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In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

Starting from a selfadjoint Schrodinger operator A=−d2/dx2+q with a gap G in its spectrum F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267–324] succeed in constructing another Schrodinger operator A =−d 2 / d x 2 + q that is unitarily equivalent (and thus isospectral) to A. As the means they apply come from the Weyl–Titchmarsh theory the connections prove to be intricate, in particular the relation between A and A . We show that a central assertion in GST's paper rests substantially on factorizations of the form (A−μ)(A−ν)=B ∗ B, A B=BA, μ, ν being numbers in G and B an invertible 2nd order differential operator generated by corresponding eigensolutions of A. Hence A =UAU ∗ where U is the unitary operator B|B|−1. The operators B and U do not occur explicitly in F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267–324].

If ƒ(z) is an entire function of order larger than one that satisfies certain additional conditions, characterizations of the growth of its Fourier transform are given, in terms of the growth of ƒ(z) . The results obtained are extensions of theorems of Paley and Wiener, and are a refinement of prior work of Gel'fand and Silov.

The known family of woven collections of frames have interesting properties and applications. Yet, they cannot be easily constructed. The present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction. The frames { φ i } i ∈ I and { ψ i } i ∈ I for a Hilbert space H are called partition-woven (or briefly P-woven), if there exists a nontrivial subset σ of I such that the family { g i } i ∈ I represented as the triple ( { φ i } , { ψ i } , σ ) and called a P-weaven frame, is a frame defined by g i = φ i if i ∈ σ and g i = ψ i , otherwise. The paper also studies the class of CP-woven collections of frames which is a subset of the P-woven collections of frames and satisfy a stronger condition. The set of all woven frames is a strict subset of the set of all CP-woven frames which is further a strict subset of the set of all P-woven frames. Several examples and counterexamples are studied to show whether or not a pair of frames is P-woven or CP-woven.

This paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems, whose unperturbed system has a center at the origin and may have invariant multiply lines. By Melnikov function method, this paper provides the upper and lower bounds for the Poincare cyclicity. In particular, the exact cyclicities of two specific cases are given using analyzing method, one of which has been discussed in the paper [7] .

In this paper, we determine the wave front sets of solutions to Schrodinger equations of a harmonic oscillator with sub-quadratic perturbation by using the representation of the Schrodinger evolution operator of a harmonic oscillator introduced in Kato, Kobayashi and Ito (2012) via the wave packet transform. In the previous paper (2011, 2014), the authors have studied the wave front sets for Schrodinger equations with a free particle and a harmonic oscillator and Schrodinger equations with sub-quadratic potential.

In this paper, we consider the model of the Landau-Lifshitz-Maxwell system coupled with spin accumulation. This system consists of the Landau-Lifshitz equation, Maxwell equation describing the magnetic field and a quasi-linear parabolic equation describing the process of spin accumulation. This paper establishes the existence of weak solutions ( M , S , H , E ) to the coupled system in three dimensions and shows the magnetization vector field M is Holder continuous except a closed set Σ, which has locally finite 3-dimensional parabolic Hausdorff measure. Finally, we deduce the Holder continuity of ∇M away from Σ in two special cases.

The first part of the paper concerns with infinite symmetric block Jacobi matrices J with p × p -matrix entries. We present new conditions for general block Jacobi matrices to be self-adjoint and have discrete spectrum. In the second part of the paper a special classes of block Jacobi matrices J X , α are investigated. Our approach here substantially relies on a close connection between Jacobi matrices J X , α from this class and symmetric 2 p × 2 p Dirac operators D X , α with point interactions in L 2 ( ( a , b ) ; C 2 p ) established in our previous papers. In particular, their deficiency indices are related by n ± ( D X , α ) = n ± ( J X , α ) and under a simple additional assumption they are discrete only simultaneously. For block Jacobi matrices of this class we present several conditions ensuring equality n ± ( J X , α ) = k with any k ≤ p . It is worth mentioning that a connection between Dirac and Jacobi operators is employed here in both directions for the first time. In particular, to prove the equality n ± ( J X , α ) = p for J X , α it is first established for Dirac operator D X , α . We also find several conditions for matrix Schrodinger and Dirac operators with point interactions on finite or infinite intervals to have discrete spectrum.

In this paper, we present a scheme for solving two-dimensional miscible displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h 1 / 2 ) in this paper.

In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ = σ ( x ) and the inner pressure p = p ( x ) . Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20] . In this paper we study existence of nonradial solutions by using the bifurcation method. Let { γ k } k = 2 ∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k ⁎ ⩾ 2 such that in the two dimension case for any k ⩾ k ⁎ , γ k is a bifurcation point, and in the three dimension case for any even k ⩾ k ⁎ , γ k is also a bifurcation point.

In the paper [10] the L p -realization L p of the matrix Schrodinger operator L u = d i v ( Q ∇ u ) + V u was studied. The generation of a semigroup in L p ( R d , C m ) and characterization of the domain D ( L p ) has been established. In this paper we perturb the operator L p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on L p ( R d , C m ) with domain embedded in W 2 , p ( R d , C m ) . We also study the analyticity, compactness, positivity and ultracontractivity of the semigroup and prove Gaussian kernel estimates. Further kernel estimates and asymptotic behaviour of eigenvalues of the matrix Schrodinger operator are investigated.

In this paper, we consider a time-delayed nonlocal model describing the dynamics of hematopoietic stem cells (HSCs), which represent the immature cells in the hematopoiesis process. By the method of characteristics, the nonlocal model is obtained from an age-structured reaction–diffusion system in bounded domain with Dirichlet boundary conditions. Along this paper, we focus on the mathematical analysis of it. Firstly, we give some results on the existence, uniqueness, positivity and boundedness of solutions. Next, we obtain a threshold value R s and prove that the trivial steady state is globally asymptotically stable when R s 1 . When R s > 1 , we prove the existence and uniqueness of positive stationary solution under the respective additional conditions on the monotonicity and non-monotonicity of the integral term. Finally, we prove the uniform weak persistence of the system when R s > 1 . Some numerical simulations are provided to verify the validity of our theoretical results.

The paper is concerned with the study of the limit behaviour of the sequences of the positive linear functionals and operators associated with integrated generalized means defined with respect to a given probability Borel measure in the framework of Borel convex subsets of a Hilbert space. The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space. Several applications are shown in the special cases of bounded and unbounded real intervals which involve the most common integrated means. Furthermore, some consequences concerning the convergence in distribution, and hence stochastic, of generalized means of vector-valued random variables are also presented. Finally the paper ends with an application related to the so-called box integral problem which refers to the problem to evaluate the limit behaviour as n → ∞ of the average distance between two points of [ 0 , 1 ] n randomly chosen according to a given distribution on [ 0 , 1 ] n .

In this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable Banach spaces. We first prove an identity characterizing invariant Gaussian measure involving its covariance operator and the infinitesimal generator of the semigroup. This gives an answer to a question raised by Rudnicki in his inspiring review paper [32] . Under suitable conditions, we use the proved identity to give an invariant mixing Gaussian measure as distribution of a Wiener integral.

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L 2 -spaces. However, in subsequent papers, we intend to extend the study to L p -spaces ( 1 p ∞ ) .

In the present paper a generalized Kahlerian space is considered, as a generalized Riemannian space GR N with almost complex structure F i h that is covariantly constant with respect to the first and the second kind of covariant derivative. In the general case of a holomorphically projective mapping f of two non-symmetric generalized Kahlerian spaces GK N and G K ‾ N it is impossible to obtain a generalization of the holomorphically projective curvature tensor. In the present paper we study the case when GK N and G K ‾ N have the same torsion in corresponding points. Such a mapping we call “equitorsion mapping”. We obtain quantities H P W θ ( θ = 1 , ⋯ , 5 ) , that are generalizations of the holomorphically projective tensor, i.e. they are invariants based on f. Among H P W θ only H P W 5 is a tensor. Using another five linearly independent curvature tensors, we can prove that there exist three holomorphically projective tensors.

In this paper, we investigate the existence and asymptotic behavior of ground state sign-changing solutions to a class of Schrodinger–Poisson systems { − △ u + V ( x ) u + μ ϕ u = λ f ( x ) u + | u | 2 u , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , where V is a smooth function, f is nonnegative, μ > 0 , λ λ 1 and λ 1 is the first eigenvalue of the problem − △ u + V ( x ) u = λ f ( x ) u in H. With the help of the sign-changing Nehari manifold, we obtain that the Schrodinger–Poisson system possesses at least one ground state sign-changing solution u μ for all μ > 0 and each λ λ 1 . Moreover, we prove that its energy is strictly larger than twice that of ground state solutions. Besides, we give a convergence property of u μ as μ ↘ 0 . This paper can be regarded as the complementary work of Shuai and Wang [23] , Wang and Zhou [24] .

It is well known that the nonlinear Schrodinger (NLS) equation is a very important integrable equation. Ablowitz and Musslimani introduced and investigated an integrable nonlocal NLS equation through inverse scattering transform. Very recently, we proposed an integrable nonlocal modified Korteweg–de Vries equation (mKdV) which can also be found in the papers of Ablowitz and Musslimani. We have constructed the Darboux transformation and soliton solutions for the nonlocal mKdV equation. In this paper, we will investigate further the nonlocal mKdV equation. We will give its exact solutions including soliton and breather through inverse scattering transformation. These solutions have some new properties, which are different from the ones of the mKdV equation.

In this paper, we study the pseudo-relativistic Lane-Emden equation ( − Δ + m 2 − m ) u = | u | p − 1 u in Ω , with the Dirichlet boundary condition u = 0 on ∂Ω. Here, Ω ⊂ R n is bounded and the operator ( − Δ + m 2 − m ) is defined in terms of spectral decomposition. In the authors' previous paper [8] , it is shown the existence and nonexistence results for the least energy solutions, depending on the range of parameter p. This paper is devoted to study the asymptotic behavior of nontrivial solutions when the mass parameter m in the equation either goes to 0 + or diverges to infinity.

This paper discusses the differentiability of a class of functions associated with eigenvalues and eigenvectors of symmetric matrices. Recursive style formulas of partial derivatives for this class of functions are derived and higher order derivatives can be easily obtained from these formulas. Meanwhile, some interesting characteristics of multiple eigenvalues are revealed. Examples involving inverse eigenvalue problems and primary matrix functions are given to illustrate the applications of the results obtained in this paper.

In this paper we present comparison results for focal points of conjoined bases Y ( t ) , Y ˆ ( t ) of two linear Hamiltonian differential systems under the majorant condition H ( t ) − H ˆ ( t ) ≥ 0 for their Hamiltonians H ( t ) , H ˆ ( t ) . Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for H ˆ ( t ) . The main result of the paper connects the difference between the number of proper focal points of Y ( t ) , Y ˆ ( t ) with the number of proper focal points of the transformed conjoined basis Z ˆ − 1 ( t ) Y ( t ) , where Z ˆ ( t ) is a symplectic fundamental solution matrix of the Hamiltonian system associated with H ˆ ( t ) . Focal points of this transformed basis coincide with focal points of the matrix Wronskian of Y ( t ) , Y ˆ ( t ) . The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.

In this paper a theory is developed for obtaining families of solutions to the KdV equation by formulating a Riemann–Hilbert problem with an appropriate shift. The theory builds on the classical work of Segal and Wilson [17] in which families of solutions are indexed on closed subspaces W of a space of functions on the unit circle admitting a direct sum decomposition H = H + ⊕ H − ( H + , H − are subspaces of functions holomorphic respectively inside and outside the unit disk). The theory developed in this paper lends itself easily to obtaining explicit solutions. Examples where the subspace W can be associated to soliton type solutions are considered. More complex systems where singularities and Riemann surfaces play a role are also presented. In the last section the connection of our results to the τ-function is analyzed. The theory developed in this paper can easily be applied to other integrable systems and, eventually, to discrete integrable systems.

One of the key estimates in the original paper [3] does not seem to hold true, which may invalidate some results of that paper. However, there is a couple of ways to fix this mistake by adding additional hypotheses in Theorem 1 [3] .

We study the 1-d isentropic Euler equations with time-dependent damping { ∂ t ρ + ∂ x ( ρ u ) = 0 , ∂ t ( ρ u ) + ∂ x ( ρ u 2 ) + ∂ x p ( ρ ) = − μ ( 1 + t ) λ ρ u , ρ | t = 0 = 1 + e ρ 0 ( x ) , u | t = 0 = e u 0 ( x ) . In a previous paper [8] , we have proven that, when λ = 1 , μ > 2 , the 1-D Euler equations have global existence of small data solutions. However in this paper, we will show that, when the damping, with respect to time, decays faster or equal to 2 1 + t , the C 1 solution of the above system will blow up in finite time. More precisely, when λ = 1 , 0 ≤ μ ≤ 2 or λ > 1 , μ ≥ 0 , we will give a finite upper bound for the lifespan. Combining the results in this paper and [8] , we see that, when the damping decays with time like μ ( 1 + t ) λ , the critical exponents for λ , μ to separate the global existence and finite-time blow up of small data solutions are λ = 1 , μ = 2 .

In our previous paper [23] it was proved that a sense-preserving homeomorphism g on the unit circle S 1 belongs to the Weil-Petersson class WP ( S 1 ) , namely, g can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincare metric if and only if g is absolutely continuous such that log ⁡ g ′ belongs to the Sobolev class H 1 2 . In this sequel to [23] , we show that the smooth Hilbert manifold structure on WP ( S 1 ) inherited from H 1 2 by the pullback g ↦ log ⁡ | g ′ | is compatible with the standard Hilbert manifold structure introduced by Takhtajan-Teo [27] . This enables us to give a fast approach to some results in our previous papers [23] and [24] .

In this paper we perform a complete study about the existence, uniqueness and simplicity of the principal eigenvalue of a class of elliptic eigenvalue problems with glued Dirichlet-Robin boundary conditions on a component of the boundary of the domain and Dirichlet boundary conditions on the other component of the boundary. Moreover, this principal eigenvalue and its normalized principal eigenfunction are approached in R × H 1 ( Ω ) by principal eigen-pairs of boundary eigenvalue problems with classical mixed boundary conditions. In some sense, the glued mixed boundary conditions analyzed in this work may be regarded as classical mixed boundary conditions with large potentials in the Robin boundary condition. The principal eigenvalues analyzed in this paper, play a crucial role in analyzing the existence, uniqueness and asymptotic behavior of the positive solutions of certain kinds of semilinear elliptic boundary value problems with nonlinear boundary conditions and spatial heterogeneities. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results in [2] , [3] and [4] play a crucial role to obtain some of the results of this work.

In this paper, we present new upper and lower bounds for the Mills ratio of the standard Gaussian law. Several different methods are used to derive these new bounds. One of the methods reproduces the bounds of several different authors in previous works as special cases and is a very general method that produces many new bounds. One of the bounds can be written in terms of hyperbolic sine and inverse hyperbolic sine functions. Some of the bounds involve exponential functions and are improved versions of previously proposed bounds or are improved versions of the new bounds introduced earlier in this paper. Some results from reliability theory and Jensen's inequality are used to improve determinantal inequalities. Some open problems are discussed, and conjectures are made.

New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. After various preceding works of the last 25 years where the behavior of the Fourier transform has been considered on specific subspaces of the space of functions of bounded variation, in this paper such problems are considered on the whole space of functions of bounded variation. The widest subspaces of the space of functions of bounded variation are studied for which the cosine and sine Fourier transforms are integrable. The main result of the paper is an asymptotic formula for the sine Fourier transform of an arbitrary locally absolutely continuous function of bounded variation. Interrelations of various function spaces are studied, in particular, the sharpness of Hardy's inequality is established and the inequality itself is strengthened in certain cases. A way to extend the obtained results to the radial case is shown.

Within the Askey scheme of hypergeometric orthogonal polynomials, Racah polynomials stay on the top of the hierarchy and they generalize all of the discrete hypergeometric orthogonal polynomials. In this paper, we investigate asymptotic behaviors of Racah polynomials with varying parameters when the polynomial degree tends to infinity. Using the difference equation technique developed in our earlier papers, we obtain an asymptotic formula in the outer region via ratio asymptotics and then derive asymptotic formulas in the oscillatory region via a matching method. Our asymptotic formulas are explicitly given in terms of the polynomial degree, variable and parameters, using elementary functions such as logarithmic, exponential and rational functions. By taking limits, our results also yield asymptotic formulas for orthogonal polynomials in the lower hierarchy of the Askey scheme such as Hahn polynomials and Krawtchouk polynomials.

The geometric Fourier transform defined using Clifford algebra plays an increasingly active role in modern data analysis, in particular for color image processing. However, due to the non-commutativity, it is hard to obtain many important analytic properties. In this paper, we prove several important sharp inequalities, including the Hausdorff-Young inequality and its converse, Pitt's inequality and Lieb's inequality for Clifford ambiguity functions. With these inequalities, several uncertainty principles with optimal constants are derived for the geometric Fourier transform. Most results in this paper are new even in the easier case, i.e. the quaternion setting. The method developed here also works for more general geometric Fourier transforms.

This paper concerns with a class of elliptic anisotropic Dirichlet problems depending of one real parameter on bounded Euclidean domains. Our approach is based on variational and topological methods. More concretely, along the paper we show the existence of at least two weak solutions for the treated problem by using a direct consequence of the celebrated Pucci and Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri. This abstract approach has been developed for equations on Carnot groups; see [15] .

The aim of this paper is to continue the geometric-analytic study of φ-curvatures initiated in [2] . These curvatures arise naturally in many geometric contexts, notably in Ricci-harmonic solitons theory. In the present paper we prove two rigidity results related to harmonic-Einstein manifolds, a generalization of the notion of Einstein manifolds to the present situation. We observe that, when we restrict our theorems to the classical case of Einstein manifolds, we obtain some new results even in this setting.

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $\Lambda$, via the infinite path space $\Lambda^\infty$ of $\Lambda$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $\Lambda^\infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $\Lambda$. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are $\zeta$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $\mu$, and show that $\mu$ is a rescaled version of the measure $M$ on $\Lambda^\infty$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2(\Lambda^\infty, M)$ which was constructed by Farsi et al., Comment: This paper is a partial replacement of arXiv:1701.05321; the latter will not be submitted for publication. v2: Section 3 has been extensively revised to include a more detailed treatment of Dixmier traces. This version to appear in J. Math. Anal. Appl

In this paper, we are concerned with the existence of infinitely many sign-changing solutions of nonlinear Schrodinger systems in R N by applying the perturbation method and the method of invariant sets of descending flow. The main feature which distinguishes this paper from other related works (such as [13] ) lies in the fact that coefficients of nonlinearities are dependent on x rather than constants.

In this paper, we investigate the effects of vorticity for smooth subsonic non-isentropic Euler flows in infinitely long axisymmetric nozzles. It is a sequel to B. Duan and Z. Luo (2013) [13] on non-isentropic Euler system. In [13] , the authors obtained the unique existence of a smooth subsonic solution with restriction on the smallness of Bernoulli's function in the upstream. The smallness assumption is sufficient to prevent the appearance of singularity, however, it leads that the vorticity is also small. The purpose of this paper is to remove such restriction on the smallness of the vorticity.

We study isentropic gas flow in a divergent nozzle with friction. It is physically known that the flow is influenced by friction. In this paper, we investigate the mathematical effect of the friction on solutions. We first construct approximate solutions by using the Godunov scheme and the fractional step procedure. We obtain the bounded estimate by the invariant region. This is the difficult point of the present paper. We use the compensated compactness to prove their convergence. Finally, we prove the existence of global solutions with arbitrary L ∞ data.

In the recent paper by Mark C. Ho (2014) the notion of a λ-Toeplitz operator on the Hardy space H 2 ( T ) over the one-dimensional torus T was introduced and it was shown (under the supplementary condition) that for λ ∈ T the essential spectrum of such an operator is invariant with respect to the rotation z ↦ λ z ; if in addition λ is not of finite order the essential spectrum is circular. In this paper, we generalize these results to the case when T is replaced by an arbitrary compact Abelian group whose dual is totally ordered.

In this paper, we study the nonexistence and existence of radial large positive solutions for a class of Schrodinger systems with a nonlinear random operator under certain growth conditions. We also establish the necessary and sufficient conditions for the existence of radial large positive solutions by a successive iteration technique. The results of the paper improve and extend many previous work in this field of research.

This paper is concerned with a system of heat equations with hysteresis and Navier–Stokes equations in two-dimensional domains. There are some studies on the system of heat equations and Navier–Stokes equations in which the temperature (the solution to heat equations) is controlled by obstacle functions. In this paper, the control of the temperature is precisely described by introducing hysteresis. This paper provides the existence of solutions along the abstract theory for evolution equations with subdifferential operators.

In a previous paper the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha}(\gamma_d)$. In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha}(\gamma_d)$. Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups., Comment: arXiv admin note: text overlap with arXiv:1108.0238

The paper deals with the class of linear differential equations of any even order 2 α + 4 , α ∈ N 0 , which are associated with the so-called ultraspherical-type polynomials. These polynomials form an orthogonal system on the interval [ − 1 , 1 ] with respect to the ultraspherical weight function ( 1 − x 2 ) α and additional point masses of equal size at the two endpoints. The differential equations of “ultraspherical-type” were developed by R. Koekoek in 1994 by utilizing special function methods. In the present paper, a new and completely elementary representation of these higher-order differential equations is presented. This result is used to deduce the orthogonality relation of the ultraspherical-type polynomials directly from the differential equation property. Moreover, we introduce two types of factorizations of the corresponding differential operators of order 2 α + 4 into a product of α + 2 linear second-order operators.

In this paper, we further investigate the problem of commutativity up to a factor (or $\lambda$-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to the problem. We finish the paper by a conjecture on the commutativity of self-adjoint operators., Comment: 08 pages (to appear in Journal of Mathematical Analysis and Applications, Elsevier)

The aim of this paper is to prove some optimal estimates of logarithmic-type in Hardy–Sobolev spaces H 1 , p , 1 ≤ p ≤ ∞ , of both the unit disk and the annulus. These estimates extend those previously established by Baratchart and Zerner in the Hardy–Sobolev space H 1 , 2 of the unit disk and by Chaabane and the first author of the paper in the case of the spaces H 1 , ∞ of the unit disk and of the annulus.

Let K 0 denote the modified Bessel function of second kind and zeroth order. In this paper we will study the function ω ˜ n ( x ) : = ( − x ) n K 0 ( n ) ( x ) n ! for positive argument. The function ω ˜ n plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives ω ˜ n ( m ) with respect to n can be bounded by O ( ( n + 1 ) m / 2 ) while for small and large arguments x the growth even becomes independent of n. These estimates are based on an integral representation of K 0 which involves the function g n ( t ) = t n n ! exp ( − t ) and its derivatives. The estimates then rely on a subtle analysis of g n and its derivatives which we will also present in this paper.

This paper is devoted to study a class of stochastic differential equations with Levy noise. In comparison to the standard Gaussian noise, Levy noise is more versatile and interesting with a wider range of applications. However, Levy noise makes the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. We propose several sufficient conditions under which we investigate the long-time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability. Also, we discuss two types of continuity of the solution: continuous in probability and continuous in the pth moment. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.

In this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the Dirichlet boundary value problems of the form − Δ u + a ( x ) u = g ( x , u ) in a bounded smooth domain. Conditions ( S 1 ) – ( S 3 ) in the present paper are somewhat weaker than the famous Ambrosetti–Rabinowitz-type superquadratic condition. Here, we assume that the primitive of the nonlinearity g is either asymptotically quadratic or superquadratic as | u | → ∞ .

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L -functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L -functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L -functions and shifted L -functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.

In the paper Generalized roundness and negative type, Lennard, Tonge, and Weston show that the geometric notion of generalized roundness in a metric space is equivalent to that of negative type. Using this equivalent characterization, along with classical embedding theorems, the authors prove that for p > 2 , L p fails to have generalized roundness q for any q > 0 . It is noted, as a consequence, that the Schatten class C p , for p > 2 , has maximal generalized roundness 0. In this paper, we prove that this result remains true for p in the interval ( 0 , 2 ) .

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Fellerʼs seminal paper. In particular, this paper extends Fellerʼs results for continuous Q-functions to measurable Q-functions and provides additional results.