In this paper,we obtain that b∈ BMO（Rn） if and only if the commutator[b,Iα]is bounded from the Morrey spaces Lp1,λ1Rn×Lp2,λ2Rnto Lq,λ（Rn）,for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip...In this paper,we obtain that b∈ BMO（Rn） if and only if the commutator[b,Iα]is bounded from the Morrey spaces Lp1,λ1Rn×Lp2,λ2Rnto Lq,λ（Rn）,for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β（R^n） if and only if the commutator[b,I_α]is bounded from the Morrey spaces Lp1,λ1）Rn×Lp2,λ（Rnto Lq,λRn,for some appropriate indices p,q,λ,μ.展开更多
In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a...In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a∈L∞Smρ,the phase functionφsatisfies some measure conditions and ∇kξφ(·,ξ)L∞≤C|ξ|−k for all k≥2,ξ≠0,and some∈>0,we obtain that Tφ,a is bounded on L^(2) if m<n2 min{ρ−1,−2}.This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators.Moreover,the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schr odinger operator.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1126105511661075)
文摘In this paper,we obtain that b∈ BMO（Rn） if and only if the commutator[b,Iα]is bounded from the Morrey spaces Lp1,λ1Rn×Lp2,λ2Rnto Lq,λ（Rn）,for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β（R^n） if and only if the commutator[b,I_α]is bounded from the Morrey spaces Lp1,λ1）Rn×Lp2,λ（Rnto Lq,λRn,for some appropriate indices p,q,λ,μ.
基金Supported by National Natural Science Foundation of China(Grant No.12071437)National key R&D program of China(Grant No.2022YFA1005703)。
文摘In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a∈L∞Smρ,the phase functionφsatisfies some measure conditions and ∇kξφ(·,ξ)L∞≤C|ξ|−k for all k≥2,ξ≠0,and some∈>0,we obtain that Tφ,a is bounded on L^(2) if m<n2 min{ρ−1,−2}.This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators.Moreover,the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schr odinger operator.
