1. Introduction In application of nonlinear boundary value problems, it is sometimes important to know that L~2-boundedness of a class of pseudo-differential operators with symbols whioh have nonsmooth coefficients. I...1. Introduction In application of nonlinear boundary value problems, it is sometimes important to know that L~2-boundedness of a class of pseudo-differential operators with symbols whioh have nonsmooth coefficients. In [2], Coifman and Meyer proved that the operator σ(x, D) is bounded in L~2(R~n) if its symbold (x, ξ) satisfies:展开更多
In this work we discuss the existence of ψ-bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of ψ--bounded solutions for the linear nonhomogeneous d...In this work we discuss the existence of ψ-bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of ψ--bounded solutions for the linear nonhomogeneous difference equation x（n＋1）=A（n）x（n）＋f（n） for every ψ-bounded sequence f（n）.展开更多
A general approach to transference principles for discrete and continuous sequence of operators (semi) groups is described. This allows one to recover the classical transference results of Calderon, Coifman and Weiss ...A general approach to transference principles for discrete and continuous sequence of operators (semi) groups is described. This allows one to recover the classical transference results of Calderon, Coifman and Weiss and of Berkson, Gilleppie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) the sequence of operators semigroups that need not be grouped. As an application, functional calculus estimates for bounded sequence of operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to -spaces and—involving the concept of γ-boundedness—to general spaces. Analogous results for strongly-continuous one-parameter (semi) groups are presented as well by Markus Haase [1]. Finally, an application is given to singular integrals for one-parameter semigroups.展开更多
We present some convergence and boundedness theorems with respect to filter convergence for lattice group-valued measures. We give a direct proof, based on the sliding hump argument. Furthermore we pose some open prob...We present some convergence and boundedness theorems with respect to filter convergence for lattice group-valued measures. We give a direct proof, based on the sliding hump argument. Furthermore we pose some open problems.展开更多
We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogen...We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogeneous inξ-variables,non-degenerate and satisfies certain conditions.Then the Fourier integral operator T is L^(p)-bounded.Applying the method of(1),we can obtain the L^(p)-boundedness of the Fourier integral operator if(2)the symbol a∈Λ_(1,δ,k,k’)^(m0),0≤δ≤1,with m0,k,k’and S given as in(1).Also,the method of(1)gives:(3)a∈Λ_(1,δ,k,k’),0≤δ<1 and k,k’given as in(1),then the L^(p)-boundedness of the pseudo-differential operators holds,1<p<∞.展开更多
We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively h...We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively homogeneous in ξ-variables.Then we use the above property,Paley’s inequality,covering lemma of Calderon and Zygmund etc.,and obtain the L~p-boundedness of Fourier integral operators if(1) the symbol a ∈ Λ_(k)^(m_(0)) and Supp a = E×R~n,with E a compact set of R~n(m_(0) =-|1/p-1/2|n,1<p≤2,k>n/2;2<p<∞,k>n/p),(2) the symbol a ∈ Λ_(0,k,k’)^(m_(0))(m_(0) =-|1/p-1/2|n,1<p ≤2,k>n/2,k’>n/p;2<p<∞,k>n/p,k’>n/2) with the phase function S(x,ξ) = xξ + h(x,ξ),x,ξ ∈ R~n non-degenerate,satisfying certain conditions and ?ξ h ∈ S_(1,0)~0(R~n× R~n),or(3) the symbol a ∈ Λ_(0,k,k’)^(m_(0)),the requirements for m_(0),k,k’ are the same as in(2),and ?_(ξ)h is not in S_(1,0)~0(R~n× R~n) but the phase function S is non-degenerate,satisfies certain conditions and is positively homogeneous in ξ-variables.展开更多
基金Project supported by the Science Fund of the Chinese Academy of Sciences.
文摘1. Introduction In application of nonlinear boundary value problems, it is sometimes important to know that L~2-boundedness of a class of pseudo-differential operators with symbols whioh have nonsmooth coefficients. In [2], Coifman and Meyer proved that the operator σ(x, D) is bounded in L~2(R~n) if its symbold (x, ξ) satisfies:
基金The NSF(Y2008A30,ZR2010AL011)of Shandong Province
文摘In this work we discuss the existence of ψ-bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of ψ--bounded solutions for the linear nonhomogeneous difference equation x（n＋1）=A（n）x（n）＋f（n） for every ψ-bounded sequence f（n）.
文摘A general approach to transference principles for discrete and continuous sequence of operators (semi) groups is described. This allows one to recover the classical transference results of Calderon, Coifman and Weiss and of Berkson, Gilleppie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) the sequence of operators semigroups that need not be grouped. As an application, functional calculus estimates for bounded sequence of operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to -spaces and—involving the concept of γ-boundedness—to general spaces. Analogous results for strongly-continuous one-parameter (semi) groups are presented as well by Markus Haase [1]. Finally, an application is given to singular integrals for one-parameter semigroups.
文摘We present some convergence and boundedness theorems with respect to filter convergence for lattice group-valued measures. We give a direct proof, based on the sliding hump argument. Furthermore we pose some open problems.
文摘We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogeneous inξ-variables,non-degenerate and satisfies certain conditions.Then the Fourier integral operator T is L^(p)-bounded.Applying the method of(1),we can obtain the L^(p)-boundedness of the Fourier integral operator if(2)the symbol a∈Λ_(1,δ,k,k’)^(m0),0≤δ≤1,with m0,k,k’and S given as in(1).Also,the method of(1)gives:(3)a∈Λ_(1,δ,k,k’),0≤δ<1 and k,k’given as in(1),then the L^(p)-boundedness of the pseudo-differential operators holds,1<p<∞.
文摘We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively homogeneous in ξ-variables.Then we use the above property,Paley’s inequality,covering lemma of Calderon and Zygmund etc.,and obtain the L~p-boundedness of Fourier integral operators if(1) the symbol a ∈ Λ_(k)^(m_(0)) and Supp a = E×R~n,with E a compact set of R~n(m_(0) =-|1/p-1/2|n,1<p≤2,k>n/2;2<p<∞,k>n/p),(2) the symbol a ∈ Λ_(0,k,k’)^(m_(0))(m_(0) =-|1/p-1/2|n,1<p ≤2,k>n/2,k’>n/p;2<p<∞,k>n/p,k’>n/2) with the phase function S(x,ξ) = xξ + h(x,ξ),x,ξ ∈ R~n non-degenerate,satisfying certain conditions and ?ξ h ∈ S_(1,0)~0(R~n× R~n),or(3) the symbol a ∈ Λ_(0,k,k’)^(m_(0)),the requirements for m_(0),k,k’ are the same as in(2),and ?_(ξ)h is not in S_(1,0)~0(R~n× R~n) but the phase function S is non-degenerate,satisfies certain conditions and is positively homogeneous in ξ-variables.
