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Denote by I d c G the lattice of all principal ℓ -ideals of an Abelian ℓ -group G. Our main result is the following. Theorem For every countable Abelian ℓ-group G, every countable completely normal distributive 0 -lattice L, and every closed 0 -lattice homomorphism φ : I d c G → L , there are a countable Abelian ℓ-group H, an ℓ-homomorphism f : G → H , and a lattice isomorphism ι : I d c H → L such that φ = ι ∘ I d c f. We record the following consequences of that result: (1) A 0-lattice homomorphism φ : K → L , between countable completely normal distributive 0-lattices, can be represented, with respect to the functor I d c , by an ℓ -homomorphism of Abelian ℓ -groups iff it is closed. (2) A distributive 0-lattice D of cardinality at most ℵ 1 is isomorphic to some I d c G iff D is completely normal and for all a , b ∈ D the set { x ∈ D | a ≤ b ∨ x } has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ 1. The bound ℵ 1 is sharp. Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height. All our results are stated in terms of vector lattices over any countable totally ordered division ring. [ABSTRACT FROM AUTHOR]

We present extensions of the Siegel integral formula ([13]), which counts the vectors of the random lattice, to the context of counting its sublattices and flags. Perhaps surprisingly, it turns out that many quantities of interest diverge to infinity. [ABSTRACT FROM AUTHOR]

We establish that the summability of the series ∑ ε n is the necessary and sufficient criterion ensuring that every (1 + ε n) -bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if n ε n → ∞ , then in ℓ 2 there exists a (1 + ε n) -bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis. [ABSTRACT FROM AUTHOR]

Following Södergren, we consider a collection of random variables on the space X n of unimodular lattices in dimension n : normalizations of the angles between the N = N (n) shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions (whose support depends on a parameter K = K (n)) evaluated at these random variables in the regime where K and N are allowed to tend to infinity with n at the rate K N = o (n 1 / 6). Our main result is that as n ⟶ ∞ , these random variables exhibit a joint Poissonian and Gaussian behavior. [ABSTRACT FROM AUTHOR]

Generalizing some clever computations for monomial algebras by J. Chuang and A. King in an unpublished article, we provide a sufficient criterion that allows in some cases to explicitly compute the A ∞ -algebra structure of the Yoneda algebra Ext A • (K , K) of a nonnegatively graded connected algebra A = T V / I in the case where the A ∞ -algebra structure of the Yoneda algebra is given as the dual of (signed) inclusions appearing in the lattice of submodules of TV representing the Tor groups Tor p A (K , K) for p ∈ N. This includes the case of monomial algebras A considered by Chuang and King, but also generalized Koszul algebras, and other nice algebras. The previous nice description of the A ∞ -(co)algebra structure is however far from universal. Indeed, we also give an example of a nonnegatively graded connected algebra A where the A ∞ -algebra structure of the Yoneda algebra can never be obtained by dualizing the (signed) inclusions of the Tor groups. [ABSTRACT FROM AUTHOR]

We show that a constant factor approximation of the shortest and closest lattice vector problem in any ℓ p -norm can be computed in time 2 (0.802 + ε) n. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem in the ℓ 2 norm. To obtain our result, we combine the latter algorithm for ℓ 2 with geometric insights related to coverings. [ABSTRACT FROM AUTHOR]

Abstract In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2. The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d. [ABSTRACT FROM AUTHOR]

Abstract Every infinite dimensional Banach space admits Riesz operators that are not finite rank. In this note we discuss conditions under which a Riesz operator, or some power thereof, is a finite rank operator. [ABSTRACT FROM AUTHOR]

The abstraction of the study of stochastic processes to Banach lattices and vector lattices has received much attention by Grobler, Kuo, Labuschagne, Stoica, Troitsky and Watson over the past fifteen years. By contrast mixing processes have received very little attention. In particular mixingales were generalized to the Riesz space setting in Kuo et al. (2013) [12] . The concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in Kuo et al. (2017) [11] . In the present work we formulate the concept of near-epoch dependence for Riesz space processes and show that if a process is near-epoch dependent and either strong or uniform mixing then the process is a mixingale, giving access to a law of large numbers. The above is applied to autoregressive processes of order 1 in Riesz spaces. [ABSTRACT FROM AUTHOR]

In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and L ∞ . Minimal topologies connect with the recent, and actively studied, subject of “unbounded convergences”. In fact, a Hausdorff locally solid topology τ on a vector lattice X is minimal iff it is Lebesgue and the τ and unbounded τ -topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, uτ , associated to τ on X . Regarding metrizability, we prove that if τ is a locally solid metrizable topology then uτ is metrizable iff there is a countable set A with I ( A ) ‾ τ = X . We prove that a minimal topology is metrizable iff X has the countable sup property and a countable order basis. In line with the idea that uo -convergence generalizes convergence almost everywhere, we prove relations between minimal topologies and uo -convergence that generalize classical relations between convergence almost everywhere and convergence in measure. [ABSTRACT FROM AUTHOR]

In this paper we find sufficient and necessary conditions under which vector lattice C ( X ) and its sublattices C b ( X ) , C 0 ( X ) and C c ( X ) have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space X . We also consider the countable sup property of C ( X × Y ) . Even when both C ( X ) and C ( Y ) have the countable sup property it is possible that C ( X × Y ) fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when C ( ∏ λ ∈ Λ X λ ) has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest. [ABSTRACT FROM AUTHOR]

For Riesz s -potentials K ( x , y ) = | x − y | − s , s > 0 , we investigate separation and covering properties of N -point configurations ω N ∗ = { x 1 , … , x N } on a d -dimensional compact set A ⊂ R ℓ for which the minimum of ∑ j = 1 N K ( x , x j ) is maximal. Such configurations are called N -point optimal Riesz s -polarization (or Chebyshev) configurations. For a large class of d -dimensional sets A we show that for s > d the configurations ω N ∗ have the optimal order of covering. Furthermore, for these sets we investigate the asymptotics as N → ∞ of the best covering constant. For these purposes we compare best-covering configurations with optimal Riesz s -polarization configurations and determine the s th root asymptotic behavior (as s → ∞ ) of the maximal s -polarization constants. In addition, we introduce the notion of “weak separation” for point configurations and prove this property for optimal Riesz s -polarization configurations on A for s > dim ( A ) , and for d − 1 ⩽ s < d on the sphere S d . [ABSTRACT FROM AUTHOR]

This paper studies a large class of continuous functions f : [ 0 , 1 ] → R d whose range is the attractor of an iterated function system { S 1 , … , S m } consisting of similitudes. This class includes such classical examples as Pólya's space-filling curves, the Riesz–Nagy singular functions and Okamoto's functions. The differentiability of f is completely classified in terms of the contraction ratios of the maps S 1 , … , S m . Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) f is nowhere differentiable; (ii) f is non-differentiable almost everywhere but with uncountably many exceptions; or (iii) f is differentiable almost everywhere but with uncountably many exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and (iii) above is calculated, and more generally, the complete multifractal spectrum of f is determined. [ABSTRACT FROM AUTHOR]

The universal completion of the Archimedean Riesz space C ( X ) of continuous, real valued functions on a completely regular space X is characterised as the space NL ( X ) of nearly finite, normal lower semi-continuous functions on X . As an application, we obtain, under additional assumptions on X , a characterisation of unbounded order convergence in C ( X ) as pointwise convergence everywhere except possibly on a set of first Baire category. This result is analogous to the situation in spaces of (real) p -summable functions, the sets of first Baire category now playing the role of null sets. We pursue this analogy further. First it is shown that, for a Baire space X , NL ( X ) is Riesz and algebra isomorphic to the space of real Borel measurable functions on X , with identification of functions differing at most on a set of first category. Secondly, through the use of density topologies and category measures, the extent to which our results can be cast in a measure-theoretic setting, and vice versa, is explored. Finally, through an application of the Maeda–Ogasawara Representation Theorem, we obtain a characterisation of those completely regular spaces X and Z such that C ( X ) and C ( Z ) have Riesz isomorphic universal completions. [ABSTRACT FROM AUTHOR]

Let L be a linear operator defined such that − L generates an associated heat semigroup e − t L . Suppose this semigroup does not satisfy a preservation condition. A proof that a generalised Riesz transform D g ( L ) satisfies L p bounds for some range 2 < p < q is provided based on two new estimates. One is a Hardy type inequality for L , the other a bound regarding how the gradient of the heat semigroup acts on a characteristic function. Applications are to cases where L is the Dirichlet Laplacian Δ Ω on inner uniform subsets of R n . [ABSTRACT FROM AUTHOR]

In 1998, Kirchberg and Wassermann constructed a separable nuclear operator system with the property that the canonical unital ⁎ -homomorphism from the maximal C ⁎ -algebra onto the C ⁎ -envelope is injective. We show that such an operator system is unique, and completely order isomorphic to the operator system associated with the noncommutative Poulsen simplex. [ABSTRACT FROM AUTHOR]

We consider the system consisting of K coupled acoustic channels with the different sound velocities c j . Channels are interacting at any point via the pressure and its time derivatives. Using the moment approach and the theory of exponential families with vector coefficients we establish two controllability results: the system is exactly controllable if (i) the control u j in the j th channel acts longer than the double travel time of a wave from the start to the end of the j -th channel; (ii) all controls u j act more than or equal to the maximal double travel time. [ABSTRACT FROM AUTHOR]

We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger–Newton equation) − Δ u + u = ( I α ⁎ | u | p ) | u | p − 2 u . Here I α stands for the Riesz potential of order α ∈ ( 0 , N ) , and N − 2 N + α < 1 p ≤ 1 2 . We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N . [ABSTRACT FROM AUTHOR]

A double phase problem with a bounded Borel measure on the right hand side is studied. We prove an optimal pointwise gradient estimate for such a measure data problem via Riesz potentials of the measure under log-Dini continuity assumption on the modulating coefficients. As a consequence, we find an optimized C 1 regularity criterion. [ABSTRACT FROM AUTHOR]

We bound an exponential sum that appears in the study of irregularities of distribution (the low-frequency Fourier energy of the sum of several Dirac measures) by geometric quantities: a special case is that for all { x 1 , … , x N } ⊂ T 2 , X ≥ 1 and a universal c > 0 ∑ i , j = 1 N X 2 1 + X 4 ‖ x i − x j ‖ 4 ≲ ∑ k ∈ Z 2 ‖ k ‖ ≤ X | ∑ n = 1 N e 2 π i 〈 k , x n 〉 | 2 ≲ ∑ i , j = 1 N X 2 e − c X 2 ‖ x i − x j ‖ 2 . Since this exponential sum is intimately tied to rather subtle distribution properties of the points, we obtain nonlocal structural statements for near-minimizers of the Riesz-type energy. For X ≳ N 1 / 2 both upper and lower bound match for maximally-separated point sets satisfying ‖ x i − x j ‖ ≳ N − 1 / 2 . [ABSTRACT FROM AUTHOR]

In this paper we characterize the boundedness of the bilinear form defined by ( f , g ) ∈ H ˙ s ( R ) × H ˙ s ( R ) → ∫ R ( − Δ ) s / 2 ( f g ) ( x ) ( − Δ ) s / 2 ( b ) ( x ) d x , in the product of homogeneous Sobolev spaces H ˙ s ( R ) × H ˙ s ( R ) , 0 < s < 1 / 2 . We deduce a characterization of the space of pointwise multipliers from H ˙ s ( R ) to its dual H ˙ − s ( R ) in terms of trace measures. [ABSTRACT FROM AUTHOR]

Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the α (strong) and φ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized L 1 and L ∞ spaces begun by Kuo, Labuschagne and Watson will be extended. [ABSTRACT FROM AUTHOR]

In this manuscript, we consider an initial-boundary-value problem governed by a ( 1 + 1 ) -dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics. The model considers the presence of a spatial Laplacian of fractional order which is defined in terms of Riesz fractional derivatives, as well as the inclusion of a generic continuously differentiable potential. It is known that the undamped regime has an associated positive energy functional, and we show here that it is preserved throughout time under suitable boundary conditions. To approximate the solutions of this model, we propose a finite-difference discretization based on fractional centered differences. Some discrete quantities are proposed in this work to estimate the energy functional, and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative. Moreover, we establish suitable computational constraints under which the discrete energy of the system is positive. The method is consistent of second order, and is both stable and convergent. The numerical simulations shown here illustrate the most important features of our numerical methodology. [ABSTRACT FROM AUTHOR]

In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space direction. The time and space fractional derivatives are considered in the senses of Caputo and Riesz, respectively. First, the centered difference approach is used to approximate the Riesz fractional derivative in space. Then, the obtained fractional ordinary differential equations are transformed into equivalent Volterra integral equations. And then, the trapezoidal rule is utilized to approximate the Volterra integral equations. The stability and convergence of our scheme are proved via mathematical induction method. Finally, numerical experiments are performed to confirm the high accuracy and efficiency of our scheme. [ABSTRACT FROM AUTHOR]

For p ≥ 2 , we study nonlinear elliptic equations of p -Laplacian type with measure data div a ( D u , u , x ) = μ in Ω when the nonlinearity depends on the variable of the possibly unbounded solution itself. We obtain the point-wise gradient bound and the gradient continuity in terms of a certain Riesz potential of the right-hand side measure. [ABSTRACT FROM AUTHOR]

The change of variable formula, or Itô's rule, is studied in a Dedekind complete vector lattice E with weak order unit E . Using the functional calculus we prove that for a Hölder continuous semimartingale X t = X a + M t + B t , t ∈ J , and a twice continuously differentiable function f , the formula (0.1) f ( X t ) = f ( X a ) + ∫ 0 t f ′ ( X s ) d M s + ∫ 0 t f ′ ( X s ) d B s + 1 2 ∫ 0 t f ″ ( X s ) d 〈 M 〉 s , 0 ≤ s ≤ t ∈ J holds. The first integral in the formula is an Itô integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Lévy's characterization of Brownian motion as being a continuous martingale with compensator tE . The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices. [ABSTRACT FROM AUTHOR]

We show that an Urysohn lattice pre-homomorphism defined on a normal sublattice D of a vector lattice E can be extended to the whole space E and the extended operator is an Urysohn lattice homomorphism. We introduce a new class of nonlinear operators which called φ -operators and describe some of their properties. Finally we investigate a structure of positive orthogonally additive operators dominated by an Urysohn lattice homomorphism defined on a vector lattice E and taking value in Dedekind complete vector lattice F . [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration of scalar functions with respect to a vector measure defined on a δ -ring and taking values in a Dedekind σ -complete vector lattice and use it for obtaining general representation theorems for Dedekind complete vector lattices. [ABSTRACT FROM AUTHOR]

According to Riesz decomposition theorem, superharmonic functions on the punctured unit ball are represented as the sum of generalized potentials and harmonic functions. In this paper we study growth properties near the origin of spherical means for generalized Riesz potentials of functions belonging to central variable Morrey spaces. We also deal with monotone Sobolev functions. [ABSTRACT FROM AUTHOR]

For a class of continuously differentiable function ϕ satisfying certain decay conditions, it is shown that if the maximum gap δ : = sup i ⁡ ( x i + 1 − x i ) between the consecutive sample points is smaller than a certain number B 0 , then any f ∈ V ( ϕ ) can be reconstructed uniquely and stably. As a consequence of this result, it is shown that if δ < 1 , then { x i : i ∈ Z } is a stable set of sampling for V ( ϕ ) with respect to the weight { w i : i ∈ Z } , where w i = ( x i + 1 − x i − 1 ) / 2 and ϕ is the scaling function associated with Meyer wavelet. Further, the maximum gap condition δ < 1 is sharp. [ABSTRACT FROM AUTHOR]

This paper provides a deeper study of the Hardy and BMO spaces associated to the Neumann Laplacian Δ N . For the Hardy space H Δ N 1 ( R n ) (which is a proper subspace of the classical Hardy space H 1 ( R n ) ) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space BMO Δ N ( R n ) (which contains the classical BMO ( R n ) ) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on L ∞ ( R n ) functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for H 1 ( R n ) and BMO ( R n ) . [ABSTRACT FROM AUTHOR]

In the setting of a Dedekind complete Riesz space on which a conditional expectation is defined, we study the Itô integral. This integral obtains the integration of a stochastic process of a continuous parameter relative to a martingale. A considerable part of the paper is devoted to aspects of integration of a vector-valued function with respect to a vector measure. Here we use the Dobrakov integral that is, in our case, a generalization of the well known Bartle integral. We discuss natural and predictable processes to prove a general version of the Doob–Meyer decomposition of a submartingale. This decomposition provides an essential tool used in the definition of the stochastic integral. We derive the properties of the stochastic integral that are useful in developing the theory of stochastic processes in Riesz spaces. [ABSTRACT FROM AUTHOR]

To strengthen the weak restriction principle discovered in both [2, Theorem 5.1] for λ = κ and [32, Lemma 2.1] for λ < κ , this paper shows that the Riesz operator I α of order α maps the Morrey–Hardy space L H p , κ to the Morrey–Radon space L μ q , λ continuously if and only if sup ( x , r ) ∈ R n × ( 0 , ∞ ) ⁡ μ ( B ( x , r ) ) / r β < ∞ provided { 0 < λ ≤ κ ≤ n ; 0 < α < n ; 1 ≤ p < κ / α ; n − α p < β ≤ n ; 0 < q = p ( β + λ − n ) / ( κ − α p ) . Moreover, this brand-new restriction principle is optimal in the sense that if n = λ > κ > 0 then the iff-statement fails. Quite remarkably, such an optimization confirms affirmatively the conjecture in [7, Remark 3.3] : if α < α p < κ < λ = n & q = p β / ( κ − α p ) then there is no continuity of I α : L p , κ → L μ q , n for an arbitrary Radon measure μ with ⦀ μ ⦀ β < ∞ . [ABSTRACT FROM AUTHOR]

We define and study order continuity, topological continuity, γ -Hölder-continuity and Kolmogorov–Čentsov-continuity of continuous-time stochastic processes in vector lattices and show that every such kind of continuous submartingale has a continuous compensator of the same kind. The notion of variation is introduced for continuous time stochastic processes and for a γ -Hölder-continuous martingale with finite variation, we prove that it is a constant martingale. The localization technique for not necessarily bounded martingales is introduced and used to prove our main result which states that the quadratic variation of a continuous-time γ -Hölder continuous martingale X is equal to its compensator 〈 X 〉 . [ABSTRACT FROM AUTHOR]

We show how the Zak kq -representation can be adapted to deal with pseudo-bosons, and under which conditions. Then we use this representation to prove completeness of a discrete set of bi-coherent states constructed by means of pseudo-bosonic operators. The case of Riesz bi-coherent states is analyzed in detail. [ABSTRACT FROM AUTHOR]

We prove boundary inequalities for conjugate differential forms in C 1 -domains. They extend the classical Riesz inequalities for conjugate harmonic functions. [ABSTRACT FROM AUTHOR]

We study two geometric properties of reproducing kernels in model spaces K θ where θ is an inner function: overcompleteness and existence of uniformly minimal systems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern–Clark point. It is shown that “uniformly minimal non-Riesz” sequences of reproducing kernels exist near each Ahern–Clark point which is not an analyticity point for θ , while overcompleteness may occur only near the Ahern–Clark points of infinite order and is equivalent to a “zero localization property”. In this context the notion of quasi-analyticity appears naturally, and as a by-product of our results we give conditions in the spirit of Ahern–Clark for the restriction of a model space to a radius to be a class of quasi-analyticity. [ABSTRACT FROM AUTHOR]

Recently, Labuschagne and Watson introduced the space L 2 ( T ) for a conditional expectation T with natural domain a Riesz space L 1 ( T ) . The main purpose of this paper is to carry on with this process by introducing and investigate the spaces L p ( T ) with p ∈ [ 1 , ∞ ] . [ABSTRACT FROM AUTHOR]

The one-dimensional Dirac operator with periodic potential V = ( 0 P ( x ) Q ( x ) 0 ) , where P , Q ∈ L 2 ( [ 0 , π ] ) subject to periodic, antiperiodic or a general strictly regular boundary condition ( bc ), has discrete spectrums. It is known that, for large enough | n | in the disk centered at n of radius 1/2, the operator has exactly two (periodic if n is even or antiperiodic if n is odd) eigenvalues λ n + and λ n − (counted according to multiplicity) and one eigenvalue μ n b c corresponding to the boundary condition ( b c ) . We prove that the smoothness of the potential could be characterized by the decay rate of the sequence | δ n b c | + | γ n | , where δ n b c = μ n b c − λ n + and γ n = λ n + − λ n − . Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if sup γ n ≠ 0 ⁡ | δ n b c | | γ n | is finite. [ABSTRACT FROM AUTHOR]

One of the most influential fundamental tools in harmonic analysis is the Riesz transforms. It maps L p functions to L p functions for any p ∈ ( 1 , ∞ ) which plays an important role in singular operators. As an application in fluid dynamics, the norm equivalence between ‖ ∇ u ‖ L p and ‖ div u ‖ L p + ‖ curl u ‖ L p is well established for p ∈ ( 1 , ∞ ) . However, since Riesz operators sent bounded functions only to BMO functions, there is no hope to bound ‖ ∇ u ‖ L ∞ in terms of ‖ div u ‖ L ∞ + ‖ curl u ‖ L ∞ . As pointed out by Hoff (2006) [11] , this is the main obstacle to obtain uniqueness of weak solutions for isentropic compressible flows. Fortunately, based on new observations, see Lemma 2.2 , we derive an exact estimate for ‖ ∇ u ‖ L ∞ ≤ ( 2 + 1 / N ) ‖ div u ‖ L ∞ for any N-dimensional radially symmetric vector functions u . As a direct application, we give an affirmative answer to the open problem of uniqueness of some weak solutions to the compressible spherically symmetric flows in a bounded ball. [ABSTRACT FROM AUTHOR]

We consider real pre-Hilbert modules H on Archimedean f -algebras A with unit e . We provide conditions on A and H such that a Riesz representation theorem for bounded/continuous A -linear operators holds. [ABSTRACT FROM AUTHOR]

In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme. [ABSTRACT FROM AUTHOR]

In this article, we explore the mapping and boundedness properties of linear and bilinear fractional integral operators acting on Lebesgue spaces with large indices. The prototype ν -order fractional integral operator is the Riesz potential I ν , and the standard estimates for I ν are from L p into L q when 1 < p < n ν and 1 p = 1 q + ν n . We show that a ν -order linear fractional integral operator can be continuously extended to a bounded operator from L p into the Sobolev- BMO space I s ( B M O ) when n ν ≤ p < ∞ and 0 ≤ s < ν satisfy 1 p = ν − s n . Likewise, we prove estimates for ν -order bilinear fractional integral operators from L p 1 × L p 2 into I s ( B M O ) for various ranges of the indices p 1 , p 2 , and s satisfying 1 p 1 + 1 p 2 = ν − s n . [ABSTRACT FROM AUTHOR]

We study expected Riesz s -energies and linear statistics of some determinantal processes on the sphere S d . In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on S d . With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2 -energies of points defined through determinantal point processes associated with isotropic kernels. As a corollary we get that the Riesz 2 -energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble (in S 2 ). [ABSTRACT FROM AUTHOR]

We investigate spectral multipliers, Bochner–Riesz means and the convergence of eigenfunction expansion corresponding to the Schrödinger operator with anharmonic potential L = − d 2 d x 2 + | x | . We show that the Bochner–Riesz profile of the operator L completely coincides with such profile of the harmonic oscillator H = − d 2 d x 2 + x 2 . It is especially surprising because the Bochner–Riesz profile of the one-dimensional standard Laplace operator is known to be essentially different and the case of operators H and L resembles more the profile of multidimensional Laplace operators. Another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for Bochner–Riesz means convergence. [ABSTRACT FROM AUTHOR]

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base Ω ⊂ R d and a time-dependent separation Λ. Under certain mild regularity assumptions on Λ, we show that for any q > 1 sufficiently close to 1, the mixed problem in L q is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space L q 1 and the Neumann data in L q , there is a unique solution and the non-tangential maximal function of its gradient is in L q on the lateral boundary of the domain. When q = 1 , a similar result is shown when the data is in the Hardy space. Under the additional condition that the boundary of the domain Ω is Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of m variables, where m = 0 , ... , d − 2 , with respect to the Hausdorff distance, we also prove the unique solvability result for any q ∈ (1 , (m + 2) / (m + 1)). In particular, when m = 0 , i.e., Λ is Reifenberg-flat of co-dimension 2, we derive the L q solvability in the optimal range q ∈ (1 , 2). For the Laplace equation, such results were established in [25,24,5] and [13]. [ABSTRACT FROM AUTHOR]

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen. [ABSTRACT FROM AUTHOR]