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Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz spaces

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Acta Mathematica Sinica,English Series 2019年第2期35卷 185-203页

作者： Sonia ACINAS Sergio FAVIER Felipe ZO Departamento de Matemática Facultad de Ciencias y Exactas NaturalesUniversidad Nacional de La Pampa Instituto de Matemática Aplicada San Luis IMASLUniversidad Nacional de San Luis and CONICET Departamento de Matemática Universidad Nacional de San Luis

In this paper we pursue the study of the best approximation operator extended from L~Φ to L~φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calder′on–Zygmund class t_m^p(x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–orlicz indices for the function φ.

关键词： Orlicz spaces extended best polynomial approximation pointwise and norm convergence weak and strong type inequalities orlicz indices

WEIGHTED INEQUALITIES FOR CERTAIN MAXIMAL FUNCTIONS IN Orlicz spaces

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Analysis in Theory and Applications 2001年第4期17卷 65-76页

作者： Zhu Xuexian (Peking University, China) Peking University China

Let Mg be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ2. It gives an A′Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the orlicz space LΦ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′Φ(g).

关键词： Orlicz spaces maximal operator maximal function orlicz space

The U* -property of Orlicz spaces

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Chinese Quarterly Journal of Mathematics 1995年第2期 8-15页

作者：王廷辅计东海

Ａｂｓｔｒａｃｔ：ＦｏｒＯｒｌｉｃｚｓｐａｃｅｓ，ｔｈｅｃｒｉｔｅｒｉａｏｆＵ＊－ｐｒｏｐｅｒｔｙａｒｅｇｉｖｅｎ．Ｋｅｙｗｏｒｄｓ：ＯｒｌｉｃｚＳｐａｃｅｓ；Ｕ－ｐｒｏｐｅｒｔｙ；Ｕ＊－ｐｒｏｐｅｒｔｙ（王廷辅）（... 详细信息

Ａｂｓｔｒａｃｔ：ＦｏｒＯｒｌｉｃｚｓｐａｃｅｓ，ｔｈｅｃｒｉｔｅｒｉａｏｆＵ＊－ｐｒｏｐｅｒｔｙａｒｅｇｉｖｅｎ．Ｋｅｙｗｏｒｄｓ：ＯｒｌｉｃｚＳｐａｃｅｓ；Ｕ－ｐｒｏｐｅｒｔｙ；Ｕ＊－ｐｒｏｐｅｒｔｙ（王廷辅）（计东海）Ｌｅｔ（Ｘ，１１＇１１）ｂ...

关键词： Orlicz spaces U-property U* -property

The　Convergences　of　Bivariate　Integral　Jackson　Interpolating　Operators　in　orlicz　spaces

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Chinese Quarterly Journal of Mathematics 1994年第2期 77-86页

作者：尚增科王有一盛保怀

ＴｈｅＣｏｎｖｅｒｇｅｎｃｅｓｏｆＢｉｖａｒｉａｔｅＩｎｔｅｇｒａｌＪａｃｋｓｏｎＩｎｔｅｒｐｏｌａｔｉｎｇＯｐｅｒａｔｏｒｓｉｎＯｒｌｉｃｚＳｐａｃｅｓ尚增科，王有一，盛保怀ＴｈｅＣｏｎｖｅｒｇｅｎｃｅｓｏｆＢｉｖ... 详细信息

ＴｈｅＣｏｎｖｅｒｇｅｎｃｅｓｏｆＢｉｖａｒｉａｔｅＩｎｔｅｇｒａｌＪａｃｋｓｏｎＩｎｔｅｒｐｏｌａｔｉｎｇＯｐｅｒａｔｏｒｓｉｎＯｒｌｉｃｚＳｐａｃｅｓ尚增科，王有一，盛保怀ＴｈｅＣｏｎｖｅｒｇｅｎｃｅｓｏｆＢｉｖａｒｉａｔｅＩｎｔｅｇｒａｌＪａｃ...

关键词： Orlicz spaces Hermite-Fejer interpolating operators Jackson interpolating operators

Asymptotic Behaviour of Solutions to the Navier-stokes Equations of a Two-dimensional Compressible Flow

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Acta Mathematicae Applicatae Sinica 2011年第4期27卷 697-712页

作者： Ying-hui ZHANG Zhong TAN Department of Mathematics Hunan Institute of Science and Technology Yueyang 414006 China School of Mathematical Sciences Xiamen University Fujian 361005 China

In this paper, we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p（g） = aglog^d（g） for large g. Here d 〉 2, a 〉 0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.

关键词： asymptotic behaviour Navier-stokes equations compressible barotropic flow Orlicz spaces

On the Boundedness of Rough Oscillatory Singular Integrals on Triebel-Lizorkin spaces

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Acta Mathematica Sinica,English Series 2011年第10期27卷 1881-1898页

作者： Leslie CHENG Yi Biao PAN Department of Mathematics Bryn Mawr College Bryn Mawr PA 19010 USA Department of Mathematics University of Pittsburgh Pittsburgh~ PA 15260 USA

We obtain appropriate sharp bounds on Triebel-Lizorkin spaces for rough oscillatory inte- grals with polynomial phase. By using these bounds and using an extrapolation argument we obtain some new and previously known ... 详细信息

关键词： Oscillatory singular integral rough kernel Orlicz spaces Block spaces extrapolation Triebel-Lizorkin spaces

Shape Optimization in Two-Dimensional Viscous Compressible Fluids

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Acta Mathematica Sinica,English Series 2010年第9期26卷 1793-1806页

作者： Zhong TAN Ying Hui ZHANG School of Mathematical Sciences Xiamen University Xiamen 361005 P. R. China Department of Mathematics Hu'nan Institute of Science and Technology Yueyang 414006 P. R. China

We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids in two space dimensions. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier-Stokes system for a general viscous barotropic fluid with the pressure satisfying p（o） = aQlog^d（o） for large Q. Here d 〉 1 and a 〉 0.

关键词： Optimization shape Orlicz spaces Navier-Stokes equations

The Regularity Estimates for the Elliptic Equations in orlicz Classes

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Chinese Quarterly Journal of Mathematics 2012年第1期27卷 29-35页

作者： TAO Xiang-xing FANG Yi School of Sciences Zhejiang University of Science TechnologyHangzhou 310023China

The purpose of this paper is to give the element proof of the regularity estimates in orlicz classes for the second order derivatives of the solutions to the general second order elliptic equations.The global regulari... 详细信息

关键词： regularity estimates Orlicz spaces ▽2-condition △2-condition