Classification of Non-Degenerate Involutive Set-Theoretic Solutions to the Yang-Baxter Equation with Multipermutation Level Two

Set-theoretic solutions to the Yang-Baxter equation of multipermutation level two are classified via transvection orbits which are abelian torsors. The relationship to square-free solutions and rack solutions is determined by using the square map and its relationship to non-degeneracy of the corresponding cycle set. The crucial role of left and right ideal powers of braces is discussed in connection with applications to higher multipermutation level. As an illustration, we give a simple proof of a recent theorem of Smoktunowicz on braces with nilpotent adjoint group, and a classification of braces with multipermutation level two, removing the finiteness in Smoktunowicz’ theorem for such braces.

     

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.     

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.     

Hardy Spaces Associated to Critical Herz Spaces with Variable Exponent

Mediterranean Journal of Mathematics - García-Cuerva (J Lond Math Soc (2) 39:499–513, 1989) has introduced Herz spaces associated to $${A^p}$$ and studied atomic decomposition and its...     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

Real Interpolation of Sobolev Spaces Associated to a Weight

We hereby study the interpolation property of Sobolev spaces of order 1 denoted by \(W^{1}_{p,V}\), arising from Schrödinger operators with positive potential. We show that for 1?≤?p ₁?p?p ₂?q ₀ with p?>?s ₀, \(W^{1}_{p,V}\) is a real interpolation space between \(W_{p_1,V}^{1}\) and \(W_{p_2,V}^{1}\) on some classes of manifolds and Lie groups. The constants s ₀, q ₀ depend on our hypotheses.     

Differential Operators Over Quantum Spaces

The topic of this paper is the development of a differential calculus on a quantum space covariant with respect to the action of a quantum group. Quantized differential operators, jets, the de Rham and Spencer complexes, etc., are constructed. Also the integration over a quantum space, adjoint operators, integral forms, Greens formula are discussed. The Wess–Zumino de Rham complex and the algebra of differential operators are treated as a basic example.     

On Duality in Spaces of Solutions to Elliptic Systems

Suppose that D is a bounded domain in (n2) with connected real-analytic boundary, A is an elliptic system with real-analytic coefficients in a neighborhood of the closure of D, and sol(A,D) is the space of solutions to the system Au=0 in D furnished with the standard Frechet–Schwartz topology. Then the dual of sol(A,D) represents the space sol(A, ) of solutions to the system Au=0 in a neighborhood of furnished with the standard inductive limit topology over some decreasing net of neighborhoods of . The corresponding pairing is generated by the inner product in the Lebesgue space L ²(D).     

Banach空间中微分包含解的存在性*

本文在无穷维Banach空间中讨论微分包含解的存在性,先给出了几个普通微分包含的比较定理,讨论了近似解与解的关系,然后得到了Banach空间中微分包含解的存在性定理.     

Quantum Cohomology of Minuscule Homogeneous Spaces

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov–Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In particular, we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties. DOI: .     

Frames Associated with Measurable Spaces

We present two results dealing with dilation and perturbation properties of general frames, which generalize the works of several authors in the theory of discrete frames. We also study the representing measures associated with frames. In particular, we give a necessary and sufficient condition for a representing measure to be an extreme point of the set of representing measures and show how it can be used in some concrete examples.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Reproducing Kernels for Dirichlet Spaces Associated to Finitely Supported Measures

We compute the reproducing kernels for the generalized Dirichlet spaces \(\mathcal {D}_{\mu }\), with \(\mu \) a finite linear combination of point mass measures on the unit circle.     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

伴随Herz空间的Hardy空间上的向量值Calderón-Zygmund算子(英文)

本文证明了一类具有向量值核的Calderon－Zygmund算子是Herz型Hard,空间HKp到向量值Herz空间KE,p有界的,应用这一结果,得到了粗糙核Calderon－Zygmund算子,极大型Calderon－Zygmund算子,极大算子等是HKp到Kp有界的．