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We prove compactness of a restricted set of real-valued, compactly supported potentials V V for which the corresponding Schrödinger operators H V H_V have the same resonances, including multiplicities. More specifically, let B R ( 0 ) B_R(0) be the ball of radius R > 0 R > 0 about the origin in R d \mathbb {R}^d , for d = 1 , 3 d=1,3 . Let I R ( V 0 ) \mathcal {I}_R (V_0) be the set of real-valued potentials in C 0 ∞ ( B ¯ R ( 0 ) ; R ) C_0^\infty ( \overline {B}_R(0); \mathbb {R}) so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as H V 0 H_{V_0} , for a fixed, but arbitrary, potential V 0 ∈ C 0 ∞ ( B ¯ R ( 0 ) ; R ) V_0 \in C_0^\infty ( \overline {B}_R(0); \mathbb {R}) . We prove that the set I R ( V 0 ) \mathcal {I}_R (V_0) is a compact subset of C 0 ∞ ( B ¯ R ( 0 ) ) C_0^\infty (\overline {B}_R(0)) in the C ∞ C^\infty -topology. An extension to Sobolev spaces of less regular potentials is discussed.

Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globally regular on $M$ for degrees $q_0\le q\le n-q_0$; and also for the top degrees $q=0$ and $q=n$ in the case $q_0=1$. We also introduce the notion of a "plurisubharmonic CR manifold" and show that it generalizes the notion of "plurisubharmonic defining function" for a a domain in $\mathbb C^N$ and implies that $M$ satisfies the weak compactness property., 29 pages

Let $B^{\sigma}_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $\alpha$-regular measure $\mu$. The {\it critical exponent} $\sigma^*$ is the supremum of the $\sigma$ such that $B^{\sigma}_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{\sigma^*}_{2, \infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied., Comment: 34 pages,15 figures

Let M M be a 3 × 3 3\times 3 integer matrix each of whose eigenvalues is greater than 1 1 in modulus and let D ⊂ Z 3 \mathcal {D}\subset \mathbb {Z}^3 be a set with | D | = | det M | |\mathcal {D}|=|\det M| , called a digit set. The set equation M T = T + D MT = T+\mathcal {D} uniquely defines a nonempty compact set T ⊂ R 3 T\subset \mathbb {R}^3 . If T T has positive Lebesgue measure it is called a 3 3 -dimensional self-affine tile. In the present paper we study topological properties of 3 3 -dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form D = { 0 , v , 2 v , … , ( | det M | − 1 ) v } \mathcal {D}=\{0,v,2v,\ldots , (|\det M|-1)v\} for some v ∈ Z 3 ∖ { 0 } v\in \mathbb {Z}^3\setminus \{0\} . We prove that the boundary of such a tile T T is homeomorphic to a 2 2 -sphere whenever its set of neighbors in a lattice tiling which is induced by T T in a natural way contains 14 14 elements. The combinatorics of this lattice tiling is then the same as the one of the bitruncated cubic honeycomb, a body-centered cubic lattice tiling by truncated octahedra. We give a characterization of 3 3 -dimensional self-affine tiles with collinear digit set having 14 14 neighbors in terms of the coefficients of the characteristic polynomial of M M . In our proofs we use results of R. H. Bing on the topological characterization of spheres.

Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.

Let X X be an irreducible m m -symmetric Markov process on E E with strong Feller property. In addition, suppose X X has a tightness property. We then show that the semigroup of X X is a compact operator on L 2 ( E ; m ) L^2(E;m) and every eigenfunction has a bounded continuous version.

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some ϵ > 0 \epsilon > 0 so that if the squeezing function on a smoothly bounded convex domain is greater than 1 − ϵ 1-\epsilon outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension d d there exists some ϵ > 0 \epsilon > 0 so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within ϵ \epsilon of − 4 / ( d + 1 ) -4/(d+1) outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to − 4 / ( d + 1 ) -4/(d+1) on the boundary).

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq \Gamma_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathcal{L}}$ which are right invariant under a minimal parabolic (Borel) subgroup $B < G$. In the special case when $G$ is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on $\hat{\mathcal{L}}$ right invariant under two distinct minimal parabolics, and therefore all of $G$, are in bijective correspondence with the holonomy-invariant ones., Comment: This paper has been accepted for publication in Transactions of the AMS

The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger’s theorem to include toral automorphisms and, more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos’s 1956 book regarding an isomorphism invariant that he proposed.

For any compact set $K \subseteq \mathbb{R}^3$ we define a number $r(K)$ that is either a nonnegative integer or $\infty$. Intuitively, $r(K)$ provides some information on how wildly $K$ sits in $\mathbb{R}^3$. We show that attractors for discrete or continuous dynamical systems have finite $r$ and then prove that certain arcs, balls and spheres cannot be attractors by showing that their $r$ is infinite.

Let ( M , g ¯ , e − f d μ ) (M,\overline {g}, e^{-f}d\mu ) be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that in M M there is no complete two-sided L f L_f -stable immersed f f -minimal hypersurface with finite weighted volume. Further, if M M is a 3 3 -manifold, we prove a smooth compactness theorem for the space of complete embedded f f -minimal surfaces in M M with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in R 3 \mathbb {R}^3 by Colding-Minicozzi.

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in ℝ2 introduced by Riviere. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces

In the same vein as the classical Stone–ˇCech compactification, we prove in this work that the maximal spectra of the rings of semialgebraic and bounded semialgebraic functions on a semialgebraic set M ⊂ Rn, which are homeomorphic topological spaces, provide the smallest Hausdorff compactification of M such that each bounded R-valued semialgebraic function on M extends continuously to it. Such compactification β∗sM, which can be characterized as the smallest compactification that dominates all semialgebraic compactifications of M, is called the semialgebraic Stone– ˇ Cech compactification of M, although it is very rarely a semialgebraic set. We are also interested in determining the main topological properties of the remainder ∂M = β∗sM \M and we prove that it has finitely many connected components and that this number equals the number of connected components of the remainder of a suitable semialgebraic compactification of M. Moreover, ∂M is locally connected and its local compactness can be characterized just in terms of the topology of M.

We show that Zorboska’s criterion for compactness of Toeplitz operators with BMO 1 \text {BMO}^1 symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on C n \mathbb {C}^n . We establish some basic properties of BMO p \text {BMO}^p for p ≥ 1 p \geq 1 and complete the characterization of bounded and compact Toeplitz operators with BMO 1 \text {BMO}^1 symbols. Via the Bargmann isometry and results of Lo and Englis̆, we also give a compactness criterion for the Gabor-Daubechies “windowed Fourier localization operators” on L 2 ( R n , d v ) L^2(\mathbb {R}^n, dv) when the symbol is in a BMO 1 \text {BMO}^1 Sobolev-type space. Finally, we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of Toeplitz operators.

We establish the existence of travelling waves to the gradient system u t = u z z − ∇ W ( u ) u_t = u_{zz} - \nabla W(u) connecting two minima of W W when u : R × ( 0 , ∞ ) ⟶ R N u : \mathbb {R} \times (0,\infty ) \longrightarrow \mathbb {R}^N ; that is, we establish the existence of a pair ( U , c ) ∈ [ C 2 ( R ) ] N × ( 0 , ∞ ) (U,c) \in [C^2(\mathbb {R})]^N \times (0,\infty ) , satisfying \[ { U x x − ∇ W ( U ) = − c U x , U ( ± ∞ ) = a ± , \left \{\begin {array}{l} U_{xx} - \nabla W ( U ) = - c\; U_x,\\ U(\pm \infty ) = a^{\pm }, \end {array}\right . \] where a ± a^{\pm } are local minima of the potential W ∈ C loc 2 ( R N ) W \in C_{\textrm {loc}}^2(\mathbb {R}^N) with W ( a − ) > W ( a + ) = 0 W(a^-)> W(a^+)=0 and N ≥ 1 N \geq 1 . Our method is variational and based on the minimization of the functional E c ( U ) = ∫ R { 1 2 | U x | 2 + W ( U ) } e c x d x E_c (U) = \int _{\mathbb {R}}\Big \{ \frac {1}{2}|U_x|^2 + W( U ) \Big \}e^{cx} dx in the appropriate space setup. Following Alikakos and Fusco (2008), we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed.

Consider a Lévy process X t X_t with quadratic variation process V t = σ 2 t + ∑ 0 > s ≤ t ( Δ X s ) 2 V_t=\sigma ^2 t+ \sum _{0>s\le t} (\Delta X_s)^2 , t > 0 t>0 , where Δ X t = X t − X t − \Delta X_t=X_t-X_{t-} denotes the jump process of X X . We give stability and compactness results, as t ↓ 0 t\,\downarrow \,0 , for the convergence both of the deterministically normed (and possibly centered) processes X t X_t and V t V_t , as well as theorems concerning the “self-normalised” process X t / V t X_{t}/\sqrt {V_t} . Thus, we consider the stochastic compactness and convergence in distribution of the 2-vector ( ( X t − a ( t ) ) / b ( t ) , V t / b ( t ) ) \left ((X_t-a(t))/b(t), V_t/b(t)\right ) , for deterministic functions a ( t ) a(t) and b ( t ) > 0 b(t)>0 , as t ↓ 0 t\,\downarrow \,0 , possibly through a subsequence; and the stochastic compactness and convergence in distribution of X t / V t X_{t}/\sqrt {V_t} , possibly to a nonzero constant (for stability), as t ↓ 0 t\,\downarrow \,0 , again possibly through a subsequence. As a main application it is shown that X t / V t ⟶ D N ( 0 , 1 ) X_{t}/\sqrt {V_t}\stackrel {\mathrm {D}}{\longrightarrow } N(0,1) , a standard normal random variable, as t ↓ 0 t\,\downarrow \,0 , if and only if X t / b ( t ) ⟶ D N ( 0 , 1 ) X_t/b(t)\stackrel {\mathrm {D}}{\longrightarrow } N(0,1) , as t ↓ 0 t\downarrow 0 , for some nonstochastic function b ( t ) > 0 b(t)>0 ; thus, X t X_t is in the domain of attraction of the normal distribution, as t ↓ 0 t\,\downarrow \,0 , with or without centering constants being necessary (these being equivalent). We cite simple analytic equivalences for the above properties, in terms of the Lévy measure of X X . Functional versions of the convergences are also given.

We deform the contact form by the (normalized) CR Yamabe flow on a closed spherical CR 3 3 -manifold. We show that if a contact form evolves with positive Tanaka-Webster curvature and vanishing torsion from initial data, then we obtain a new Li-Yau-Hamilton inequality for the CR Yamabe flow. By combining this parabolic subgradient estimate with a compactness theorem of a sequence of contact forms, it follows that the CR Yamabe flow exists for all time and converges smoothly to, up to the CR automorphism, a unique limit contact form of positive constant Webster scalar curvature on a closed CR 3 3 -manifold, which is CR equivalent to the standard CR 3 3 -sphere with positive Tanaka-Webster curvature and vanishing torsion.

LetΩ\Omegabe a bounded domain inRn,\mathbf {R}^n,n≥3,n \ge 3,with a boundary∂Ω∈C2.\partial \Omega \in C^2.We consider the following singularly perturbed nonlinear elliptic problem onΩ\Omega:\[ε2Δu−u+f(u)=0, u>0 on Ω,u=0 on ∂Ω,\varepsilon ^2 \Delta u - u + f(u) = 0, \ \ u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega ,\]where the nonlinearityffis of subcritical growth. Under rather strong conditions onf,f,it has been known that for smallε>0,\varepsilon > 0,there exists a mountain pass solutionuεu_\varepsilonof above problem which exhibits a spike layer near a maximum point of the distance functionddfrom∂Ω\partial \Omegaasε→0.\varepsilon \to 0.In this paper, we construct a solutionuεu_\varepsilonof above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions onff, which we believe to be almost optimal.

We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, \xi}(\partial D)$ existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_1: H^{1}(D)\to L^{2}(D)$ and $I_{2}:H^{1}(D)\to L^{2,\xi}(\partial D)$ i.e. on type of singularities. We obtain an exact description of the weights $\xi$ for bounded domains with 'outside peaks' on its boundaries. This result allows us to formulate correctly the corresponding Robin boundary-value problems for elliptic operators., Comment: Correction of misprints: pages 5, 6, 11. We found more compact of Theorem 1, part 3, 4

In this paper we prove a priori estimates for a class of quasi-linear elliptic equations. To make the proofs clear and transparent we concentrate on the p-Laplacian. We focus on L p -estimates for weak solutions of the problem with all standard boundary conditions on non-smooth domains. As an application we prove existence, continuity and compactness of the resolvent operator. We finally prove estimates for solutions to equations with non-linear source and show that, under suitable growth conditions, all solutions are globally bounded.

We generalize the so-called Weighted Konig Lemma, due to Mate, for a submultiplicative function on a subset of the union ∪ n∈ℕ Σ n , where Σ is a set and Σ n is the Cartesian product of n copies of Σ. Instead of a combinatorial argument as done by Mate, our proof uses Tychonoff's compactness theorem to show the existence of a Konig chain for a submultiplicative function. As a consequence, we obtain an extension of the Daubechies—Lagarias theorem concerning a finite set Σ of matrices with right convergent products: Here we replace matrices by Banach algebra elements, and we substitute compactness for finiteness of Σ. The last result yields new generalizations of the Kelisky— Rivlin theorem on iterates of the Bernstein operators on the Banach space C[0, 1].

In their famous paper, Schweizer and Smítal introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case. In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in R 3 \mathbb {R}^3 . Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics. We also prove that any interval map with positive topological entropy contains two invariant subsets X , Y ⊂ I X,Y \subset I such that f | X f|_X has positive topological entropy and f | Y f|_Y displays distributional chaos of type 1 1 , but not conversely.

We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This model lacks strong compactness or contraction properties. Our purpose is to prove the existence of an entropy solution obtained, as usual, in passing to the limit in a sequence of solutions to the parabolic approximation. The method consists in the derivation of a kinetic formulation for the weak limit. The specific structure of the limiting kinetic equation allows for a ‘rigidity theorem’, which identifies some property of the solution (which might be nonunique) to this kinetic equation. This is enough to deduce a posteriori the strong convergence of a subsequence.

Let $\Lambda$ be a compact locally maximal invariant set of a $C^2$-diffeomorphism $f:M\to M$ on a smooth Riemannian manifold $M$. In this paper we study the topological pressure $P_{\rm top}(\phi)$ (with respect to the dynamical system $f|\Lambda$) for a wide class of H\"older continuous potentials and analyze its relation to dynamical, as well as geometrical, properties of the system. We show that under a mild nonuniform hyperbolicity assumption the topological pressure of $\phi$ is entirely determined by the values of $\phi$ on the saddle points of $f$ in $\Lambda$. Moreover, it is enough to consider saddle points with ``large'' Lyapunov exponents. We also introduce a version of the pressure for certain non-continuous potentials and establish several variational inequalities for it. Finally, we deduce relations between expansion and escape rates and the dimension of $\Lambda$. Our results generalize several well-known results to certain non-uniformly hyperbolic systems., Comment: 19 pages, Replaced with revised version, Accepted for publication in Trans. Amer. Math. Soc

In this note, composition operators on Bergman spaces of a simply connected domain are studied characterizing boundedness and compactness of such operators whenever the domain is Lavrentiev.

For a compact set $\Gamma\subset\Bbb{R}^2$ and a point $x$, we define the visible part of $\Gamma$ from $x$ to be the set $\Gamma_x = \{u \in\Gamma : [x, u] \cap\Gamma = \{u\}\}.$ (Here $[x, u]$ denotes the closed line segment joining $x$ to $u$.) In this paper, we use energies to show that if $\Gamma$ is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point $x\in\Bbb{R}^2$, the Hausdorff dimension of $\Gamma_x$ is strictly less than the Hausdorff dimension of $\Gamma$. In fact, for almost every $x$, $\dim_H(\Gamma_x)\leq \frac{1}{2}+\sqrt{\dim_H(\Gamma){-}\frac{3}{4}}.$ We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than $\sigma+\frac{1}{2}+\sqrt{{\dim_H}{(\Gamma)}{-}{\frac{3}{4}}}$ for $\sigma > 0$.

We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS).We prove that for an AC RDS the Ω-limit set ΩB(ω) of any bounded set B is nonempty, compact, strictly invariant and attracts the set B. We establish that the 2D Navier Stokes Equations (NSEs) in a domain satisfying the Poincar´e inequality perturbed by an additive irregular noise generate an AC RDS in the energy space H. As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.

Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).

We consider a class of kinetic equations, equipped with a single conservation law, which generate L 1 -contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the "dissipative" sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in L 1 . The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.

Let A be a unital simple C*-algebra, with tracial rank zero and let X be a compact metric space. Suppose that h 1 ,h 2 : C(X) → A are two unital monomorphisms. We show that h 1 and h 2 are approximately unitarily equivalent if and only if [h 1 ] = [h 2 ] in KL(C(X), A) and τ o h 1 (f) = τ o h 2 (f) for every f ∈ C(X) and every trace r of A. Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let X be a compact metric space and let α, β: X → X be two minimal homeomorphisms. Using the above-mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a K-theoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with the strong orbit equivalence of Giordano, Putnam and Skau, and the K-theoretical condition is equivalent to saying that the associate crossed product C*-algebras are isomorphic. Another application of the above-mentioned result is given for C*-dynamical systems related to a problem of Kishimoto. Let A be a unital simple AH-algebra with no dimension growth and with real rank zero, and let a ∈ Aut(A). We prove that if α r fixes a large subgroup of K 0 (A) and has the tracial Rokhlin property, then Ax α Z is again a unital simple AH-algebra with no dimension growth and with real rank zero.

We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If μ \mu is a continous measure on the locally compact abelian group G G and f ∈ L ∞ ( μ ) f\in L^\infty (\mu ) , then { γ f : γ ∈ G ^ } \{\gamma f:\gamma \in \widehat G\} is not relatively weakly compact. ii) If G G is a discrete abelian group and f ∈ ℓ ∞ ( G ) ∖ C o ( G ) f\in \ell ^\infty (G)\backslash C_o(G) , then { γ f : γ ∈ E } \{\gamma f:\gamma \in E\} is not relatively weakly compact if E ⊂ G ^ E\subset \widehat G has non-empty interior. That result will follow from an existence theorem for I o I_o -sets, as follows. iii) Every infinite subset of a discrete abelian group Γ \Gamma contains an infinite I o I_o -set such that for every neighbourhood U U of the identity of Γ ^ \widehat \Gamma the interpolation (except at a finite subset depending on U U ) can be done using at most 4 point masses. iv) A new proof that B ( G ) ⊂ W A P ( G ) B(G)\subset WAP(G) for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms. v) Analogous results for C o ( G ) C_o(G) , A p ( G ) A_p(G) , and M p ( G ) M_p(G) are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of G ^ \widehat G in L ∞ ( μ ) L^\infty (\mu ) , for abelian G G ; and when ρ \rho is a continuous homomorphism of the locally compact group Γ \Gamma into the unitary elements of a von Neumann algebra M \mathcal {M} , the weak* closure of ρ ( Γ ) \rho (\Gamma ) is studied.

A continuous map f : X → Y f:X\to Y of topological spaces X , Y X, Y is said to be almost 1 1 -to- 1 1 if the set of the points x ∈ X x\in X such that f − 1 ( f ( x ) ) = { x } f^{-1}(f(x))=\{x\} is dense in X X ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and σ \sigma -compact spaces (e.g., n n -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if f f is a minimal self-mapping of a 2-manifold M M , then point preimages under f f are tree-like continua and either M M is a union of 2-tori, or M M is a union of Klein bottles permuted by f f .

We prove global well-posedness of the initial value problem for a modified Kadomtsev-Petviashvili II (mKP II) equation in the energy space. The proof proceeds in three main steps and involves several different techniques. In the first step, we make use of several linear estimates to solve a fourth-order parabolic regularization of the mKP II equation by a fixed point argument, for regular initial data (one estimate is similar to the sharp Kato smoothing effect proved for the KdV equation by Kenig, Ponce, and Vega, 1991). Then, compactness arguments (the energy method performed through the Miura transform) give the existence of a local solution of the mKP II equation for regular data. Finally, we approximate any data in the energy space by a sequence of smooth initial data. Using Bourgain's result concerning the global well-posedness of the KP II equation in L 2 and the Miura transformation, we obtain convergence of the sequence of smooth solutions to a solution of mKP II in the energy space.

Let X be a real reflexive Banach space with dual X* and G C X open and bounded and such that 0 ∈ G. Let T: X ⊃ D(T) → 2 X* be maximal monotone with 0 ∈ D(T) and 0 ∈ T(0), and C: X ⊃ D(C) → X* with 0 E D(C) and C(0) ≠ 0. A general and more unified eigenvalue theory is developed for the pair of operators (T, C). Further conditions are given for the existence of a pair (A, x) ∈ (0, ∞ ) x (D(T + C) n ∂G) such that (**) Tx+λCx? 0. The "implicit" eigenvalue problem, with C(λ, x) in place of λCx, is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators T, C. No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω 1 \omega _1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman’s (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah’s club–guessing sequences) that shows similar results do not hold for closed pre–images of ω 2 \omega _2 .

We show that Brownian motion on any unbounded submanifold P in an ambient manifold N with a pole p is transient if the following conditions are satisfied: The p-radial mean curvatures of P are sufficiently small outside a compact set and the p-radial sectional curvatures of N are sufficiently negative. The 'sufficiency' conditions are obtained via comparison with explicit transience criteria for radially drifted Brownian motion in warped product model spaces.

The Schrödinger operator − Δ + V -\Delta +V , of a compact Riemannian manifold M M , has pure point spectrum. Suppose that V 0 V_0 is a smooth reference potential. Various criteria are given which guarantee the compactness of all V V satisfying spec ⁡ ( − Δ + V ) = spec ⁡ ( − Δ + V 0 ) \operatorname {spec}(-\Delta +V)=\operatorname {spec}(-\Delta +V_0) . In particular, compactness is proved assuming an a priori bound on the W s , 2 ( M ) W_{s,2}(M) norm of V V , where s > n / 2 − 2 s>n/2-2 and n = dim ⁡ M n=\dim M . This improves earlier work of Brüning. An example involving singular potentials suggests that the condition s > n / 2 − 2 s>n/2-2 is appropriate. Compactness is also proved for non–negative isospectral potentials in dimensions n ≤ 9 n\le 9 .

We give a simple example of a variety V \mathbf {V} of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety R R A \mathbf {RRA} of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences. Using probabilistic methods of Erdős, we construct an infinite sequence G 0 , G 1 , … G_0,G_1,\ldots of finite graphs with arbitrarily large chromatic number, such that each G n G_n is a bounded morphic image of G n + 1 G_{n+1} and has no odd cycles of length at most n n . The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from the G n G_n satisfies arbitrarily many axioms from a certain axiomatisation of V ( R R A ) \mathbf {V}\ (\mathbf {RRA}) , while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that V ( R R A ) \mathbf {V}\ (\mathbf {RRA}) has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.

The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel′ on compactness of saddle surfaces in a Euclidean space.

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ ( z ) \mu (z) has the norm ‖ μ ‖ ∞ = 1. \|\mu \|_\infty =1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of μ . \mu . A uniqueness theorem is also proved when the singular set Sing ⁡ ( μ ) \operatorname {Sing} (\mu ) of μ \mu is contained in a totally disconnected compact set with an additional thinness condition on Sing ⁡ ( μ ) . \operatorname {Sing}(\mu ).

In this paper we investigate the zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) + (1/epsilon) D(W(x,t)) + E(W(x,t)). We analyse the singular convergence, as epsilon tends to 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming the existence of a symmetrizer having the so called block structure and by assuming ``dissipativity conditions'' on B. (iii) We perform the convergence analysis by using generalizations of Compensated Compactness due to Tartar and Gerard. Finally we include examples which show how to use our theory to approximate prescribed general quasilinear parabolic systems, satisfying Petrowski parabolicity condition, or general reaction diffusion systems., Comment: 26 pages, preliminary version Dec.00

In this paper we define and discuss tight surfaces - smooth or polyhedral - in three-dimensional compact Euclidean space forms and prove existence and non-existence results. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.

The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author in [5, 9], is an invariant of foliated homotopy type with values in {1,2,...,1}. A foliation with all leaves compact and Hausdorff leaf space M/F is called compact Hausdorff. The transverse saturated category cat\| M of a compact Hausdorff foliation is always finite. In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category cat\| (M) in terms of the geometry of F and the Epstein filtration of the exceptional set E. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that max{cat(M/F),cat\| (E)} � cat\| (M) � cat\| (E) + q We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

Let X t \texttt {X}_t be a symmetric α \alpha -stable process killed on exiting an open subset D D of R n \mathbb R^n . We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set B B in the complement of D D in the first exit moment from D D increases when D D and B B are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets K K in R n \mathbb R^n with given volume, the balls have the least α \alpha -capacity ( 0 > α > 2 0>\alpha >2 ).

We prove that the topological statement: “Any compact covering mapping between two Borel sets is inductively perfect" is equivalent to the set-theoretical statement: “ ∀ α ∈ ω ω , ℵ 1 L ( α ) > ℵ 1 “\,\forall \alpha \in \omega ^\omega ,\; \aleph _1^{L(\alpha )}>\aleph _1 ".

Let (X, d) be a metric space, and f : X → X be a continuous map. In this paper we prove that if R(f) is compact, and w(x, f) ¬= O for all x ∈ X, then f is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism h and a non-expanding map g that is pointwise convergent to a fixed point v 0 such that f is uniformly conjugate to a subsystem (h x g)|S of the product map h x g. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

We consider the hyperbolic Hardy class Q H P (B), 0 < p < ∞. It consists of Φ holomorphic in the unit complex ball B for which |Φ| < 1 and sup √{ Q (Φ(rζ),0)} p dσ(ξ) < ∞, 0

Let N be an isolating neighborhood for a map f. If we can decompose N into the disjoint union of compact sets N 1 and N 2 , then we can relate the dynamics on the maximal invariant set Inv N to the shift on two symbols by noting which component of N each iterate of a point x ∈ Inv N lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to N. In essence, we measure the difference between the Conley index of Inv N and the sum of the indices of Inv N 1 and Inv N 2 .

If X is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line R into X an element of H 1 (X, U/U 0 ), where U is the space of uniformly continuous functions from R to R and U 0 is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to H 1 (X, U/U 0 ) is surjective. If X is the n-dimensional torus, it is bijective, while if X is a compact orientable surface of genus > 1, it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds P to compact smooth manifolds X. With each such Lipschitz homotopy class we associate an element of H n (X, B + /B + 0 ), where n is the dimension of P, B is the space of bounded continuous functions from the positive real axis to R, and B + 0 is the set of all f ∈ B + such that lim t→ ∞ f(t) = 0.