In this paper we give the (L^p(ω~p),L^q(ω~q)) boundedness for a class of multilinear operators, which is similar to the higher-order commutator for the rough fractional integral.In our results the kernel function is...In this paper we give the (L^p(ω~p),L^q(ω~q)) boundedness for a class of multilinear operators, which is similar to the higher-order commutator for the rough fractional integral.In our results the kernel function is merely assumed on a size condition.展开更多
For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \...For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \right)dy,$$where Rm (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m m 1 in $\dot \Lambda_\beta $(0 < # < 1), OHgr;(x) ] L^s(S^nm1)($s \ge {n \over {n - \beta }}$) is homogeneous of degree zero, the authors prove that TA is bounded from L^p(A^n) to L^q) (A^n) (${1 \over p} - {1 \over q} = {\beta \over n},\,1 < p < {n \over \beta }$) and from L^1 (A^n) to L^n/(nm#), ^X (A^n) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\|\left\| {D^\gamma A} \right\|\right\|_{\dot \Lambda_\beta} $. And if Q has vanishing moments of order m m 1 and satisfies some kinds of Dini regularity otherwise, then TA is also bounded from L^p (A^n) to ${\dot F}^{\beta,\infty}_p$ (A^n)(1 < s' < p < X) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\| \left\|{D^\gamma A} \right\|\right\|_{\dot \Lambda _\beta } $.展开更多
The purpose of this paper is to propose and study a class of quasi-interpolating operators in multivariate spline space S-1/2(Delta(mn)(2*)) on non-uniform type-2 triangulation. Based on the operators, we construct cu...The purpose of this paper is to propose and study a class of quasi-interpolating operators in multivariate spline space S-1/2(Delta(mn)(2*)) on non-uniform type-2 triangulation. Based on the operators, we construct cubature formula for two-dimensional hypersingular integrals. Some computing work have been done and the results are quite satisfactory.展开更多
We study some basic properties of weak Orlicz spaces and their applications to harmonic analysis.We first discuss the absolute continuity of the quasi-norm and its normality,then prove the boundedness of several maxim...We study some basic properties of weak Orlicz spaces and their applications to harmonic analysis.We first discuss the absolute continuity of the quasi-norm and its normality,then prove the boundedness of several maximal operators.We also establish a kind of Marcinkiewicz-type interpolation theorem between weak Orlicz spaces.As applications,the weak type analogues of several classical inequalities in harmonic analysis is obtained.展开更多
In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,Ф along a surface Г = {x = Ф（｜y｜）y/｜y｜）} on the Triebel-Lizorkin space Fq,q^α（R^n ） for Ω belonging to H1 （Sn-...In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,Ф along a surface Г = {x = Ф（｜y｜）y/｜y｜）} on the Triebel-Lizorkin space Fq,q^α（R^n ） for Ω belonging to H1 （Sn-1） and some class WFα（S^n-1）, which relates to Grafakos-Stefanov class. Some previous results are extended and improved.展开更多
Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits...Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.展开更多
A new gradient operator was derived in recent studies of topological structures and shape transi- tions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the oper...A new gradient operator was derived in recent studies of topological structures and shape transi- tions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the operator’s general mathematical characteristics should be investigated. This paper explores the integral characteristics of the operator. The second divergence and the differential properties of the operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem. These new theorems provide a mathematical basis for the use of this operator in many disciplines.展开更多
Since the Leibniz-Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh-analysis, or harmonic analysis on local fields. On the basis of i...Since the Leibniz-Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh-analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one-dimensional dyadic derivative and integral are bounded from the dyadic Hardy space Hp,q to Lp,q, of weak type （L1,L1）, and the corresponding maximal operators of the two-dimensional case are of weak type （Hi, L1）. In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces Hp and the hybrid Hardy spaces H^#p 0〈p≤1.展开更多
In this paper,we give an order-preserving,isometric and isomorphic embeddingoperator from the fuzzy number space E^1 to a classical Banach space and show some necessary andsufficient integrability conditions of fuzzy ...In this paper,we give an order-preserving,isometric and isomorphic embeddingoperator from the fuzzy number space E^1 to a classical Banach space and show some necessary andsufficient integrability conditions of fuzzy integrals which were defined by M.Matloka and O.Kaleva by means of such an operator.展开更多
Under appropriate conditions on Young's functions Φ1 and Φ2,we give necessary and sufficient conditions in order that weighted integral inequalities hold for Doob's maximal operator M on martingale Orlicz se...Under appropriate conditions on Young's functions Φ1 and Φ2,we give necessary and sufficient conditions in order that weighted integral inequalities hold for Doob's maximal operator M on martingale Orlicz setting.When Φ1 = tp and Φ2 = tq,the inequalities revert to the ones of strong or weak(p,q)-type on martingale space.展开更多
This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operator...This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operators and fractional integral operators with homogeneous kernels on block-type spaces.展开更多
Provides information on a study which proposed a spline method for solving two-dimensional Fredholm Integral Equations of second kind space with hypersingular kernels. Details on the quasi-interpolating operators; Inf...Provides information on a study which proposed a spline method for solving two-dimensional Fredholm Integral Equations of second kind space with hypersingular kernels. Details on the quasi-interpolating operators; Information on the cubature formulas; Formulas of the approximation method.展开更多
After reviewing Grunsky operator and Faber operator acting on Dirichlet space,we discuss the boundedness of Faber operator on BMOA,a new subject which turns out to be closely related to the BMO theory of the universal...After reviewing Grunsky operator and Faber operator acting on Dirichlet space,we discuss the boundedness of Faber operator on BMOA,a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space.In particular,we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space.Meanwhile,we obtain several results on quasiconformal mappings,BMOTeichm¨uller space and chord-arc curves as well.As by-products,this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.展开更多
Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfa...Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss cur- vature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.展开更多
We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for th...We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.展开更多
In this paper, the author gives the weighted weak Lipschitz boundedness with power weight for rough multilinear integral operators. A simple way is obtained that is closely linked with a class of rough fractional inte...In this paper, the author gives the weighted weak Lipschitz boundedness with power weight for rough multilinear integral operators. A simple way is obtained that is closely linked with a class of rough fractional integral operators.展开更多
基金This research is supported by the NNSF (Grant:19971010)National 973 Project of China.
文摘In this paper we give the (L^p(ω~p),L^q(ω~q)) boundedness for a class of multilinear operators, which is similar to the higher-order commutator for the rough fractional integral.In our results the kernel function is merely assumed on a size condition.
文摘For a class of multilinear singular integral operators TA,$$T_A f\left( x \right) = \int {_{\Ropf^n} } {{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + m - 1} }}R_m \left( {A;x,y} \right)f\left( y \right)dy,$$where Rm (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m m 1 in $\dot \Lambda_\beta $(0 < # < 1), OHgr;(x) ] L^s(S^nm1)($s \ge {n \over {n - \beta }}$) is homogeneous of degree zero, the authors prove that TA is bounded from L^p(A^n) to L^q) (A^n) (${1 \over p} - {1 \over q} = {\beta \over n},\,1 < p < {n \over \beta }$) and from L^1 (A^n) to L^n/(nm#), ^X (A^n) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\|\left\| {D^\gamma A} \right\|\right\|_{\dot \Lambda_\beta} $. And if Q has vanishing moments of order m m 1 and satisfies some kinds of Dini regularity otherwise, then TA is also bounded from L^p (A^n) to ${\dot F}^{\beta,\infty}_p$ (A^n)(1 < s' < p < X) with the bound $C\sum\nolimits_{\left| \gamma \right| = m - 1} {} \left\| \left\|{D^\gamma A} \right\|\right\|_{\dot \Lambda _\beta } $.
文摘The purpose of this paper is to propose and study a class of quasi-interpolating operators in multivariate spline space S-1/2(Delta(mn)(2*)) on non-uniform type-2 triangulation. Based on the operators, we construct cubature formula for two-dimensional hypersingular integrals. Some computing work have been done and the results are quite satisfactory.
基金supported by National Natural Science Foundation of China (Grant No.11071190)
文摘We study some basic properties of weak Orlicz spaces and their applications to harmonic analysis.We first discuss the absolute continuity of the quasi-norm and its normality,then prove the boundedness of several maximal operators.We also establish a kind of Marcinkiewicz-type interpolation theorem between weak Orlicz spaces.As applications,the weak type analogues of several classical inequalities in harmonic analysis is obtained.
基金partially supported by Grant-in-Aid for Scientific Research(C)(No.23540228),Japan Society for the Promotion of Science
文摘In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,Ф along a surface Г = {x = Ф（｜y｜）y/｜y｜）} on the Triebel-Lizorkin space Fq,q^α（R^n ） for Ω belonging to H1 （Sn-1） and some class WFα（S^n-1）, which relates to Grafakos-Stefanov class. Some previous results are extended and improved.
文摘Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
文摘A new gradient operator was derived in recent studies of topological structures and shape transi- tions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the operator’s general mathematical characteristics should be investigated. This paper explores the integral characteristics of the operator. The second divergence and the differential properties of the operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem. These new theorems provide a mathematical basis for the use of this operator in many disciplines.
基金the Preliminary Research Foundation of National Defense (No,002,2BQ) the Foundation of Fuzhou University (No.0030824649)
文摘Since the Leibniz-Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh-analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one-dimensional dyadic derivative and integral are bounded from the dyadic Hardy space Hp,q to Lp,q, of weak type （L1,L1）, and the corresponding maximal operators of the two-dimensional case are of weak type （Hi, L1）. In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces Hp and the hybrid Hardy spaces H^#p 0〈p≤1.
基金This work is partially supported by the National Natural Science Foundation of China
文摘In this paper,we give an order-preserving,isometric and isomorphic embeddingoperator from the fuzzy number space E^1 to a classical Banach space and show some necessary andsufficient integrability conditions of fuzzy integrals which were defined by M.Matloka and O.Kaleva by means of such an operator.
基金supported by National Natural Science Foundation of China (Grant Nos.10671147,11071190)
文摘Under appropriate conditions on Young's functions Φ1 and Φ2,we give necessary and sufficient conditions in order that weighted integral inequalities hold for Doob's maximal operator M on martingale Orlicz setting.When Φ1 = tp and Φ2 = tq,the inequalities revert to the ones of strong or weak(p,q)-type on martingale space.
文摘This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operators and fractional integral operators with homogeneous kernels on block-type spaces.
基金the National Natural Science Foundation of China, and the Foundation for Doctoralprogram of the State Education Commission of
文摘Provides information on a study which proposed a spline method for solving two-dimensional Fredholm Integral Equations of second kind space with hypersingular kernels. Details on the quasi-interpolating operators; Information on the cubature formulas; Formulas of the approximation method.
文摘After reviewing Grunsky operator and Faber operator acting on Dirichlet space,we discuss the boundedness of Faber operator on BMOA,a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space.In particular,we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space.Meanwhile,we obtain several results on quasiconformal mappings,BMOTeichm¨uller space and chord-arc curves as well.As by-products,this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.
文摘Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss cur- vature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.
基金supported by National Natural Science Foundation of China (Grant Nos.10931001, 10871173)
文摘We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.
基金Supposed by Zhejiang Provincial Natural Science Foundtion of China under Grant(No.M103069)Supoorted by the Education Dept. of Zhejiang Province (20021022).
文摘In this paper, the author gives the weighted weak Lipschitz boundedness with power weight for rough multilinear integral operators. A simple way is obtained that is closely linked with a class of rough fractional integral operators.
