Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovski? may be written in the form L¹≡E⊙E^′$L^1\equiv E\odot E^{\prime }$, where ⊙ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E  , i.e., when F≡E⊙M(E,F)$F\equiv E\odot M(E,F)$, where M(E,F)$M(E,F)$ is the space of pointwise multipliers from E to F  . Properties of M(E,F)$M(E,F)$ were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E⊙F$E\odot F$. The formulas for pointwise product of Calderón–Lozanovski? _Eφ$E_{\phi }$-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through a Marcinkiewicz space.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

C^n单位球上Q_p空间的点态乘子

该文研究了C^n单位球上Q_p空间的点态乘子空间M(Q_p),分别给出了M(Q_p)与α Bloch 空间及Dirichlet型空间之间严格精确的包含关系.并相应地讨论了单位球上Q_{p,0}空间的点态乘子.     

Cn中单位球上两个全纯函数空间之间的点乘子

在Cn中单位球上讨论了加权Bergman空间Aαp和Bloch型空间βq之间的点乘子.根据α,p,q的不同情况,得到了Aαp空间到βq空间的所有点乘子,并研究了βq空间到Apα空间的点乘子的性质.     

Asymptotic behavior of spreading fronts in the anisotropic Allen–Cahn equation on Rn

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on ${\mathbb{R}}^n$ with $n\ge 2$ and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow $V_{\gamma }={\kappa }_{\gamma }+c$ with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.     

A representation of the characteristic function of a stable probability measure on certain TV spaces

In this paper, a Lévy-Khintchine type representation of the characteristic function of a K-regular stable probability measure on real locally convex topological vector spaces, satisfying certain conditions, is presented.