LP(Rn) boundedness is considered for the multilinear singular integral operator defined by where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives...LP(Rn) boundedness is considered for the multilinear singular integral operator defined by where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈Lipαt(Sn-1) (0 <α≤1) and implies the LP(Rn) (1 < p <∞) boundedness for the operator TA.Some endpoint estimates are also established.展开更多
In this paper, the authors discuss a class of multilinear singular integralsand obtain that the operators are bounded from H^1 (R^n) to weak L^1 (R^n). Using this result, wecan directly prove a main theorem in [5].
A new approach to domain-specific reasoning is presented that is based on a type-theoretic logical framework (LF) but does not require the user to be an expert in type theory. The concepts of the domain and its relate...A new approach to domain-specific reasoning is presented that is based on a type-theoretic logical framework (LF) but does not require the user to be an expert in type theory. The concepts of the domain and its related reasoning systems are formalized in LF, but the user works with the system through a syntax and interface appropriate to his/her work. A middle layer provides translation between the user syntax and LF, and allows additional support for reasoning (e.g., model checking). Thus, the complexity of the logical framework is hidden but the benefits of using type theory and its related tools are retained, such as precision and machine-checkable proofs. This approach is investigated through a number of case studies: here, the authors consider the verification of properties of concurrency. The authors have formalized a specification language (CCS) and logic (μ-calculus) in LF, together with useful lemmas, and a user-oriented syntax has been designed. The authors demonstrate the approach with simple examples. However, applying lemmas to objects introduced by the user may result in framework-level objects which cannot be translated back to the user level. The authors discuss this problem, define a notion of adequacy, and prove that in this case study, translation can always be reversed.展开更多
The commutators of singular integral operators with homogeneous kernel(Ω(x))/(|x|~n)are studied, where Ω is homogeneous of degree zero,and has mean value zero on the unit sphere.It is proved that Ω ∈ L(logL)^(k+1)...The commutators of singular integral operators with homogeneous kernel(Ω(x))/(|x|~n)are studied, where Ω is homogeneous of degree zero,and has mean value zero on the unit sphere.It is proved that Ω ∈ L(logL)^(k+1)(S^(n-1))is a sufficient condition such that the κ-th order commutator is bounded on L^2(R^n).展开更多
The L^2(A^n) boundedness for the multilinear singular integral operators defined by$T_A f\left( x \right) = \int_{Ropf^n } {{{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + 1} }}} \left( {A\left( x ...The L^2(A^n) boundedness for the multilinear singular integral operators defined by$T_A f\left( x \right) = \int_{Ropf^n } {{{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + 1} }}} \left( {A\left( x \right) - A\left( y \right) - \nabla A\left( y \right)\left( {x - y} \right)} \right)f\left( y \right)dy$is considered, whereQ is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO(A^n). A sufficient condition based on the Fourier transform estimate and implying the L^2(A^n) boundedness for the multilinear operator TA is given.展开更多
文摘LP(Rn) boundedness is considered for the multilinear singular integral operator defined by where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈Lipαt(Sn-1) (0 <α≤1) and implies the LP(Rn) (1 < p <∞) boundedness for the operator TA.Some endpoint estimates are also established.
基金supported by Professor Xu Yuesheng's research grant in the program of "One hundred Distinguished Young Scientists"of the Chinese Academy of Sciences.
文摘In this paper, the authors discuss a class of multilinear singular integralsand obtain that the operators are bounded from H^1 (R^n) to weak L^1 (R^n). Using this result, wecan directly prove a main theorem in [5].
文摘A new approach to domain-specific reasoning is presented that is based on a type-theoretic logical framework (LF) but does not require the user to be an expert in type theory. The concepts of the domain and its related reasoning systems are formalized in LF, but the user works with the system through a syntax and interface appropriate to his/her work. A middle layer provides translation between the user syntax and LF, and allows additional support for reasoning (e.g., model checking). Thus, the complexity of the logical framework is hidden but the benefits of using type theory and its related tools are retained, such as precision and machine-checkable proofs. This approach is investigated through a number of case studies: here, the authors consider the verification of properties of concurrency. The authors have formalized a specification language (CCS) and logic (μ-calculus) in LF, together with useful lemmas, and a user-oriented syntax has been designed. The authors demonstrate the approach with simple examples. However, applying lemmas to objects introduced by the user may result in framework-level objects which cannot be translated back to the user level. The authors discuss this problem, define a notion of adequacy, and prove that in this case study, translation can always be reversed.
基金supported by the NSF of China(19701039)partially supported by the NFS of China(19971010)Visiting Scholar Foundation of Key Lab.in Peking University
文摘The commutators of singular integral operators with homogeneous kernel(Ω(x))/(|x|~n)are studied, where Ω is homogeneous of degree zero,and has mean value zero on the unit sphere.It is proved that Ω ∈ L(logL)^(k+1)(S^(n-1))is a sufficient condition such that the κ-th order commutator is bounded on L^2(R^n).
文摘The L^2(A^n) boundedness for the multilinear singular integral operators defined by$T_A f\left( x \right) = \int_{Ropf^n } {{{\Omega \left( {x - y} \right)} \over {\left| {x - y} \right|^{n + 1} }}} \left( {A\left( x \right) - A\left( y \right) - \nabla A\left( y \right)\left( {x - y} \right)} \right)f\left( y \right)dy$is considered, whereQ is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO(A^n). A sufficient condition based on the Fourier transform estimate and implying the L^2(A^n) boundedness for the multilinear operator TA is given.
