The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i｜x｜^aΩ（x）｜x｜^-n is studied,where a∈R,a≠0,1 and Ω∈L^1（S^n-1） is homogeneous of degree zero and satisfie...The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i｜x｜^aΩ（x）｜x｜^-n is studied,where a∈R,a≠0,1 and Ω∈L^1（S^n-1） is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω（x＇ ）∈Llog＋L（S^n-1 ）, the Fp^a,q（R^n） boundedness of the above operator is obtained. Meanwhile ,when Ω（x） satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 （R^n）.展开更多
In this paper, we study certain Hausdorff operators in the high-dimensional product spaces. We obtain their power weighted boundedness from Lp to Lq and characterize the necessary and sufficient conditions for their b...In this paper, we study certain Hausdorff operators in the high-dimensional product spaces. We obtain their power weighted boundedness from Lp to Lq and characterize the necessary and sufficient conditions for their boundedness on the power weighted Lp spaces. Moreover, in the case p = q, we obtain the sharp bound constants.展开更多
In this paper, we consider the rough singular integral operators on product Triebel-Lizorkin spaces and prove certain boundedness properties on the Triebel-Lizorkin spaces. We also use the same method to study the fra...In this paper, we consider the rough singular integral operators on product Triebel-Lizorkin spaces and prove certain boundedness properties on the Triebel-Lizorkin spaces. We also use the same method to study the fractional integral operator and the Littlewood-Paley functions. The results extend some known results.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 obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for th...We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.展开更多
We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon...We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon equation with nonlinear term|u|~ku in modulation space,where k is not an integer.Moreover,we also study the global solution with small initial value for the Klein-Gordon-Hartree equation.The results show some advantages of modulation space both in high and low regularity cases.展开更多
文摘The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i｜x｜^aΩ（x）｜x｜^-n is studied,where a∈R,a≠0,1 and Ω∈L^1（S^n-1） is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω（x＇ ）∈Llog＋L（S^n-1 ）, the Fp^a,q（R^n） boundedness of the above operator is obtained. Meanwhile ,when Ω（x） satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 （R^n）.
基金supported by National Natural Science Foundation of China(Grant Nos.11271330 and 10931001)Education Foundation of Zhejiang Province(Grant No.Y201225707)Natural Science Foundation of Zhejiang Province of China(Grant No.Y604563)
文摘In this paper, we study certain Hausdorff operators in the high-dimensional product spaces. We obtain their power weighted boundedness from Lp to Lq and characterize the necessary and sufficient conditions for their boundedness on the power weighted Lp spaces. Moreover, in the case p = q, we obtain the sharp bound constants.
基金supported by the Program for New Century Excellent Talents in Fujian Province, National Natural Science Foundation of China (Grant No. 10601040)Tian Yuan Foundation of China(Grant No. 10526033)China Postdoctoral Science Foundation (Grant No. 2005038639)
文摘In this paper, we consider the rough singular integral operators on product Triebel-Lizorkin spaces and prove certain boundedness properties on the Triebel-Lizorkin spaces. We also use the same method to study the fractional integral operator and the Littlewood-Paley functions. The results extend some known results.
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 National Natural Science Foundation of China (Grant Nos.10931001, 10871173)
文摘We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671363 and 11471288)Natural Science Foundation of Zhejiang Province (Grant No. LQ15A010003)
文摘We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon equation with nonlinear term|u|~ku in modulation space,where k is not an integer.Moreover,we also study the global solution with small initial value for the Klein-Gordon-Hartree equation.The results show some advantages of modulation space both in high and low regularity cases.
