The authors mainly study the Hausdorff operators on Euclidean space R^n. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and ...The authors mainly study the Hausdorff operators on Euclidean space R^n. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and Herz type spaces. The results reveal that the Hausdorff operators have better performance on the Herz type Hardy spaces H Kα,pq (R^n) than their performance on the Hardy spaces H^p(R^n) when 0 < p < 1. Also, the authors obtain some new results and reprove or generalize some known results for the high dimensional Hardy operator and adjoint Hardy operator.展开更多
In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L^p(R)and the...In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L^p(R)and the Hardy space H^1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H^p(R) with 0 < p < 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H^p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H^1(R).展开更多
We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp mult...We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp multiplier, we obtain the optimal range of exponents (γ,β,p)to ensure the H^p(R^n) boundedness of a^βS^γ1f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p(R^n) spaces.展开更多
For n = 2 or 3 and x ∈ Rn, we study the oscillatory hyper Hilbert transform Tα,β f(x)=∫Rf(x-Г(t,x))e^-i|t|-β|t|^-1-α dt along an appropriate variable curveГ(t, x) in R^n (namely,Г(t, x) is a curve in R^n for ...For n = 2 or 3 and x ∈ Rn, we study the oscillatory hyper Hilbert transform Tα,β f(x)=∫Rf(x-Г(t,x))e^-i|t|-β|t|^-1-α dt along an appropriate variable curveГ(t, x) in R^n (namely,Г(t, x) is a curve in R^n for each fixed x), where α〉β〉0. We obtain some LP boundedness theorems of Tα,β, under some suitable conditions on α and β. These results are extensions of some earlier theorems. However, Tα,β f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.展开更多
We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means ...We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means and some other well-known operators.We provide a simple proof so that we are able to extend and improve many results published in recent papers.展开更多
In this paper, we consider the embedding relations between any two a-modulation spaces. Based on an observation that the a-modulation space with smaller a can be regarded as a corresponding decomposition space associa...In this paper, we consider the embedding relations between any two a-modulation spaces. Based on an observation that the a-modulation space with smaller a can be regarded as a corresponding decomposition space associated with a-covering for larger a, we give a complete characterization of the Fourier multipliers between a-modulation spaces with different a. Then we establish a full version of optimal embedding relations between a-modulation spaces.展开更多
We consider the boundedness of the n-dimension oscillatory hyper- Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. T...We consider the boundedness of the n-dimension oscillatory hyper- Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems significantly improve some known results.展开更多
基金supported by the National Natural Science Foundation of China (Nos. 10931001, 10871173)
文摘The authors mainly study the Hausdorff operators on Euclidean space R^n. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and Herz type spaces. The results reveal that the Hausdorff operators have better performance on the Herz type Hardy spaces H Kα,pq (R^n) than their performance on the Hardy spaces H^p(R^n) when 0 < p < 1. Also, the authors obtain some new results and reprove or generalize some known results for the high dimensional Hardy operator and adjoint Hardy operator.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671363, 11471288 and 11601456)
文摘In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L^p(R)and the Hardy space H^1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H^p(R) with 0 < p < 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H^p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H^1(R).
基金the National Natural Science Foundation of China (Grant Nos. 1177138& 11371316, 1147128&11601456).
文摘We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp multiplier, we obtain the optimal range of exponents (γ,β,p)to ensure the H^p(R^n) boundedness of a^βS^γ1f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p(R^n) spaces.
基金Project supported by the National Natural Science Foundation of China (Nos.10931001 and 10871173)the Educational Science Foundation of Zhejiang (No.Z201017584)the Science Foundation of Zhejiang University of Science and Technology (No.F501108A02)
文摘The authors prove the certain de Leeuw type theorems on some non-convolution operators,and give some applications on certain known results.
基金This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11671363, 11371316, 11771388).
文摘For n = 2 or 3 and x ∈ Rn, we study the oscillatory hyper Hilbert transform Tα,β f(x)=∫Rf(x-Г(t,x))e^-i|t|-β|t|^-1-α dt along an appropriate variable curveГ(t, x) in R^n (namely,Г(t, x) is a curve in R^n for each fixed x), where α〉β〉0. We obtain some LP boundedness theorems of Tα,β, under some suitable conditions on α and β. These results are extensions of some earlier theorems. However, Tα,β f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.
基金supported by National Natural Science Foundation of China(Grant Nos.11971295,11871108 and 11871436)Natural Science Foundation of Shanghai(No.19ZR1417600).
文摘We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means and some other well-known operators.We provide a simple proof so that we are able to extend and improve many results published in recent papers.
基金supported by National Natural Science Foundation of China(Grant Nos.11601456,11771388,11371316,11701112 and 11671414)China Postdoctoral Science Foundation(Grant No.2017M612628)
文摘In this paper, we consider the embedding relations between any two a-modulation spaces. Based on an observation that the a-modulation space with smaller a can be regarded as a corresponding decomposition space associated with a-covering for larger a, we give a complete characterization of the Fourier multipliers between a-modulation spaces with different a. Then we establish a full version of optimal embedding relations between a-modulation spaces.
基金Acknowledgements The authors are thankful to the referees for their careful reading and useful comments. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11501516, 11471288) and the Natural Science Foundation of Zhejiang Province (No. LQ15A010003).
文摘We consider the boundedness of the n-dimension oscillatory hyper- Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems significantly improve some known results.