Ⅰ. INTRODUCTIONWhittaker first studied the relation between the first integral and integral invariant, and proved that an integral invariant can be determined by a given first integral. Recently, Mei Fengxiang furthe...Ⅰ. INTRODUCTIONWhittaker first studied the relation between the first integral and integral invariant, and proved that an integral invariant can be determined by a given first integral. Recently, Mei Fengxiang further extended this result into the nonholonomic dynamical systems. In view展开更多
A variation of parameters formula with Green function type and Gronwall type integral inequality are proved for impulsive differential equations involving piecewise constant delay of generalized type. Some results for...A variation of parameters formula with Green function type and Gronwall type integral inequality are proved for impulsive differential equations involving piecewise constant delay of generalized type. Some results for piecewise constant linear and nonlinear delay differential equations with impulsive effects are obtained.They include existence and uniqueness theorems, a variation of parameters formula, integral inequalities, the oscillation property and some applications.展开更多
In this paper,we investigate the boundedness and compactness for variation operators of CalderónZygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces.To be precise,letρ>...In this paper,we investigate the boundedness and compactness for variation operators of CalderónZygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces.To be precise,letρ>2 and K be a standard Calderón-Zygmund kernel.Denote by V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))(m≥1)theρ-variation operators of Calderón-Zygmund singular integrals and their m-th iterated commutators,respectively.By assuming that V_(ρ)(T_(K))satisfies an a priori estimate,i.e.,the map V_(ρ)(T_(K)):L^(p0)(R^(n))→L^(p0)(R^(n))is bounded for some p0∈(1,∞),the bounds for V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))on weighted Morrey spaces and Sobolev spaces are established.Meanwhile,the compactness properties of V_(ρ)(T_(K,b)^(m))on weighted Lebesgue and Morrey spaces are also discussed.As applications,the corresponding results for the Hilbert transform,the Hermite Riesz transform,Riesz transforms and rough singular integrals as well as their commutators on the above function spaces are presented.展开更多
文摘Ⅰ. INTRODUCTIONWhittaker first studied the relation between the first integral and integral invariant, and proved that an integral invariant can be determined by a given first integral. Recently, Mei Fengxiang further extended this result into the nonholonomic dynamical systems. In view
文摘A variation of parameters formula with Green function type and Gronwall type integral inequality are proved for impulsive differential equations involving piecewise constant delay of generalized type. Some results for piecewise constant linear and nonlinear delay differential equations with impulsive effects are obtained.They include existence and uniqueness theorems, a variation of parameters formula, integral inequalities, the oscillation property and some applications.
基金supported by National Natural Science Foundation of China(Grant No.11701333)。
文摘In this paper,we investigate the boundedness and compactness for variation operators of CalderónZygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces.To be precise,letρ>2 and K be a standard Calderón-Zygmund kernel.Denote by V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))(m≥1)theρ-variation operators of Calderón-Zygmund singular integrals and their m-th iterated commutators,respectively.By assuming that V_(ρ)(T_(K))satisfies an a priori estimate,i.e.,the map V_(ρ)(T_(K)):L^(p0)(R^(n))→L^(p0)(R^(n))is bounded for some p0∈(1,∞),the bounds for V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))on weighted Morrey spaces and Sobolev spaces are established.Meanwhile,the compactness properties of V_(ρ)(T_(K,b)^(m))on weighted Lebesgue and Morrey spaces are also discussed.As applications,the corresponding results for the Hilbert transform,the Hermite Riesz transform,Riesz transforms and rough singular integrals as well as their commutators on the above function spaces are presented.
