Let Q 2 = [0, 1]2 be the unit square in two dimension Euclidean space ?2. We study the L p boundedness properties of the oscillatory integral operators T α,β defined on the set S(?3) of Schwartz test functions f by ...Let Q 2 = [0, 1]2 be the unit square in two dimension Euclidean space ?2. We study the L p boundedness properties of the oscillatory integral operators T α,β defined on the set S(?3) of Schwartz test functions f by $$ \mathcal{T}_{\alpha ,\beta } f(x,y,z) = \int_{Q^2 } {f(x - t,y - s,z - t^k s^j )e^{ - it^{ - \beta _1 } s^{ - \beta 2} } t^{ - 1 - \alpha _1 } s^{ - 1 - \alpha _2 } dtds} , $$ where β1 > α1 ? 0, β2 > α2 ? 0 and (k, j) ∈ ?2. As applications, we obtain some L p boundedness results of rough singular integral operators on the product spaces.展开更多
We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especial...We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especially, we establish the analogue of the sharp Sobolev embedding theorem on weighted modulation spaces.展开更多
We prove that the fundamental semi-group e^it（m^2│△│）^1/2 （m≠ 0） of the Klein-Gordon equation is bounded on the modulation space M^8p,q（R^n） for all 0 〈 p, q ≤∞ and s ∈ R. Similarly, we prove that the wa...We prove that the fundamental semi-group e^it（m^2│△│）^1/2 （m≠ 0） of the Klein-Gordon equation is bounded on the modulation space M^8p,q（R^n） for all 0 〈 p, q ≤∞ and s ∈ R. Similarly, we prove that the wave semi-group e^it│△│^1/2 is bounded on the Hardy type modulation spaces μ^εp,q（R^n） for all 0 〈 p, q ≤ ∞, and s ∈R. All the bounds have an asymptotic factor t^n│1/p-1/2│ as t goes to the infinity. These results extend some known results for the case of p ≥ 1. Also, some applications for the Cauchy problems related to the semi-group eit（m^2I＋│△│）1/2 are obtained. Finally we discuss the optimum of the factor t^n│1/p-1/2│ and raise some unsolved problems.展开更多
We study certain square functions on product spaces Rn×Rm,whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log+L)away from Rn×{0}and{0}×Rm...We study certain square functions on product spaces Rn×Rm,whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log+L)away from Rn×{0}and{0}×Rm by means of polynomial distortions in the radial variable.As a model case,we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn×Rm)(P>1)forΩ∈e Llog+L(Sn-1×Sm-1)satisfying the cancellation condition.展开更多
A boundedness criterion is set up for some convolution operators on a compact Lie group.By this criterion a Hormander multiplier theorem is proved in the Hardy spaces on SU(2).
基金the National Natural Science Foundation of China (Grant Nos. 10571122, 10371046)the Natural Science Foundation of Fujian Province of China (Grant No. Z0511004)
文摘Let Q 2 = [0, 1]2 be the unit square in two dimension Euclidean space ?2. We study the L p boundedness properties of the oscillatory integral operators T α,β defined on the set S(?3) of Schwartz test functions f by $$ \mathcal{T}_{\alpha ,\beta } f(x,y,z) = \int_{Q^2 } {f(x - t,y - s,z - t^k s^j )e^{ - it^{ - \beta _1 } s^{ - \beta 2} } t^{ - 1 - \alpha _1 } s^{ - 1 - \alpha _2 } dtds} , $$ where β1 > α1 ? 0, β2 > α2 ? 0 and (k, j) ∈ ?2. As applications, we obtain some L p boundedness results of rough singular integral operators on the product spaces.
基金supported by National Natural Science Foundation of China(Grant Nos.113712951102610411471041 and 11471288)
文摘We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especially, we establish the analogue of the sharp Sobolev embedding theorem on weighted modulation spaces.
基金supported by National Natural Science Foundation of China (Grant Nos.11271330 and 10931001)
文摘We prove that the fundamental semi-group e^it（m^2│△│）^1/2 （m≠ 0） of the Klein-Gordon equation is bounded on the modulation space M^8p,q（R^n） for all 0 〈 p, q ≤∞ and s ∈ R. Similarly, we prove that the wave semi-group e^it│△│^1/2 is bounded on the Hardy type modulation spaces μ^εp,q（R^n） for all 0 〈 p, q ≤ ∞, and s ∈R. All the bounds have an asymptotic factor t^n│1/p-1/2│ as t goes to the infinity. These results extend some known results for the case of p ≥ 1. Also, some applications for the Cauchy problems related to the semi-group eit（m^2I＋│△│）1/2 are obtained. Finally we discuss the optimum of the factor t^n│1/p-1/2│ and raise some unsolved problems.
基金supported by the National Natural Science Foundation of China(Grant No.10571156)National Fund of 973 Project(Grant No.G1999075105)+1 种基金Zhejiang Provincial Natural Science Foundation of China(Grant No.RC97017)Research Fund for the Dectoral Program of Higher Education(Grant No.20030335019).
文摘We study certain square functions on product spaces Rn×Rm,whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log+L)away from Rn×{0}and{0}×Rm by means of polynomial distortions in the radial variable.As a model case,we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn×Rm)(P>1)forΩ∈e Llog+L(Sn-1×Sm-1)satisfying the cancellation condition.
文摘A boundedness criterion is set up for some convolution operators on a compact Lie group.By this criterion a Hormander multiplier theorem is proved in the Hardy spaces on SU(2).
