We study the image of the model subspace ${{K}_{\theta }}$ under the composition operator ${{C}_{\varphi }}$, where $\varphi $ and $\theta $ are inner functions, and find the smallest model subspace which contains the linear manifold ${{C}_{\varphi }}{{K}_{\theta }}$. Then we characterize the case when ${{C}_{\varphi }}$ maps ${{K}_{\theta }}$ into itself. This case leads to the study of the inner functions $\varphi $ and $\psi $ such that the composition $\psi \,\text{o}\,\varphi $ is a divisor of $\psi $ in the family of inner functions.

In this paper we study sharp localized ${{L}^{q}}\,\to \,{{L}^{p}}$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

Let ${{F}_{2n,2}}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf{P}$ denote the affine automorphism group of ${{F}_{2n,2}}$. In this article the theory of continuous wavelet transform on ${{F}_{2n,2}}$ associated with $\mathbf{P}$ is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on ${{F}_{2n,2}}$ is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if $n\,=\,1$, ${{F}_{2,2}}$ is the 3-dimensional Heisenberg group ${{H}^{1}}$, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on ${{F}_{2,2}}$. This result cannot be extended to the case $n\,\ge \,2$.

We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of ${{\mathbb{S}}^{n}}\,\times \,\mathbb{R}$, extending the classification of umbilical surfaces in ${{\mathbb{S}}^{2}}\times \mathbb{R}$ by Souam and Toubiana as well as the local description of umbilical hypersurfaces in ${{\mathbb{S}}^{n}}\,\times \,\mathbb{R}$ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic ${{\mathbb{S}}^{1}}\times \mathbb{R}$ or ${{\mathbb{S}}^{2}}\times \mathbb{R}$, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor $\mathbb{R}$ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$ having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$.

“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius $\left( \text{QcF} \right)$ coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is $\text{QcF}$ if it is weakly isomorphic to its (left, or right) rational dual $\text{Rat}\left( {{C}^{*}} \right)$ in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, $\text{QcF}$, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra $A$ that is isomorphic to its complete topological dual ${{A}^{\vee }}$. We show that $A$ is a (quasi)Frobenius algebra if and only if $A$ is the dual ${{C}^{*}}$ of a (quasi)co-Frobenius coalgebra $C$. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer $q$-homological algebra, topology or graph theory, showing the importance of the concept.

In this paper, we consider a generalized Kähler–Einstein equation on a Kähler manifold $M$. Using the twisted $\mathcal{K}$–energy introduced by Song and Tian, we show that the existence of generalized Kähler–Einstein metrics with semi–positive twisting (1, 1)–form $\theta$ is also closely related to the properness of the twisted $\mathcal{K}$-energy functional. Under the condition that the twisting form $\theta$ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler–Einstein metric implies a Moser–Trudinger type inequality.

For parabolic linear operators $L$ of second order in divergence form, we prove that the solvability of initial ${{L}^{p}}$ Dirichlet problems for the whole range $1\,<\,p\,<\,\infty $ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of $L$ satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of $p\,>\,1$, the initial ${{L}^{p}}$ Dirichlet problem associated with $Lu\,=\,0$ over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

We prove that the two-variable $\text{BMW}$ algebra embeds into an algebra constructed from the $\text{HOMFLY-PT}$ polynomial. We also prove that the $\mathfrak{s}{{\mathfrak{O}}_{2N}}-\text{BMW}$ algebra embeds in the $q$-Schur algebra of type $A$. We use these results to suggest a schema providing categorifications of the $\mathfrak{s}{{\mathfrak{D}}_{2N}}-\text{BMW}$ algebra.

In this paper, we show that the failure of the unique branch hypothesis $\left( \text{UBH} \right)$ for tame trees implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing Ord $\cup \mathbb{R}$ such that $M\,\vDash \,\text{A}{{\text{D}}^{+}}\,+\,\Theta \,>\,{{\theta }_{0}}$. In particular, this implies the existence (in $V$) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees.

Consider a Shimura curve $X_{0}^{D}\left( N \right)$ over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on ${{\mathbf{Q}}_{p}}$ points when $p|D$ and for the first time give criteria for ${{\mathbf{Q}}_{p}}$ points when $p|N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.