Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions.

This paper is concerned with the existence and asymptotical behavior of positive ground state solutions for a class of critical quasilinear Schrodinger equation. By using a change of variables and variational argument, we prove the existence of positive ground state solution and discuss their asymptotical behavior.

We prove that Sp(2k+l) admits at least two non-naturally reductive Einstein metrics which are Ad(Sp(k)×Sp(k)×Sp(l))-invariant if k<l.It implies that every compact simple Lie group Sp(n) for n≥4 admits at least 2[(n−1)/3] non-naturally reductive left-invariant Einstein metrics.

Let K₃ be a non-normal cubic extension over \mathbb{Q}.We study the higher moment of the coefficients aK₃ (n) of Dedekind zeta function over sum of two squares {\displaystyle {\sum }_{n_{\mathrm{1}}^{\mathrm{2}}+n_{\mathrm{2}}^{\mathrm{2}}\le x}{a}_{K_{\mathrm{3}}}^l(n_{\mathrm{1}}^{\mathrm{2}}+n_{\mathrm{2}}^{\mathrm{2}})},where 2≤l≤8 and n1,n2,l∈\mathbb{Z}.

In the study of the collapsed manifolds with bounded sectional curvature, the following two results provide basic tools: a (singular) fibration theorem by K. Fukaya [J. Differential Geom., 1987, 25(1): 139–156] and J. Cheeger, K. Fukaya, and M. Gromov [J. Amer. Math. Soc., 1992, 5(2): 327–372], and the stability for isometric compact Lie group actions on manifolds by R. S. Palais [Bull. Amer. Math. Soc., 1961, 67(4): 362–364] and K. Grove and H. Karcher [Math. Z., 1973, 132: 11–20]. The main results in this paper (partially) generalize the two results to manifolds with local bounded Ricci covering geometry.

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvaturedimension condition RCD(0;N) with N 2 R and N>1: In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K;N); where K;N 2 R and N>1: Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carathéodory spaces which may have independent interests.

We prove Auslander's defect formula in an exact category, and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander{Reiten duality.

We mainly establish the distortion theorems of Jacobi determinant for three subclasses of starlike mappings on B_P^n; where : In particular, the above distortion theorems are sharp if B_P^n is the unit polydisk in {ℂ}^n: Our results reduce to the corresponding classical results in one dimension of complex function theory.

We propose a mixture network regression model which considers both response variables and the node-specific random vector depend on the time. In order to estimate and compare the impacts of various connections on a response variable simultaneously, we extend it into p different types of connections. An ordinary least square estimators of the effects of different types of connections on a response variable is derived with its asymptotic property. Simulation studies demonstrate the effectiveness of our proposed method in the estimation of the mixture autoregressive model. In the end, a real data illustration on the students' GPA is discussed.

We define a vector representation V (u) of elliptic Ding-Iohara algebra U (q; t; p): Furthermore, we construct the tensor products of the vector representations and the Fock modules ℱ(u) by taking the inductive limit of certain subspaces in the finite tensor products of vector representations.

For the maximal space-like hypersurface defined on 2-dimensional space forms, based on the regularity and the strict convexity of the level sets, the steepest descents are well defined. In this paper, we come to estimate the curvature of its steepest descents by deriving a dierential equality.

We study the mean square of the error term of the mean value for binary Egyptian fractions. We get an asymptotic formula under the Riemann Hypothesis.

Let f be a holomorphic Hecke cusp form with even integral weight k\ge \mathrm{2} for the full modular group, and let \chi be a primitive Dirichlet character modulo q. Let {{\displaystyle L}}_f(s,\chi ) be the automorphic L-function attached to f and \chi. We study the mean-square estimate of {{\displaystyle L}}_f(s,\chi ) and establish an asymptotic formula.

We will prove that for and , the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator {{\displaystyle M}}_{\gamma }^c equals that of the centered Hardy-Littlewood maximal operator for all . When p = 1 and , it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator {{\displaystyle M}}_{\gamma }^c equals that of the centered Hardy-Littlewood maximal operator for all . Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.

Extriangulated category was introduced by H. Nakaoka and Y. Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (resp., cotilting) subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (resp., cotilting) subcategories and obtain an Auslander-Reiten correspondence between tilting (resp., cotilting) subcategories and coresolving covariantly (resp., resolving contravariantly) finite subcatgories which are closed under direct summands and satisfy some cogenerating (resp., generating) conditions. Applications of the results are given: we show that tilting (resp., cotilting) subcategories defined here unify many previous works about tilting modules(subcategories) in module categories of Artin algebras and in abelian categories admitting a cotorsion triples; we also show that the results work for the triangulated categories with a proper class of triangles introduced by A. Beligiannis.