This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of...This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameterγchanges,we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.展开更多
In the paper,we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Lit tiewood maximal function Mf and the centered Hardy-Lit tiewood maximal function Mcf on R^n...In the paper,we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Lit tiewood maximal function Mf and the centered Hardy-Lit tiewood maximal function Mcf on R^n.As two applications,we can easily deduce that Mcf and Mf are continuous if f is continuous,and Mf is continuous if f is of local bounded variation on R.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, the authors point out that the methods used by Li(2004, 2005,2007) can be applied to study maximal functions on weighted harmonic AN groups.
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
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
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
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
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
We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of in...We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.展开更多
基金Supported by NSF of Zhejiang Province of China(LQ18A010002,LQ17A010002)。
文摘This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameterγchanges,we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.
基金This paper is supported by NSF of Zhejiang Province of China(Grant No.LQ18A010002 and No.LQ17A010002)in part by National Natural Foundation of China(Grant Nos.11871452 and 12001488).
文摘In the paper,we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Lit tiewood maximal function Mf and the centered Hardy-Lit tiewood maximal function Mcf on R^n.As two applications,we can easily deduce that Mcf and Mf are continuous if f is continuous,and Mf is continuous if f is of local bounded variation on R.
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, the authors point out that the methods used by Li(2004, 2005,2007) can be applied to study maximal functions on weighted harmonic AN groups.
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.
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.
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We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.
