational curvature may not be satisfied. The main L^p estimates are proved by interpolating L² bounds with suitable bounds in Hardy spaces on product domains. The L² bounds are derived by almost-orthogonality arguments. In an appendix we derive an estimate for the Hilbert transform along the radial vector field and prove an interpolation lemma related to restricted weak type inequalities.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.