Showing papers in "Journal D Analyse Mathematique in 2011"

A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains

Abstract: This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian $$ - \Delta {|_{C_0^\infty (\Omega )}}$$ in L 2(Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C 1,r , for r > 1/2. Second, we establish Kreĭn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces

Abstract: We investigate the lifting property of modulation spaces and construct explicit isomorpisms between them. For each weight function \omega and suitable window function \varphi, the Toeplitz operator ...

Linear forms and quadratic uniformity for functions on ℤ N

Abstract: A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b], [GW10c], we gave a complete answer to this question for groups of the form G = Fpn, provided p is not too small. In ℤN, substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm \({\left\| \cdot \right\|_{{U^3}}}\) requires a more sophisticated “local” formulation. When N is prime, ℤN is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a]. Moreover, we obtain a doubly exponential bound.

On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data II. Perturbations with Finite Moments

Abstract: We solve the Cauchy problem for the Korteweg-de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of finite derivatives with finite moments.

Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation

Abstract: The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H −7/4, using the ideas of an $${\overline F ^s}$$ -type space [8]. Then we show that the equation is globally well posed in H s for s ≥ −7/4, using the ideas of the “I-method” [7].

On boundary behavior of generalized quasi-isometries

Abstract: We establish a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in $$\overline {{^n}} = {^{^n}} \cup \{ \infty \} ,n \ge 2$$ , under integral constraints of the type ∫ Φ(Q n−1(x))dm(x) < ∞ with a convex non-decreasing function Φ: [0,∞]→[0,∞]. Integral conditions on Φ are found that are necessary and sufficient for a continuous extension of f to the boundary. Our results are applied to finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries as well as quasi-isometries in ℝ n . In particular, a generalization and strengthening of the well-known theorem of Gehring-Martio on homeomorphic extension to boundaries of quasi-conformal mappings between QED (quasi-extremal distance) domains is obtained.

Exponential sum estimates over a subgroup in an arbitrary finite field

Abstract: Let F q be the finite field consisting of q = p r elements and yy an additive character of the field F q . Take an arbitrary multiplicative subgroup H of size |H| > q C/(log log q) for some constant C > 0 not largely contained in any multiplicative shift of a subfield. We show that |Σ h∈H yy(h)| = o(|H|). This means that H is equidistributed in F q .

Universally L 1 -bad arithmetic sequences

Abstract: We present a modified version of Buczolich and Mauldin’s proof that the sequence of square numbers is universally L1-bad. We extend this result to a large class of sequences, including the dth powers and the set of primes. Furthermore, we show that any subsequence of the averages taken along these sequences is also universally L1-bad.

Weighted Fourier inequalities: Boas’ conjecture in ℝ n

Abstract: Weighted Lp(ℝn) → Lq(ℝn) Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function. Extensions to general weights are also given.

Bilipschitz embedding of self-similar sets

Abstract: In this paper, we prove that each self-similar set satisfying the strong separation condition can be bilipschitz embedded into each self-similar set with larger Hausdorff dimension. A bilipschitz embedding between two self-similar sets of the same Hausdorff dimension both satisfying the strong separation condition is only possible if the two sets are bilipschitz equivalent.

Limit-periodic Schrödinger operators with uniformly localized eigenfunctions

Abstract: We exhibit limit-periodic Schrodinger operators that are uniformly localized in the strongest sense possible. That is, for these operators there are uniform exponential decay rates such that every element of the hull has a complete set of eigenvectors that decay exponentially off their centers of localization at least as fast as prescribed by the uniform decay rate. Consequently, these operators exhibit uniform dynamical localization.

Analog of a theorem of forelli for boundary values of holomorphic functions on the unit ball of ℂ n

Abstract: Let f ∈ Cω(∂Bn), where Bn is the unit ball of ℂn. We prove that if \(a,b \in {\overline B ^n}\), a ≠ b, for every complex line L passing through one of a or b, the restricted function \(f{|_{L \cap \partial {B^n}}}\) has a holomorphic extention to the cross-section L∩Bn, then f is the boundary value of a holomorphic function in Bn.

Asymptotic behavior and uniqueness of blow-up solutions of quasilinear elliptic equations

Abstract: In this paper, we study the asymptotic behavior of solutions of the problem Δpu = f (u) in Ω, u = ∞ on ∂Ω, under general conditions on the function f, where Ωp is the p-Laplace operator. We show that the technique used by the author for the special case p = 2 works in this more general setting, and that the behavior described by various authors for the case p = 2 is easily derived from this technique for the general case.

Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Abstract: Let Ct = {z ∈ ℂ: |z − c(t)| = r(t), t ∈ (0, 1)} be a C1-family of circles in the plane such that limt→0+Ct = {a}, limt→1−Ct = {b}, a ≠ b, and |c′(t)|2 + |r′(t)|2 ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation c′(t)w2 + 2r′(t)w + c′(t) = 0 with |w| < 1, if such a root exists.

Modulus of curve families and extremality of spiral-stretch maps

Abstract: We develop a method using the modulus of curve families to study minimisation problems for the mean distortion functional in the class of finite distortion homeomorphisms. We apply our method to prove extremality of the spiral-stretch mappings defined on annuli in the complex plane. This generalises results of Gutlyanskiĭ and Martio [12] and Strebel [23].

Multiscale analysis for ergodic schrödinger operators and positivity of Lyapunov exponents

Abstract: A variant of multiscale analysis for ergodic Schrodinger operators is developed. This enables us to prove positivity of Lyapunov exponents, given initial scale estimates and an initial Wegner estimate. This postivivity is then applied to high-dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur–Figotin formalism.

Uniform geometric estimates of sublevel sets

Abstract: This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright [7], Phong, Stein, and Sturm [23], and Carbery and Wright [8] from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators, which generalize mixed derivatives in ℝ d . As a consequence, it is shown that, in comparison to these previous works, improved uniform estimates are possible in all but certain explicitly “flat” situations.

Quasiconformal Homogeneity of Genus Zero Surfaces

Abstract: A Riemann surface M is said to be K-quasiconformally homogeneous if, for every two points p, q ∈ M, there exists a K-quasiconformal homeomorphism f: M→M such that f(p) = q. In this paper, we show there exists a universal constant K > 1 such that if M is a K-quasiconformally homogeneous hyperbolic genus zero surface other than ⅅ^2, then K ≥ K. This answers a question by Gehring and Palka [10]. Further, we show that a non-maximal hyperbolic surface of genus g ≥ 1 is not K-quasiconformally homogeneous for any finite K ≥ 1.

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Abstract: Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.

Invariant measures for non-primitive tiling substitutions

Abstract: We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpinski gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.

Stokeswaves with vorticity

Abstract: The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the maximum principle.

Asymptotic behavior of solutions of a biharmonic Dirichlet problem with large exponents

Abstract: We analyze blow-up phenomena of bounded integrable solutions of a semilinear fourth order elliptic problem with a large exponent under Dirichlet boundary conditions. We extend the results obtained by Ren-Wei in [26] and [27] to the biharmonic case.

Parametric Poincaré-Perron theorem with applications

Abstract: We prove a parametric generalization of the classical Poincare-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

Polynomials of the best uniform approximation to sgn(x) on two intervals

Abstract: We describe polynomials of best uniform approximation to sgn(x) on the union of two intervals [−A,−1] ⊂ [1, B] in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error in this approximation.

On the spectrum of Bargmann-Toeplitz operatorswith symbols of a variable sign

Abstract: This paper discusses the spectrum of Toeplitz operators on Bargmann spaces. The Toeplitz operators that we study have real symbols with variable sign and compact support. A class of examples is considered in which the asymptotics of the eigenvalues of such operators can be computed. These examples show that the asymptotics depends on the geometry of the support of the positive and negative parts of the symbol. Applications to the perturbed Landau Hamiltonian are also given.

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

Abstract: We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity as the Ginzburg-Landau parameter e tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

Orders of accumulation of entropy on manifolds

Abstract: For a continuous self-map T of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold M and any countable ordinal α, we construct a continuous, surjective self-map ofM having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism.

Semigroups versus evolution families in the loewner theory

Abstract: We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.

Universally measure-preserving homeomorphisms of cantor minimal systems

Abstract: In 1995, T. Giordano, I. Putnam, and C. Skau [GPS] made a significant breakthrough in Cantor minimal system theory. They completely classified Cantor minimal systems in the sense of topological orbit equivalence by using C*-algebra and homological algebra techniques. Since then, a dynamical proof of their theorem has been conjectured. Such a proof is presented in this paper. We establish orbit equivalence theory based on a finite coordinate change relation arising from an ordered Bratteli diagram, which is known from [HK] in the finitary case of ergodic probability measure-preserving transformations. We obtain the Orbital Extension Theorem. This theorem is considered a topological version of the Copying Lemma of Y. Katznelson and B. Weiss [KW], which has played an important role in measured orbit equivalence theory.

Erratum to: “semiclassical second microlocal propagation of regularity and integrable systems”

Abstract: We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifoldX with respect to a Lagrangian submanifold of T X. The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of T R, and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hörmander’s theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g. eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.