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Abstract Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space ℓ p of all p -summable sequences. The ℓ p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlinear embeddings between Banach spaces. More specifically, the goal of this paper is to study weaker versions of coarse and uniform embeddability, and to provide suggestive evidences that those weaker embeddings may be stronger than one would think. We do such by proving that many known results regarding coarse and uniform embeddability remain valid for those weaker notions of embeddability. [ABSTRACT FROM AUTHOR]

In this paper we develop a structure theory of Einstein manifolds or manifolds with lower Ricci curvature bounds for certain singular spaces that arise as geometric limits of sequences of Riemannian manifolds. This theory generalizes the results that were obtained by Cheeger, Colding and Naber in the smooth setting. In the course of the paper, we will carefully characterize the assumptions that we have to impose on this sequence of Riemannian manifolds in order to guarantee that the individual results hold. An important aspect of our approach is that we don't need impose any Ricci curvature bounds on the sequence of Riemannian manifolds leading to the singular limit. The Ricci curvature bounds will only be required to hold on the regular part of the limit and we will not impose any (synthetic) curvature condition on its singular part. The theory developed in this paper will have applications in the blowup analysis of certain geometric equations in which we study scales that are much larger than the local curvature scale. In particular, this theory will have applications in the study of Ricci flows of bounded scalar curvature, which we will describe in a subsequent paper. [ABSTRACT FROM AUTHOR]

In this paper, we shall introduce h -expansiveness and asymptotical h -expansiveness for actions of sofic groups. By definition, each h -expansive action of a sofic group is asymptotically h -expansive. We show that each expansive action of a sofic group is h -expansive, and, for any given asymptotically h -expansive action of a sofic group, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy. Observe that asymptotically h -expansive property was first introduced and studied by Misiurewicz for Z -actions using the language of tail entropy. And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz's ideas when the group is amenable. It turns out that these two definitions are equivalent in this setting. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the Cauchy problem for a parabolic–elliptic system in modeling chemotaxis as well as self-attracting particles. In the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying ω-limit sets. According to the results of this paper, any nonnegative radially symmetric bounded solution either stabilizes to a steady-state as , or oscillates between two steady-states. Moreover, a rather general class of nonnegative initial data, not necessarily radially symmetric, for which the associated solutions exhibit a complex oscillatory behavior is constructed; their ω-limit sets consist of a nontrivial topological continuum of steady-states. Besides the technical difficulties inherent to the lack of compactness of the resolvent operators, one has to add the challenge that the problem is utterly non-local. Consequently, thought the basic ideas on the foundations of this paper might be considered classical, most of the proofs throughout are extremely sophisticated and absolutely new in their full generality. [Copyright &y& Elsevier]

Abstract: This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials . For the first result, we obtain the estimate of weak solutions in for and , which is an improvement over estimates by Fournier and Guerin [10]. For the second result, we have the estimate of weak solutions in , , which extends part of results by Fournier and Guerin [10] and Alexandre, Liao and Lin [1]. As an application, we deduce some global well-posedness results for . Our estimates include the case , which is the key point in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, the authors first show that the classical Hardy space can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups and , respectively, where is an elliptic operator with real constant coefficients of homogeneous order 2m (). Moreover, the authors also prove that can be characterized by the Riesz transforms if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space and give its characterizations by area integrals related to the semigroups and , respectively. Finally, authors give the boundedness of Riesz transforms, square functions and maximal functions associated with L. [Copyright &y& Elsevier]

Abstract: In this paper we characterize bounded differences of composition operators from standard weighted Bergman spaces into Lebesgue spaces when and . In the process we also find an interesting connection between the difference operator and certain weighted composition operators answering a question raised in Saukko (2011) . By combining the results of this paper with those of Saukko (2011) , we obtain a complete characterization of bounded and compact differences between standard weighted Bergman spaces. [Copyright &y& Elsevier]

Abstract: In the present paper we define Hardy spaces with variable exponents on by the grand maximal function, and then investigate their several properties. The present paper will connect harmonic analysis with function spaces with variable exponents. We obtain the atomic decomposition and the molecular decomposition. With these decomposition proved, we investigate the Littlewood–Paley characterization. Also, we specify the dual spaces of Hardy spaces with variable exponents. They will turn out to be Campanato spaces with variable growth conditions. The present paper covers local Hardy spaces with variable exponents. [Copyright &y& Elsevier]

Abstract In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian Δ N + on the upper half space R + n and the reflection Neumann Laplacian Δ N on R n with respect to the weights associated to Δ N + and Δ N respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to Δ N is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt–Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via the area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on R n and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt A p weights with p ∈ (1 , 2 ] while the previously known related results cover only p ∈ (1 , n + 1 n ]. We also point out that this two weight commutator theorem might not be true in the setting of general operators L , and in particular we show that it is not true when L is the Dirichlet Laplacian Δ D + on R + n. [ABSTRACT FROM AUTHOR]

Abstract We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R 2 , where the intraspecies interaction (− a 1 , − a 2) and the interspecies interaction − β are both attractive, i. e , a 1 , a 2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L 2 -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β ↗ β ⁎ = a ⁎ + (a ⁎ − a 1) (a ⁎ − a 2) , where 0 < a i < a ⁎ : = ‖ w ‖ 2 2 (i = 1 , 2) is fixed and w is the unique positive solution of Δ w − w + w 3 = 0 in R 2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. [ABSTRACT FROM AUTHOR]

Abstract: Let be a closed subgroup of a locally compact group G and let be the quotient space of left cosets. Let be the corresponding --algebra where . Suppose that Γ is a closed abelian subgroup of and let Ψ be a 2-cocycle on the dual group . Let be the Rieffel deformation of . Using the results of the previous paper of the author we may construct --algebra – the Rieffel deformation of . On the other hand we may perform the Rieffel deformation of the subgroup obtaining the closed quantum subgroup , which in turn, by the results of S. Vaes, leads to the --algebra . In this paper we show that . We also consider the case where is not a subgroup of , for which we cannot construct the subgroup . Then generically cannot be identified with a quantum quotient. What may be shown is that it is a -simple object in the category of --algebras. [Copyright &y& Elsevier]

This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in R N to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory. [ABSTRACT FROM AUTHOR]

Abstract: Let n be a large integer and be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that is singular. For a constant and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of satisfy this property, then the probability that is singular is at most . All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of are “fair coin flips” (taking the values each with probability 1/2), our general bound implies that the probability that is singular is at most , improving on the previous best upper bound of , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In the special case where the entries of are “lazy coin flips” (taking values each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that is singular is at most , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223–240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628], which was obtained using tools from additive combinatorics. [Copyright &y& Elsevier]

Abstract: This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on -rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as -rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for “fully elliptic differential a-operators,” which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition. [Copyright &y& Elsevier]

Abstract: This paper mainly concerns Beurling type quotient modules of over the bidisk. By establishing a theorem of function theory over the bidisk, it is shown that a Beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most . Furthermore, we apply this result to the study of boundary representations of Toeplitz algebras over quotient modules. It is proved that the identity representation of is a boundary representation of in all nontrivial cases. This extends a result of Arveson to Toeplitz algebras on Beurling type quotient modules over the bidisk (cf. [W. Arveson, Subalgebras of -algebras, Acta Math. 123 (1969) 141–224; W. Arveson, Subalgebras of -algebras II, Acta Math. 128 (1972) 271–308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk. [Copyright &y& Elsevier]

Abstract: In this paper, we consider solutions of the one-dimensional Kuramoto–Sivashinsky equation, i.e. which are L-periodic in x and have vanishing spatial average. Numerical simulations show that for , solutions display complex spatio-temporal dynamics. The statistics of the pattern, in particular its scaled power spectrum, is reported to be extensive, i.e. not to depend on L for . More specifically, after an initial layer, it is observed that the spatial quadratic average of all fractional derivatives of u is bounded independently of L. In particular, the time-space average is observed to be bounded independently of L. The best available result states that for all . In this paper, we prove that for . To our knowledge, this is the first result in favor of an extensive behavior—albeit only up to a logarithm and for a restricted range of fractional derivatives. As a corollary, we obtain , which improves the known bounds. [Copyright &y& Elsevier]

Abstract: In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball where is the algebra of all bounded linear operators on a Hilbert space . Several classical results from complex analysis have analogues in this noncommutative multivariable setting. We present basic properties for sub-pluriharmonic curves, characterize the class of sub-pluriharmonic curves that admit free pluriharmonic majorants and find, in this case, the least free pluriharmonic majorants. We show that, for any free holomorphic function Θ on , the map is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators on the full Fock space with n generators. We prove that Θ is in the noncommutative Hardy space if and only if φ has a free pluriharmonic majorant. In this case, we find Herglotz–Riesz and Poisson type representations for the least pluriharmonic majorant of φ. Moreover, we obtain a characterization of the unit ball of and provide a parametrization and concrete representations for all free pluriharmonic majorants of φ, when Θ is in the unit ball of . In the second part of this paper, we introduce a generalized noncommutative commutant lifting (GNCL) problem which extends, to our noncommutative multivariable setting, several lifting problems including the classical Sz.-Nagy–Foiaş commutant lifting problem and the extensions obtained by Treil–Volberg, Foiaş–Frazho–Kaashoek, and Biswas–Foiaş–Frazho, as well as their multivariable noncommutative versions. We solve the GNCL problem and, using the results regarding sub-pluriharmonic curves and free pluriharmonic majorants on noncommutative balls, we provide a complete description of all solutions. In particular, we obtain a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem. [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: In this paper, we initiate the study of a class of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space , where , , and p is a positive regular polynomial in n noncommutative indeterminates. These domains are defined by certain positivity conditions on p, i.e., Each such a domain has a universal model of weighted shifts acting on the full Fock space with n generators. The study of is close related to the study of the weighted shifts , their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra , the Hardy algebra , and the -algebra . A good part of this paper deals with these issues. The main tool, which we introduce here, is a noncommutative Berezin type transform associated with each n-tuple of operators in . The study of this transform and its boundary behavior leads to Fatou type results, functional calculi, and a model theory for n-tuples of operators in . These results extend to noncommutative varieties generated by classes of noncommutative polynomials. When , , , and , the elements of the corresponding variety can be seen as multivariable noncommutative analogues of Agler''s m-hypercontractions. Our results apply, in particular, when consists of the noncommutative polynomials , . In this case, the model space is a symmetric weighted Fock space , which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in , and the universal model is the n-tuple of multipliers by the coordinate functions. In this particular case, we obtain a model theory for commuting n-tuples of operators in , recovering several results already existent in the literature. [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]

Abstract: In this paper we extend some results of the paper [M. Gromov, G. Henkin, M. Shubin, Holomorphic -functions on coverings of pseudoconvex manifolds, Geom. Funct. Anal. 8 (1998) 552–585]. [Copyright &y& Elsevier]

Abstract: In this paper we consider automorphisms of the domains of closed *-derivations of C*-algebras and show that they extend to automorphisms of C*-algebras, so we call them diffeomorphisms. The diffeomorphisms generate transformations of the sets of closed *-derivations of C*-algebras. In this paper we study the subgroups of diffeomorphisms that define “bounded” shifts of derivations and the subgroups of the stabilizers of derivations. [Copyright &y& Elsevier]

Abstract: The paper studies unbounded reflexive *-derivations of C*-algebras of bounded operators on Hilbert spaces whose domains are weekly dense in and contain compact operators. It describes a one-to-one correspondence between these derivations and pairs , where are symmetric densely operators on and are J-orthogonal -reflexive lattices of subspaces in the deficiency spaces of . The domains of these *-derivations are associated with some non-selfadjoint reflexive algebras of bounded operators on . The paper analyzes the structure of the lattices of invariant subspaces of and of the normalizers of -the largest Lie subalgebras of such that are their Lie ideals. [Copyright &y& Elsevier]

In this paper we consider two von Neumann subalgebras B0 and B of a type II1 factor N. For a map φ on N, we defineand we measure the distance between B0 and B by the quantity ||EB0-EB||∞,2. Under the hypothesis that the relative commutant in N of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the ||·||2-norm. This hypothesis is satisfied, in particular, by masas and subfactors of trivial relative commutant. A general version with a slightly weaker conclusion is also proved. As a consequence, we show that if A is a masa and u∈N is a unitary such that A and uAu* are close, then u must be close to a unitary which normalizes A. These qualitative statements are given quantitative formulations in the paper. [Copyright &y& Elsevier]

In this paper we study some analytic properties of bi-free additive convolution, both scalar- and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of operator-valued distributions, simpler formulas for bi-free convolutions can be derived. We use these formulas in order to prove several results about atoms of bi-free additive convolutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove a C 2 , α -estimate for the solution to the complex Monge–Ampère equation det ⁡ ( u i j ¯ ) = f with 0 < f ∈ C α , under the assumption that u ∈ C 1 , β for some β < 1 which depends on n and α . [ABSTRACT FROM AUTHOR]

In this paper, we systematically study weighted jump and variational inequalities for rough operators. More precisely, we show some weighted jump and variational inequalities for the families T : = { T ε } ε > 0 of truncated singular integrals and M Ω : = { M Ω , t } t > 0 of averaging operators with rough kernels, which are defined respectively by T ε f ( x ) = ∫ | y | > ε Ω ( y ′ ) | y | n f ( x − y ) d y and M Ω , t f ( x ) = 1 t n ∫ | y | < t Ω ( y ′ ) f ( x − y ) d y , where the kernel Ω belongs to L q ( S n − 1 ) for q > 1 . [ABSTRACT FROM AUTHOR]

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections. [ABSTRACT FROM AUTHOR]

In this short note, we revisit the blow-up of solution for the initial boundary value problem of semilinear pseudo-parabolic equations with low/critical initial energy stated in Xu and Su (2013) [4] , and amend the proofs of the original paper. [ABSTRACT FROM AUTHOR]

In this paper, the notion of conditionally bi-free independence for pairs of faces is introduced. The notion of conditional ( ℓ , r ) -cumulants is introduced and it is demonstrated that conditionally bi-free independence is equivalent to the vanishing of mixed cumulants. Furthermore, limit theorems for the additive conditionally bi-free convolution are studied using both combinatorial and analytic techniques. In particular, a conditionally bi-free partial R -transform is constructed and a conditionally bi-free analogue of the Lévy–Hinčin formula for planar Borel probability measures is derived. [ABSTRACT FROM AUTHOR]

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation i ∂ ∂ t ψ ε ( t ) = − Δ ψ ε ( t ) + g ( ε , μ , | ( ρ ε , ψ ε ( t ) ) | 2 μ ) ( ρ ε , ψ ε ( t ) ) ρ ε where ρ ε → δ 0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Thompson groups T and V are not inner amenable. In particular, their group von Neumann algebras do not have property Γ. Moreover, we prove that if the reduced group C ⁎ -algebra of T is simple, then the Thompson group F is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of F . [ABSTRACT FROM AUTHOR]

The main result of this paper is that conditional measures of generalized Ginibre point processes, with respect to the configuration in the complement of a bounded open subset on C , are orthogonal polynomial ensembles with weights found explicitly. An especially simple formula for conditional measures is obtained in the particular case of radially-symmetric determinantal point processes, including the classical Ginibre point process. [ABSTRACT FROM AUTHOR]

In this paper, we derive a sharp version of the singular Trudinger–Moser inequality, which was originally established by Adimurthi and Sandeep [2] . Moreover, extremal functions for those singular Trudinger–Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work ( [32] ), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behavior of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classification theorems of Chen and Li [6,7] to get the desired bubble. [ABSTRACT FROM AUTHOR]

This paper studies weakly mixing (singular) and mixing masas in type II 1 factors from a bimodule point of view. Several necessary and sufficient conditions characterizing the normalizing algebra of a masa are presented. We study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures arising from semidirect products which are absolutely continuous, but not Lebesgue. We also show that there exist uncountably many pairwise non-conjugate mixing masas in the free group factors each with Pukánszky invariant { 1 , ∞ } . [ABSTRACT FROM AUTHOR]

The use of Lyapunov conditions for proving functional inequalities was initiated in [5] . It was shown in [4,30] that there is an equivalence between a Poincaré inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by • showing that strong functional inequalities are equivalent to Lyapunov type conditions; • showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and integrability property of the invariant probability measure and as such transportation inequalities, and we show that some “unbounded Lyapunov conditions” can lead to uniform ergodicity, and coming down from infinity property. [ABSTRACT FROM AUTHOR]

The Morse–Sard theorem requires that a mapping v : R n → R m is of class C k , k > max ⁡ ( n − m , 0 ) . In 1957 Dubovitskiĭ generalized this result by proving that almost all level sets for a C k mapping have H s -negligible intersection with its critical set, where s = max ⁡ ( n − m − k + 1 , 0 ) . Here the critical set, or m -critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) < m } . Another generalization was obtained independently by Dubovitskiĭ and Federer in 1966, namely for C k mappings v : R n → R d and integers m ≤ d they proved that the set of m -critical values v ( Z v , m ) is H q ∘ -negligible for q ∘ = m − 1 + n − m + 1 k . They also established the sharpness of these results within the C k category. Here we prove that Dubovitskiĭ's theorem can be generalized to the case of continuous mappings of the Sobolev–Lorentz class W p , 1 k ( R n , R d ) , p = n k (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin N -property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a bridge theorem that includes all the above results as particular cases. As a limiting case in this bridge theorem we also establish a new coarea type formula: if E ⊂ { x ∈ R n : rank ∇ v ( x ) ≤ m } , then ∫ E J m v ( x ) d x = ∫ R d H n − m ( E ∩ v − 1 ( y ) ) d H m ( y ) . The mapping v is R d -valued, with arbitrary d , and the formula is obtained without any restrictions on the image v ( R n ) (such as m -rectifiability or σ -finiteness with respect to the m -Hausdorff measure). These last results are new also for smooth mappings, but are presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015). [ABSTRACT FROM AUTHOR]

In this paper we study Johnson–Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space E ( M ) of a finite von Neumann algebra M without making any assumption on the Boyd indices of E . We show that we can obtain Johnson–Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when E = L 2 ( 0 , 1 ) . As an application, we partly resolve a problem stated by Randrianantoanina and Wu in [46] . We also show that we can obtain sharp Φ-moment analogues for Orlicz functions satisfying p -convexity and q -concavity for 1 ≤ p ≤ 2 , q = 2 and p = 2 , 2 < q < ∞ . This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder–Gundy inequalities in symmetric spaces and in the Φ-moment case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of weak and strong convergence of derivations, and of the flows associated to them, when dealing with a sequence of metric measure structures ( X , d , m n ) , m n weakly convergent to m . In particular, under curvature assumptions, either only on the limit metric structure ( X , d , m ) or on the whole sequence of metric measure spaces, we provide several stability results. [ABSTRACT FROM AUTHOR]

It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms λ u of the Cuntz algebra O n such that λ u ( F n ) ⊆ F n but u ∉ F n , and a question was raised if for such a u there must always exist a unitary v ∈ F n with λ u | F n = λ v | F n . In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms λ u for which the relative commutant λ u ( F n ) ′ ∩ F n is finite dimensional. [ABSTRACT FROM AUTHOR]

In this paper we prove different functional inequalities extending the classical Rogers–Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets. [ABSTRACT FROM AUTHOR]

The author of the above mentioned paper found that pointwise ergodic theorem for bounded additive processes, cf. Theorem 7.11 needs an additional assumption. Including the assumption of approximability, we show that the ergodic theorem is correct. Since we have not found a proof for Poisson point processes being approximable, we withdraw Corollary 7.12 which stated the pointwise almost-everywhere convergence for those processes. A minor flaw in the proof of the abstract mean ergodic theorem (Theorem 5.7) is fixed as well. [ABSTRACT FROM AUTHOR]

In this work we consider the weakly coupled Schrödinger cubic system { − Δ u i + λ i u i = μ i u i 3 + u i ∑ j ≠ i b i j u j 2 u i ∈ H 1 ( R N ; R ) , i = 1 , … , d , where 1 ≤ N ≤ 3 , λ i , μ i > 0 and b i j = b j i > 0 for i ≠ j . This system admits semitrivial solutions, that is solutions u = ( u 1 , … , u d ) with null components. We provide optimal qualitative conditions on the parameters λ i , μ i and b i j under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the d = 2 equations case. For d ≥ 3 equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case λ i ≡ λ and b i j ≡ b . We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d = 2 case. [ABSTRACT FROM AUTHOR]

This paper deals with the global existence and uniqueness results for the three-dimensional incompressible Euler equations with a particular structure for initial data lying in critical spaces. In this case the BKM criterion is not known. [ABSTRACT FROM AUTHOR]

This paper is concerned with the renormalized stochastic entropy solutions of stochastic scalar conservation law forced by a multiplicative noise on a bounded domain with a non-homogeneous boundary condition. We first introduce a notion of renormalized stochastic entropy solutions and then establish the existence and uniqueness of the renormalized stochastic entropy solutions for a general L 1 -data. Our results allow us to give a positive answer to an open problem posed by Bauzet, Vallet and Wittbold in [4] . [ABSTRACT FROM AUTHOR]

Based on the idea of a recent paper by Ambrosio–Gigli–Savaré (2014) [5] , we show that the L 2 -gradient flow of the q -Cheeger energy, called q -heat flow, solves a generalized gradient flow problem of the Renyi entropy functional in the p -Wasserstein. For that, a further study of the q -heat flow is presented including a condition for its mass preservation. Under a convexity assumption on the upper gradient, which holds for all q ≥ 2 , one gets uniqueness of the gradient flow and the two flows can be identified. Smooth solutions of the q -heat flow are solutions to the parabolic q -Laplace equation, i.e. ∂ t f t = Δ q f t . [ABSTRACT FROM AUTHOR]

In this note some flaws in the proofs of the paper P. Hästö (2015) [2] are corrected. [ABSTRACT FROM AUTHOR]

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain Ω ⊂ R n , with L ∞ Robin coefficient, L 2 Neumann data and conductivity of class W 1 , r ( Ω ) , r > n . We show that uniqueness of the Robin coefficient on a subpart of the boundary, given Cauchy data on the complementary part, does hold in dimension n = 2 but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem div ( ϕ N ( ∇ v ) ) + λ f ( | x | , v ) = 0 in B ( R ) , v = 0 on ∂ B ( R ) , where ϕ N ( y ) = y 1 − | y | 2 , y ∈ R N , λ is a positive parameter, B ( R ) = { x ∈ R N : | x | < R } , and | ⋅ | denotes the Euclidean norm in R N . All results, depending on the behavior of nonlinear term f near 0, are obtained by using global bifurcation techniques. [ABSTRACT FROM AUTHOR]