The authors introduce the homogeneous Morrey-Herz spaces and the weak homo- geneous Morrey-Herz spaces on non-homogeneous spaces and establish the boundedness in ho- mogeneous Morrey-Herz spaces for a class of subline...The authors introduce the homogeneous Morrey-Herz spaces and the weak homo- geneous Morrey-Herz spaces on non-homogeneous spaces and establish the boundedness in ho- mogeneous Morrey-Herz spaces for a class of sublinear operators including Hardy-Littlewood maximal operators,Calderón-Zygmund operators and fractional integral operators.Further- more,some weak estimate of these operators in weak homogeneous Morrey-Herz spaces are also obtained.Moreover,the authors discuss the boundedness in homogeneous Morrey-Herz spaces of the maximal commutators associated with Hardy-Littlewood maximal operators and multilinear commutators generated by Calderón-Zygmund operators or fractional integral operators with RBMO（μ）functions.展开更多
The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consisten...The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.展开更多
Let N be a sufficiently large integer.In this paper,it is proved that with at most O(N17/18+ε)exceptions,all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_...Let N be a sufficiently large integer.In this paper,it is proved that with at most O(N17/18+ε)exceptions,all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_(1)^(3)+p_(2)^(4)+p_(3)^(4)+p_(5)^(4)+p_(6)^(4)+p_(7)^(4)+p_(8)^(4)+p_(9)^(4)+p_(10)^(4),where p1,p2,…,P_(10)are prime numbers.展开更多
This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtai...This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro oper- ators （with symbols in central BMO spaces） are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.展开更多
This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from...This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.展开更多
Under certain weak local size conditions, the boundedness of linear commutators on Herz-Morrey spaces on spaces of homogeneous type are studied. In addition, the boundedness of Hardy-Littlewood maximum commutators on ...Under certain weak local size conditions, the boundedness of linear commutators on Herz-Morrey spaces on spaces of homogeneous type are studied. In addition, the boundedness of Hardy-Littlewood maximum commutators on Herz-Morrey spaces on spaces of homogeneous type are obtained.展开更多
We will prove that for 1<p<∞and 0<λ<n,the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<γ...We will prove that for 1<p<∞and 0<λ<n,the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<γ<+∞.When p=1 and 0<λ<n,it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<λ<+∞.Moreover,the same results are true for the truncated uncentered Hardy-Littlewood maximal operator.Our work extends the previous results of Lebesgue spaces to Morrey spaces.展开更多
In this paper, it is shown that Hardy-Hilbert＇s integral inequality with parameter is improved by means of a sharpening of Hoeder＇s inequality. A new inequality is established as follows：∫^∞α∫^∞α f（x）g（y）...In this paper, it is shown that Hardy-Hilbert＇s integral inequality with parameter is improved by means of a sharpening of Hoeder＇s inequality. A new inequality is established as follows：∫^∞α∫^∞α f（x）g（y）/（x＋y＋2β）dxdy〈π/sin（π/p）{∫^∞α f^p（x）dx}1/p·{∫^∞αgq（x）dx}1/q·（1-R）^m,where R=（Sp （F, h） - Sq （G, h））^2, m= min （1/p, 1/q）. As application; an extension of Hardy-Littlewood＇s inequality is given.展开更多
In this paper, we study one class of n-dimensional Hardy-Steklov operators which has important applications in the technical analysis in equity markets. We establish their weighted boundedness and the corresponding op...In this paper, we study one class of n-dimensional Hardy-Steklov operators which has important applications in the technical analysis in equity markets. We establish their weighted boundedness and the corresponding operator norms on both LP（Rn） and BMO（Rn）.展开更多
The authors prove the Hardy-Littlewood-Sobolev theorems for generalized fractional integrals L?α/2 for 0 < α < n/m, where L is a complex elliptic operator of arbitrary order 2m on Rn.
Using the circle method and sieves, the author proves that a positive proportion of positive integers can be represented as the sum of four cubes of primes.
. In this paper,the characterization of boundedness of Hardy-Littlewood maximal operators in Orlicz-Morrey spaces LΦφ（X,μ） of homogeneous type is founded.
In this paper, we give the following dominated theorem: Let φ(g) ∈ L1(G//K),φε(t)=ε> 0, and the least radical decreasing dominatedfunction φ(t) = sup |φ(y)| ∈L1(G//K). If shtφ(t) is monotonically decreasin...In this paper, we give the following dominated theorem: Let φ(g) ∈ L1(G//K),φε(t)=ε> 0, and the least radical decreasing dominatedfunction φ(t) = sup |φ(y)| ∈L1(G//K). If shtφ(t) is monotonically decreasingon (0, ∞), then for any f∈L1loc(G//K) , the following inequality holds:sup |φε * f(x)| ≤ Cmf(x),where mf(x) is the Hardy-Littlewood maximal function of f, and C = ||φ||1.An application of this dominated theorem is also given.展开更多
Let r(n) denote the number of representations of n as the sums of 5 cubes and 3 biquadra-tes of natural numbers. Then for all sufficiently large n, one has r(n)(?)n17/12, which is the expected order of magnitude of r(n).
We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove th...We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.展开更多
In this note, we give a short proof for the DiPerna-Lions flows associated to ODEs following the method of Crippa and De Lellis [3]. More precisely, assume that [divb] ∈ Ll∞oc（Rd）, ｜b｜/（1 ＋ ｜x｜ log ｜x｜） ...In this note, we give a short proof for the DiPerna-Lions flows associated to ODEs following the method of Crippa and De Lellis [3]. More precisely, assume that [divb] ∈ Ll∞oc（Rd）, ｜b｜/（1 ＋ ｜x｜ log ｜x｜） ∈ L∞（Rd） and ｜ b｜ φ（｜ b｜） ∈ Ll1oc（Rd）, where φ（r） = log log（r ＋ c）, c 0. Then, there exists a unique regular Lagrangian flow associated with the ODE X˙（t, x） = b（X（t, x））, X（0, x） = x.展开更多
Let q greater than or equal to 2,f is measurable function on R-n such that f(x)\x\(n(1-2/q)) is an element of L-q(R-n), then its Fourier transform (f) over cap can be defined and there exists a constant A(q) such that...Let q greater than or equal to 2,f is measurable function on R-n such that f(x)\x\(n(1-2/q)) is an element of L-q(R-n), then its Fourier transform (f) over cap can be defined and there exists a constant A(q) such that the inequality parallel to (f) over cap parallel to(q) less than or equal to A(q) parallel to f\.\(n(1-2/q))parallel to(q) holds. This is the Hardy-Littlewood Theorem. This paper considers the corresponding result for the Fourier-Bessel transform and Fourier-Jacobi transform. It is interesting that we can deal with these two cases in the same way, and the function corresponding to \x\(n) is tw(t), where w(t) is the weight, w(t) = t(2 alpha+1) for Fourier-Bessel transform, and w(t) = (2 sinh t)(2 alpha+1)(2 cosh t)(2 beta+1) for Fourier-Jacobi transform.展开更多
基金the North China Electric Power University Youth Foundation(No.200611004)the Renmin University of China Science Research Foundation(No.30206104)
文摘The authors introduce the homogeneous Morrey-Herz spaces and the weak homo- geneous Morrey-Herz spaces on non-homogeneous spaces and establish the boundedness in ho- mogeneous Morrey-Herz spaces for a class of sublinear operators including Hardy-Littlewood maximal operators,Calderón-Zygmund operators and fractional integral operators.Further- more,some weak estimate of these operators in weak homogeneous Morrey-Herz spaces are also obtained.Moreover,the authors discuss the boundedness in homogeneous Morrey-Herz spaces of the maximal commutators associated with Hardy-Littlewood maximal operators and multilinear commutators generated by Calderón-Zygmund operators or fractional integral operators with RBMO（μ）functions.
基金Supported by the Russian Science Foundation(Grant No.14-21-00162)
文摘The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.
文摘Let N be a sufficiently large integer.In this paper,it is proved that with at most O(N17/18+ε)exceptions,all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_(1)^(3)+p_(2)^(4)+p_(3)^(4)+p_(5)^(4)+p_(6)^(4)+p_(7)^(4)+p_(8)^(4)+p_(9)^(4)+p_(10)^(4),where p1,p2,…,P_(10)are prime numbers.
基金supported by Vietnam National Foundation for Science and Technology Development(101.02-2014.51)
文摘This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro oper- ators （with symbols in central BMO spaces） are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.
基金supported by National Natural Science Foundation of China(Grant No.11661075)
文摘This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.
基金Supported by Natural Science Foundation of Xinjiang University Supported by the NNSF of Chlna（10861010） Supported by Research Starting Foundation for Doctors of Xinjiang University（BS090102）
文摘Under certain weak local size conditions, the boundedness of linear commutators on Herz-Morrey spaces on spaces of homogeneous type are studied. In addition, the boundedness of Hardy-Littlewood maximum commutators on Herz-Morrey spaces on spaces of homogeneous type are obtained.
基金the National Natural Science Foundation of China(Grant No.11871452)the Project of Henan Provincial Department of Education(No.18A110028)the Nanhu Scholar Program for Young Scholars of XYNU.
文摘We will prove that for 1<p<∞and 0<λ<n,the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<γ<+∞.When p=1 and 0<λ<n,it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mcγequals that of the centered Hardy-Littlewood maximal operator for all 0<λ<+∞.Moreover,the same results are true for the truncated uncentered Hardy-Littlewood maximal operator.Our work extends the previous results of Lebesgue spaces to Morrey spaces.
文摘In this paper, it is shown that Hardy-Hilbert＇s integral inequality with parameter is improved by means of a sharpening of Hoeder＇s inequality. A new inequality is established as follows：∫^∞α∫^∞α f（x）g（y）/（x＋y＋2β）dxdy〈π/sin（π/p）{∫^∞α f^p（x）dx}1/p·{∫^∞αgq（x）dx}1/q·（1-R）^m,where R=（Sp （F, h） - Sq （G, h））^2, m= min （1/p, 1/q）. As application; an extension of Hardy-Littlewood＇s inequality is given.
文摘In this paper, we study one class of n-dimensional Hardy-Steklov operators which has important applications in the technical analysis in equity markets. We establish their weighted boundedness and the corresponding operator norms on both LP（Rn） and BMO（Rn）.
基金Project supported by the National Natural Science Foundation of China (No.10171111, No.10371734)and the Foundation of Advanced Research Center, Zhongshan University.
文摘The authors prove the Hardy-Littlewood-Sobolev theorems for generalized fractional integrals L?α/2 for 0 < α < n/m, where L is a complex elliptic operator of arbitrary order 2m on Rn.
基金Project supported by the National Natural Science Foundation of China (No.10041004) and the ThansCentury naming Programme Foun
文摘Using the circle method and sieves, the author proves that a positive proportion of positive integers can be represented as the sum of four cubes of primes.
文摘. In this paper,the characterization of boundedness of Hardy-Littlewood maximal operators in Orlicz-Morrey spaces LΦφ（X,μ） of homogeneous type is founded.
文摘In this paper, we give the following dominated theorem: Let φ(g) ∈ L1(G//K),φε(t)=ε> 0, and the least radical decreasing dominatedfunction φ(t) = sup |φ(y)| ∈L1(G//K). If shtφ(t) is monotonically decreasingon (0, ∞), then for any f∈L1loc(G//K) , the following inequality holds:sup |φε * f(x)| ≤ Cmf(x),where mf(x) is the Hardy-Littlewood maximal function of f, and C = ||φ||1.An application of this dominated theorem is also given.
基金Project supported by the National Natural Science Foundation of China.
文摘Let r(n) denote the number of representations of n as the sums of 5 cubes and 3 biquadra-tes of natural numbers. Then for all sufficiently large n, one has r(n)(?)n17/12, which is the expected order of magnitude of r(n).
文摘We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type （1, 1） inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated with parallelotopes do not decrease with the dimension.
文摘In this note, we give a short proof for the DiPerna-Lions flows associated to ODEs following the method of Crippa and De Lellis [3]. More precisely, assume that [divb] ∈ Ll∞oc（Rd）, ｜b｜/（1 ＋ ｜x｜ log ｜x｜） ∈ L∞（Rd） and ｜ b｜ φ（｜ b｜） ∈ Ll1oc（Rd）, where φ（r） = log log（r ＋ c）, c 0. Then, there exists a unique regular Lagrangian flow associated with the ODE X˙（t, x） = b（X（t, x））, X（0, x） = x.
文摘Let q greater than or equal to 2,f is measurable function on R-n such that f(x)\x\(n(1-2/q)) is an element of L-q(R-n), then its Fourier transform (f) over cap can be defined and there exists a constant A(q) such that the inequality parallel to (f) over cap parallel to(q) less than or equal to A(q) parallel to f\.\(n(1-2/q))parallel to(q) holds. This is the Hardy-Littlewood Theorem. This paper considers the corresponding result for the Fourier-Bessel transform and Fourier-Jacobi transform. It is interesting that we can deal with these two cases in the same way, and the function corresponding to \x\(n) is tw(t), where w(t) is the weight, w(t) = t(2 alpha+1) for Fourier-Bessel transform, and w(t) = (2 sinh t)(2 alpha+1)(2 cosh t)(2 beta+1) for Fourier-Jacobi transform.
