On the Spectral Multiplicity of Some Volterra Operators in Sobolev Spaces

Mathematical Notes -     

Modular inequalities in martingale Orlicz–Karamata spaces

In this paper, some new martingale inequalities in the framework of Orlicz‐Karamata spaces are provided. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions.     

Necessary and sufficient conditions for existence of blow‐up solutions for elliptic problems in Orlicz–Sobolev spaces

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem where either with is a smooth bounded domain or . The function ϕ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz–Sobolev spaces.     

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies or the gradient field of velocity satisfies then we show that the solution remains smooth on [0,T]. In view of the embedding with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.     

一类广义Lipschitz非线性算子的带误差的Ishikawa迭代程序

借助于周海云和陈东青［４］新近引入的广义Ｌｉｐｓｃｈｉｔｚ概念,研究了实Ｂａｎａｃｈ空间中广义Ｌｉｐｓｃｈｉｔｚ　 －强伪压缩算子的不动点和广义Ｌｉｐｓｃｈｉｔｚ －强增算子方程解的迭代逼近问题,所得结果改进和扩展了近期许多相关的结果,并部分地回答了周海云［３］提出的一个问题．     

Gateaux differentiability in Orlicz–Lorentz spaces and applications

Let Λ_w,? be the Orlicz–Lorentz space. We study Gateaux differentiability of the functional ψ_w,? (f) = ? (f *)w and of the Luxemburg norm. More precisely, we obtain the one‐sided Gateaux derivatives in both cases and we characterize those points where the Gateaux derivative of the norm exists. We give a characterization of best ψ_w,? ‐approximants from convex closed subsets and we establish a relation between best ψ_w,? ‐approximants and best approximants from a convex set. A characterization of best constant ψ_w,? ‐approximants and the algorithm to construct the best constant for maximum and minimum ψ_w,? ‐pproximants are given. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

Weighted Boundedness for Generalized Maximal andSingular Integral Operators in Orlicz-Morrey Spaces

51.IntroductionTheweightedboundednessformaximalandsingularintegraloperatorsonLp(p>1)andBMofunctionspacesarenowunderstood(see[lj)-AsMorreyspacemaybeconsideredasanextensionofthespaces,itisnaturalandimportanttostudytheweightedbou11dednessformax-imalandsingularintegraloperatorsonMorreyspaces.Thepurposeofthispaperistostu`lythequestion,theweightedboundednessfortheoperatorsinOrlicz-Morreyspacesareobtained,andthecharacterizationsfornon-weightedboundednessofmaximaloperatorarealsoobtained.Someworkint…     

Real‐variable characterizations of Musielak–Orlicz–Hardy spaces associated with Schrödinger operators on domains

Let n≥3, Ω be a strongly Lipschitz domain of and L_Ω:=?Δ+V a Schrödinger operator on L²(Ω) with the Dirichlet boundary condition, where Δ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class for some q₀>n/2. Assume that the growth function satisfies that ?(x,·) is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index , where and μ₀∈(0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Δ on Ω. In this article, the authors first show that the heat kernels of L_Ω satisfy the Gaussian upper bound estimates and the Hölder continuity. The authors then introduce the ‘geometrical’ Musielak–Orlicz–Hardy space via , the Hardy space associated with on , and establish its several equivalent characterizations, respectively, in terms of the non‐tangential or the vertical maximal functions or the Lusin area functions associated with L_Ω. All the results essentially improve the known results even on Hardy spaces with p∈(n/(n + δ),1] (in this case, ?(x,t):=t^p for all x∈Ω and t∈[0,∞)). Copyright © 2016 John Wiley & Sons, Ltd.     

Modified Ishikawa Iteration Process for Nonlinear Lipschitz Generalized Strongly Pseudo-Contractive Operators in Arbitrary Banach Spaces

In this paper, we modify the Ishikawa iteration process and show that such process, associated with a nonlinear Lipschitzian generalized strongly pseudo-contractive operator with a fixed point in a (not necessarily uniformly smooth) Banach space, converges strongly to the unique fixed point of this operator.     

On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Generalized Orlicz–Lorentz sequence spaces λ_φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ^λ ₂ for φ is defined in such a way that it guarantees many positive topological and geometric properties of λ_φ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λ_φ are considered. Moreover, some embeddings between λ_φ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λ_φ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一个增算子不动点定理及其在Banach 空间非线性方程的应用

给出了Banach空间的一个增算子不动点定理,将这一定理应用到Banach空间的积分-微分方程,给出了一类积分-微分方程的连续可微最大解和连续可微最小解的存在性定理.     

Absolutely continuous Hardy–Sobolev functions in the unit disc

Let f(z) be an analytic function defined in the unit disc whose fractional derivative of order belongs to H^p, 0<p1. We show that as a consequence of a monotonicity condition on the decay of the Taylor coefficients, it is possible to improve the usual radial boundary growth estimate for H^p functions by a logarithmic factor. As a consequence we show that under certain regularity conditions imposed on the decay and oscillations of the absolute values of the function's Taylor coefficients, it is possible to estimate the function's modulus of continuity and modulus of absolute continuity and that a consequence of this is that as p→0, these functions will be generally smoother. Examples are also given of Hardy–Sobolev functions having modulus of absolute continuity different than modulus of continuity.     

Banach空间中广义集值拟变分包含的灵敏性分析

研究了Banach空间中一类广义集值拟变分包含问题的灵敏性分析．利用预解算子的技巧,在对给定条件没有假设可微性和单调性下,建立了这类问题与广义预解方程类的等价性．     

On One Class of Matrix Differential Operators

We consider one class of matrix differential operators in the whole space. For this class of operators we establish the isomorphic properties in some special scales of weighted Sobolev spaces and study the regularity properties for solutions to the system of differential equations defined by these operators. The class of operators under consideration contains the stationary Navier–Stokes operator.     

I‐convexity and Q‐convexity in Orlicz–Bochner function spaces endowed with the Orlicz norm

Some characterizations of I‐convexity and Q‐convexity of Banach space are obtained. Moreover, the criteria is shown for Orlicz–Bochner function spaces $L_M(\mu ,X)$ endowed with the Orlicz norm being I‐convex as well as being Q‐convex.     

On the Spectrum of Frobenius–Perron Operators

We prove more results on the spectrum of the Frobenius–Perron operator P: L¹ → L¹ associated with a nonsingular transformation S: X → X on a σ-finite measure space (X, Σ, μ).     

Noncoercive Problems in Nonlinear Elasticity with Stored Energies Having Fast or Slow Growth: Orlicz—Sobolev Space Setting

This paper is concerned with a noncoercive unilateral problem in nonlinear elasticity with stored energy function having fast or slow growth. We prove an existence result for the associated minimization problem by using a recession approach in Orlicz—Sobolev space.     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

非齐性广义Morrey空间上双线性强奇异Calder\''{o}n-Zygmund算子及交换子

本文的主要建立非齐性度量测度空间上双线性强奇异积分算子$\widetilde{T}$及交换子$\widetilde{T}_{b_{1},b_{2}}$在广义Morrey空间$M^{u}_{p}(\mu)$上的有界性. 在假设Lebesgue可测函数$u, u_{1}, u_{2}\in\mathbb{W}_{\tau}$, $u_{1}u_{2}=u$,且$\tau\in(0,2)$. 证明了算子$\widetilde{T}$是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到空间$M^{u}_{p}(\mu)$有界的, 也是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到广义弱Morrey空间$WM^{u}_{p}(\mu)$有界的,其中$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$及$1相似文献