Rearrangement transformations on general measure spaces

For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.     

Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces

We establish the boundedness and weak boundedness of the maximal operator and generalized fractional integral operators on generalized Morrey spaces over metric measure spaces without the assumption of the growth condition on μ. The results are generalization and improvement of some known results. We also give the vector‐valued boundedness. Moreover we prove the independence of the choice of the parameter in the definition of generalized Morrey spaces by using the geometrically doubling condition in the sense of Hytönen.     

Lipschitz estimates for commutator of fractional integral operators on non-homogeneous metric measure spaces

In this paper, the authors establish the(L~p(μ), L~q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 p 1/(α + β) and 1/q = 1/p-(α + β), and obtain their weak(L~1(μ), L~(1/(1-α-β))(μ))-type. Moreover, the authors also consider the boundedness in the case that 1/(α+β) p 1/α,1/α≤ p ≤∞ and the endpoint cases, namely, p = 1/(α + β).     

一类广义Marcinkiewicz积分算子在乘积空间上的加权有界性

在乘积空间上定义了一类径向权Ap(Rn×Rm),并进一步研究了一类广义Marcinkiewicz积分算子在乘积空间上的加权有界性.     

Stochastic porous media equation on general measure spaces with increasing Lipschitz nonlinearities

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E,?\left(E\right),\mu )$, and the Laplacian replaced by a negative definite self-adjoint operator $L$. In the case of Lipschitz nonlinearities $\Psi$, we in particular generalize previous results for open $E?{\mathbb{R}}^d$ and $L=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $\Psi$ at infinity for $L^2\left(\mu \right)$-initial data.     

Linear transformations on tensor spaces

Let U⊗V denote the tensor product of two finite dimensional vector spaces U and V over an infinite field. Let k be a positive integer such that k≤dim U and k≤ dim V Let D_k denote the set of all non-zero elements of U⊗V of rank less than k. In this paper we study linear transformations T on U⊗V such that (TD_k)⊆D_k.     

Borel transformations on Dirichlet spaces

We study the growth of an entire functionƒ, whose Borel transformγƒ is holomorphic outside a bounded convex regionD _ƒ with boundary curvature bounded away from 0 and ∞. The functionγƒ is assumed to belong to the Dirichlet space, i.e., it satisfies , wheredv(ξ) is the area element. It is shown that forγƒ to satisfy the above conditions, it is necessary and sufficient to have and the growth indicatrixh _ƒ ofƒ satisfies the relation 0<m≤h″(ϕ)+h(ϕ)≤M<∞. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 58–65, July, 1996.     

Marcinkiewicz integral on hardy spaces

In this paper we prove that the Marcinkiewicz integral is an operator of type (H ¹,L ¹) and of type (H ^1,,L ^1,). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of , but the smoothness condition assumed on is weaker than Stein's condition.The research was supported partly by Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.The author was supported partly by NSF of China (Grant No. 19971010).The author was supported partly by NSF of China (Grant No. 19131080).     

Fuzzy integral on credibility measure

Fuzzy measure and credibility measure are two dissimilar concepts, and fuzzy integral based on fuzzy measure has been researched. This paper focuses on introduction of fuzzy integral on credibility measure and discusses its properties, since it has been not studied yet so far.     

Adams inequalities on measure spaces

In 1988 Adams obtained sharp Moser–Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser–Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser–Trudinger trace inequalities, sharp Adams/Moser–Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.     

Besov spaces via wavelets on metric spaces endowed with doubling measure,singular integral,and the T1 type theorem

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the T1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.