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In the paper, we construct compact embedded convex λ -hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. As the special case of our result, we solve Sun's problem (Int. Math. Res. Not. (2021) 11818–11844). In this sense, one can not expect to have Alexandrov type theorem for λ -hypersurfaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D''Angelo on such analytic maps in . Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables. [Copyright &y& Elsevier]

Abstract: This is the final one in the series of papers where we introduce and study the -algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated -algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as -algebras associated with topological graphs. [Copyright &y& Elsevier]

Abstract: Let be a class of unital C*-algebras. The class of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in is studied (Lin considered the case that consists of finite-dimensional C*-algebras or the tensor products of such with ). In particular, the question is considered whether, for any simple separable , there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in , such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.) [Copyright &y& Elsevier]

In a recent paper, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove a conjecture of Jorgensen and Pedersen by Keller''s criterion. Different techniques to prove these facts have also been developed by Kolountzakis, Iosevich and Pedersen. The primary aim of this paper is to present an elementary method of describing certain characterizations of spectra and tilings. To illustrate this method, we first give a simple proof of this characterization. We then use the method to derive some characteristic results connected with the dual Fuglede''s spectral-set conjecture. The results here extend several known conclusions in a simple manner. [Copyright &y& Elsevier]

For 2&les;m&les;l/2, let G be a simply connected Lie group with g0=so(2m,2l−2m) as Lie algebra, let g=k⊕p be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra k∩g0, and let U(g) be the universal enveloping algebra of g. This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g.The roots with respect to the usual compact Cartan subalgebra are all ±ei±ej with 1&les;i. In the usual positive system of roots, the simple root em−em+1 is noncompact and the other simple roots are compact. Let q=l⊕u be the parabolic subalgebra of g for which em−em+1 contributes to u and the other simple roots contribute to l, let L be the analytic subgroup of G with Lie algebra l∩g0, let LC=Intg(l), let 2δ(u) be the sum of the roots contributing to u, and let q¯=l⊕u¯ be the parabolic subalgebra opposite to q.The members of u∩p are nilpotent members of g. The group LC acts on u∩p with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on u∩p that carries a fully reducible representation of LC.For s∈Z, let λs=∑lower limit k=1, upper limit m (−l+s/2)ek. Then λs defines a one-dimensional (l,L) module Cλs. Extend this to a (q¯,L) module by having u¯ act by 0, and define N(λs+2δ(u))=U(g)⊗q¯Cλs+2δ(u). Let N′(λs+2δ(u)) be the unique irreducible quotient of N(λs+2δ(u)). The representations under study are πs=ΠS(N(λs+2δ(u))) and πs′=ΠS(N′(λs+2δ(u))), where S=dim(u∩k) and ΠS is the Sth derived Bernstein functor.For s>2l−2, it is known that πs=πs′ and that πs′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m&les;s&les;2l−2 that πs=πs′ and that πs′ is still unitary. The present paper shows that πs′ is unitary for 0&les;s&les;m−1 even though πs≠πs′, and it relates the K spectrum of the representations πs′ to the representation of LC on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in πs′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on πs′ does not make πs′ unitary for s<0 and that the K spectrum of πs′ in these cases is not related in the above way to the representation of LC on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra g0=so(2m,2l−2m+1), 2&les;m&les;l/2. [Copyright &y& Elsevier]

In our previous papers (Adv. in Math. 138 (1) (1998) 182; Potential Anal. 12 (2000) 419), we have obtained a decomposition of | f |, where f is a function defined on Rn, that is analogous to the one proved by H. Tanaka for martingales (the so-called “Tanaka formula”). More precisely, the decomposition has the form | f |=f˜+D&ast;0( f ), where D&ast;0( f ) is (a variant of ) the density of the area integral associated with f. This functional (introduced by R.F. Gundy in his 1983 paper (The density of area integral, Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth, Belmont, CA, 1983, pp. 138–149.)) can be viewed as the counterpart of the local time in Euclidean harmonic analysis. In this paper, we are interested in boundedness and continuity properties of the mapping f↦f˜ (which we call the Le´vy transform in analysis) on some classical function or distribution spaces. As was shown in [4,5], the above (non-linear) decomposition is bounded in Lp for every p∈[1,+∞[, i.e. one has || f˜ ||p&les;Cp|| f||p, where Cp is a constant depending only on p. Nevertheless our methods (roughly speaking, the Caldero´n–Zygmund theory in [4], stochastic calculus and martingale inequalities in [5]) both gave constants Cp whose order of magnitude near 1 is O(1/(p−1)). The aim of this paper is two-fold: first, we improve the preceding result and we answer a natural question, by proving that the best constants Cp are bounded near 1. Second, we prove that the Le´vy transform f↦f˜ is continuous on the Hardy spaces Hp with p>n/(n+1). [Copyright &y& Elsevier]

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1–1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures. [Copyright &y& Elsevier]

We consider the periodic weighted operator Ty=−ρ−2(ρ2y′)′+1/4 ρ−4 in L2(R,ρ2 dx) where ρ is a 1-periodic positive function satisfying q=ρ′/ρ∈L2(0,1). The spectrum of T consists of intervals separated by gaps. In the first part of the paper we construct the Marchenko–Ostrovski mapping q→h(q) and solve the corresponding inverse problem. For our approach it is essential that the mapping h has the factorization h(q)=h0(V(q)), where q→V(q) is a certain nonlinear mapping and V→h0(V) is the Marchenko–Ostrovski mapping for the Hill operator. Moreover, we solve the inverse problem for the gap length mapping. In the second part of this paper we derive the trace formula for T. [Copyright &y& Elsevier]

In this paper, we establish a moderate deviation principle for two-dimensional stochastic Navier–Stokes equations driven by multiplicative Lévy noises. The weak convergence method introduced by Budhiraja, Dupuis and Ganguly in [3] plays a key role. [ABSTRACT FROM AUTHOR]

In this paper, we give a formal solution of some nonlinear evolution equations. By the formal solution, we can obtain the blow-up solution of the heat equations, even in the supercritical case. [ABSTRACT FROM AUTHOR]

Abstract: We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in Du and Lin (2010) [12], Du and Guo (2011) [9]. In both cases, a spreading–vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading–vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods. [Copyright &y& Elsevier]

In this paper, we study existence of isometric embedding of S q m into S p n , where 1 ≤ p ≠ q ≤ ∞ and n ≥ m ≥ 2. We show that for all n ≥ m ≥ 2 if there exists a linear isometry from S q m into S p n , where (q , p) ∈ (1 , ∞ ] × (1 , ∞) ∪ (1 , ∞) ∖ { 3 } × { 1 , ∞ } and p ≠ q , then we must have q = 2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever S q embeds isometrically into S p for (q , p) ∈ (1 , ∞) × [ 2 , ∞) ∪ [ 4 , ∞) × { 1 } ∪ { ∞ } × (1 , ∞) ∪ [ 2 , ∞) × { ∞ } with p ≠ q , we must have q = 2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative L p -spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m ≥ 2 and 1 < q < 2 , S q m embeds isometrically into S ∞ n , was left open in Bull. London Math. Soc. 52 (2020) 437-447. [ABSTRACT FROM AUTHOR]

Abstract: The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or -diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study which led to discovering not only sharp geometric conditions for their validity but also an incredible rich family of tools, techniques and equivalent concepts ranging from maximum principles at infinity, function theoretic tests (Khasʼminskii criterion), comparison techniques etc. The present paper aims to move a number of steps forward in the development of a similar apparatus for the Feller property. [Copyright &y& Elsevier]

Abstract: In the paper we generalize the main results presented in Bentkus and Paulauskas (2004) by giving rates of approximation of some semigroups of operators of the order , . Also two classes of operators, generalizing sectorial and quasi-sectorial operators, are introduced and their properties are studied. [Copyright &y& Elsevier]

Abstract: In the paper, we first show the existence of global periodic conservative solutions to the Cauchy problem for a periodic modified two-component Camassa–Holm equation. Then we prove that these solutions, which depend continuously on the initial data, construct a semigroup. [Copyright &y& Elsevier]

Abstract: Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials. [ABSTRACT FROM AUTHOR]

Abstract: A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss a new method for computing the first Dirichlet eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in if and for any bounded domain in the particular case . For the method is validated numerically for the square. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the isomorphic structure of symmetrizations of quasi-Banach ideal function or sequence lattices. The symmetrization of a quasi-Banach ideal lattice E of measurable functions on , , or , consists of all functions with decreasing rearrangement belonging to E. For an order continuous E we show that every subsymmetric basic sequence in which converges to zero in measure is equivalent to another one in the cone of positive decreasing elements in E, and conversely. Among several consequences we show that, provided E is order continuous with Fatou property, contains an order isomorphic copy of if and only if either E contains a normalized -basic sequence which converges to zero in measure, or contains the function . We apply these results to the family of two-weighted Orlicz–Lorentz spaces defined on or , . This family contains usual Orlicz–Lorentz spaces when and Orlicz–Marcinkiewicz spaces when . We show that for a large class of weights , it is equivalent for the space , and for the non-weighted Orlicz space to contain a given sequential Orlicz space isomorphically as a sublattice in their respective order continuous parts. We provide a complete characterization of order isomorphic copies of in these spaces over or exclusively in terms of the indices of φ. If we show that the set of exponents p for which lattice embeds in the order continuous part of is the union of three intervals determined respectively by the indices of φ and by the condition that the function belongs to the space. [Copyright &y& Elsevier]

Abstract: Let be an elliptic operator with unbounded and sufficiently smooth coefficients and let μ be a (sub)-invariant measure of the operator . In this paper we give sufficient conditions guaranteeing that the closure of the operator generates a sub-Markovian strongly continuous semigroup of contractions in . Applications are given in the case when is a generalized Schrödinger operator. [Copyright &y& Elsevier]

Abstract: Let , , and . In this paper, we introduce a new class of function spaces which unify and generalize the Triebel–Lizorkin spaces with both and and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and , J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where , , , , and denotes the conjugate index of ; as an application of this, we further introduce certain Hardy–Hausdorff spaces and prove that the dual space of is just when . [Copyright &y& Elsevier]

Abstract: In 1960 Pukánszky introduced an invariant associating to every masa in a separable II1 factor a non-empty subset of . This invariant examines the multiplicity structure of the von Neumann algebra generated by the left-right action of the masa. In this paper it is shown that any non-empty subset of arises as the Pukánszky invariant of some masa in a separable McDuff II1 factor containing a masa with Pukánszky invariant . In particular the hyperfinite II1 factor and all separable McDuff II1 factors with a Cartan masa satisfy this hypothesis. In a general separable McDuff factor we show that every subset of containing ∞ is obtained as a Pukánszky invariant of some masa. [Copyright &y& Elsevier]

Abstract: In this paper, we give a complete solution to the gap labelling conjecture for quasi-crystals. The method adopted relies on the index theory for laminations, and the main tools are the Connes–Skandalis longitudinal K-theory index morphism together with the measured index formula. [Copyright &y& Elsevier]

Abstract: In this short paper I prove a Harnack type inequality of the blowing-up solutions for a class of fourth-order equations with exponential growth on a compact four manifold. The main method I use is the moving-plane method. [Copyright &y& Elsevier]

Abstract: By a recent result of M. De La Rosa and C. Read, there exist hypercyclic Banach space operators which do not satisfy the Hypercyclicity Criterion. In the present paper, we prove that such operators can be constructed on a large class of Banach spaces, including or . [Copyright &y& Elsevier]

Abstract: In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball. [Copyright &y& Elsevier]

Abstract: In this paper, we give necessary and sufficient conditions for Talagrand''s like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimension-free concentration of measure property. Another byproduct is the characterization of modified Log-Sobolev inequalities for log-concave probability measures on . [Copyright &y& Elsevier]

Abstract: In this paper, firstly we find best constants for two convolution problems on the unit circle via a variational method. Then we apply the best constants on a nonlinear integrable shallow water equation (the DGH equation) to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Finally, we discuss the blow-up phenomena for the nonperiodic case. [Copyright &y& Elsevier]

Abstract: We study multivariate generalisations of the classical Wiener–Hopf algebra, which is the C∗-algebra generated by the Wiener–Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C∗-algebra is known to be isomorphic to the reduced C∗-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener–Hopf C∗-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in K-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices. [Copyright &y& Elsevier]

Abstract: Using the Sz.-Nagy–Foias functional model it was shown in [L. Kérchy, Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes, Acta Sci. Math. (Szeged) 61 (1995) 443–476] that under certain conditions on a contraction T the natural embedding of a Hardy space of vector-valued functions into the corresponding space can be factored into the product of two transformations, intertwining T with a unilateral shift and with an absolutely continuous unitary operator, respectively. The norm estimates in the Factorization Theorem of this paper are sharpened to their best possible form by essential improvements in the proof. As a consequence we obtain that if the residual set of a contraction covers the whole unit circle then those invariant subspaces, where the restriction is similar to the unilateral shift with a similarity constant arbitrarily close to 1, span the whole space. Furthermore, the hyperinvariant subspace problem for asymptotically non-vanishing contractions is reduced to these special circumstances. [Copyright &y& Elsevier]

Abstract: In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg–de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference of two solutions of (1.1) at two different times and which guarantee that . [Copyright &y& Elsevier]

Abstract: Let B be a Douglas algebra and let be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that is generated by and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra generated by Toeplitz operators (on the Bergman space) with symbols in has a canonical decomposition for some R in the commutator ideal ; and S is in iff the Berezin transform vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the set of trivial Gleason parts. [Copyright &y& Elsevier]

Abstract: In this paper we study several qualitative properties of the Degasperis–Procesi equation. We first established the precise blow-up rate and then determine the blow-up set of blow-up strong solutions to this equation for a large class of initial data. We finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions. [Copyright &y& Elsevier]

Abstract: The Glauber dynamics investigated in this paper are spatial birth and death processes in a continuous system having a grand canonical Gibbs measure of Ruelle type as an invariant measure. We prove that such processes, when appropriately scaled, have as scaling limit a generalized Ornstein–Uhlenbeck process. First we prove convergence of the corresponding Dirichlet forms. This convergence requires only very weak assumptions. The interaction potential ϕ only has to be stable (S), integrable (I), and we have to assume the low activity high temperature regime. Under a slightly stronger integrability condition () and a conjecture on the Percus–Yevick equation we even can prove strong convergence of the corresponding generators. Finally, we prove that the scaled processes converge in law. Here the hardest part is to show tightness of the scaled processes (note that the processes only have càdlàg sample path). For the proof we have to assume that the interaction potential is positive (P). The limiting process then is identified via the associated martingale problem. For this the above mentioned strong convergence of generators is essential. [Copyright &y& Elsevier]

Abstract: The Lévy–Khintchine formula or, more generally, Courrège''s theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric. [Copyright &y& Elsevier]

Abstract: In the first part of the paper we discuss a multi-dimensional analogue of the well-known construction by D. Clark that allows one to study families of spectral measures of perturbations of the model contraction. In the second part we present extensions of the relevant results on the boundary behavior of pseudocontinuable functions. We show that, although the most direct analogue of the scalar theorem on the existence of boundary values for pseudocontinuable functions with respect to Clark measures fails in the non-scalar situation, suitable vector-valued versions of such results can be found. [Copyright &y& Elsevier]

Abstract: This paper is devoted to inequalities of Lieb–Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues . We prove that there exists a positive constant , such that, if , then and determine the optimal value of . Such an inequality is interesting for studying the stability of mixed states with occupation numbers. We show how the infimum of on all possible potentials V, which is a lower bound for , corresponds to the optimal constant of a subfamily of Gagliardo–Nirenberg inequalities. This explains how (∗) is related to the usual Lieb–Thirring inequality and why all Lieb–Thirring type inequalities can be seen as generalizations of the Gagliardo–Nirenberg inequalities for systems of functions with occupation numbers taken into account. We also state a more general inequality of Lieb–Thirring type where F and G are appropriately related. As a special case corresponding to , (∗∗) is equivalent to an optimal Euclidean logarithmic Sobolev inequality where , is any nonnegative sequence of occupation numbers and is any sequence of orthonormal functions. [Copyright &y& Elsevier]

Abstract: In this paper we study the problem where are positive radial functions, and . We give existence and nonexistence results, depending on the parameters p and λ. It turns out that is a critical value for the existence of solutions. [Copyright &y& Elsevier]

Abstract: The factorization of almost-periodic triangular symbols, G, associated to finite-interval convolution operators is studied for two classes of operators whose Fourier symbols are almost periodic polynomials with spectrum in the group (, , ). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann–Hilbert problem in and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the Riemann–Hilbert problem. Some unexpected, but interesting, results are obtained concerning the Fourier spectrum of the solutions of . In particular it is shown that a solution exists with Fourier spectrum in the additive group whether this group contains or not. Possible application of the method to more general classes of symbols is considered in the last section of the paper. [Copyright &y& Elsevier]

Abstract: In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and -concentration are proved. The proof relies on linear and nonlinear profile decompositions. [Copyright &y& Elsevier]

Abstract: We study properties of solutions with isolated singularities to general conformally invariant fully nonlinear elliptic equations of second order. The properties being studied include radial symmetry and monotonicity of solutions in the punctured Euclidean space and the asymptotic behavior of solutions in a punctured ball. Some results apply to more general situations including more general fully nonlinear elliptic equations of second order, and some have been used in a companion paper to establish comparison principles and Liouville type theorems for degenerate elliptic equations. [Copyright &y& Elsevier]

Abstract: In this paper we prove the following: let be a continuous function, increasing in and . Then there exists a series of the formwith the following property: for each a weighted function can be constructed, so that the series is universal in the weighted space with respect to rearrangements. [Copyright &y& Elsevier]

Abstract: In this paper we establish a direct connection between stable approximate unitary equivalence for -homomorphisms and the topology of the KK-groups which avoids entirely -algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner''s topology and the Brown–Salinas topology. We study the generalized Rørdam group , and prove that if a separable exact residually finite dimensional -algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type . In particular such an embedding exists for the -algebra of a second countable amenable locally compact maximally almost periodic group. [Copyright &y& Elsevier]

Abstract: We reformulate some of Moeglin''s results on the correspondence for the dual pairs , with p and q even, and fill in the cases where p and q are both odd. We arrive at a complete and detailed description, in terms of Langlands parameters, of the dual pair correspondence for the cases and . In addition, we point out and suggest a way to correct an error in Moeglin''s paper. [Copyright &y& Elsevier]

Abstract: In two recent papers (Foias and Pearcy, J. Funct. Anal., in press, Hamid et al., Indiana Univ. Math. J., to appear), the authors reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every -BCP-operator that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). In this note, we continue this study by showing, with the help of a new equivalence relation, that every operator whose spectrum is uncountable, as well as every nonalgebraic operator with finite spectrum, has a hyperlattice (i.e., lattice of hyperinvariant subspaces) that is isomorphic to the hyperlattice of a , quasidiagonal, (BCP)-operator whose spectrum is the closed unit disc. [Copyright &y& Elsevier]

In this paper, we introduce a new equivalence relation, ampliation quasisimilarity, on L(H), more general than quasisimilarity, that preserves the existence of nontrivial hyperinvariant subspaces. We show that if T does not have nontrivial hyperinvariant subspaces for elementary reasons, then T is ampliation quasisimilar to a (BCP)-operator in the class C00. This reduces the hyperinvariant subspace problem for operators in L(H) to a very special subcase of itself. [Copyright &y& Elsevier]

In this paper new Lαp→Lβq estimates are proved for translation-invariant Radon transforms along curves for α&les;β and p. For a fixed α and β, if p is sufficiently close to 2 the best possible q is obtained, up to &epsiv;. The method is related to that of Greenblatt (Ph.D. Thesis, Princeton University, 1998). [Copyright &y& Elsevier]

In the paper we prove an extension theorem for matrices with entries in H∞(U) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras. [Copyright &y& Elsevier]