This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary...This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.展开更多
In this paper, we obtain the boundedness of the fractional integral operators, the bilineax fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.
In the present paper,we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral.We use the expansion formula to obtain approximations for the fractional integral of orders...In the present paper,we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral.We use the expansion formula to obtain approximations for the fractional integral of ordersα,1+α,2+α,3+αand 4+α.The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.展开更多
In this paper the boundedness for the multilinear fractional integral operator Iα^（m） on the product of Herz spaces and Herz-Morrey spaces are founded, which improves the Hardy- Littlewood-Sobolev inequality for cl...In this paper the boundedness for the multilinear fractional integral operator Iα^（m） on the product of Herz spaces and Herz-Morrey spaces are founded, which improves the Hardy- Littlewood-Sobolev inequality for classical fractional integral Iα. The method given in the note is useful for more general multilinear integral operators.展开更多
In this paper, the authors get the Coifman type weighted estimates and weak weighted LlogL estimates for vector-valued generalized commutators of multilinear fractional integral with w ∈ A∞. Furthermore, both the bo...In this paper, the authors get the Coifman type weighted estimates and weak weighted LlogL estimates for vector-valued generalized commutators of multilinear fractional integral with w ∈ A∞. Furthermore, both the boundedness of vector-valued multilinear frac- tional integral and the weak weighted LlogL estimates for vector-valued multilinear fractional integral are also obtained.展开更多
Let L be the infinitesimal generator of an analytic semigroup on L^2 （R^n） with Gaussian kernel bound, and let L^-α/2 be the fractional integrals of L for 0 〈 α 〈 n. In this paper, we will obtain some boundedn...Let L be the infinitesimal generator of an analytic semigroup on L^2 （R^n） with Gaussian kernel bound, and let L^-α/2 be the fractional integrals of L for 0 〈 α 〈 n. In this paper, we will obtain some boundedness properties of commutators [b, L^-α/2] on weighted Morrey spaces L^p,k（w） when the symbol b belongs to BMO（Rn） or the homogeneous Lipschitz space.展开更多
Let ∠= -△Hn＋ V be the Schrdinger operator on the Heisenberg groups Hn,where V is a nonnegative function satisfying the reverse Hlder inequality. In this article, the author obtains the BMO_∠ and BLO_∠ estimates o...Let ∠= -△Hn＋ V be the Schrdinger operator on the Heisenberg groups Hn,where V is a nonnegative function satisfying the reverse Hlder inequality. In this article, the author obtains the BMO_∠ and BLO_∠ estimates of the fractional integrals associated to ∠.展开更多
In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional ...In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.展开更多
The fractional integral operators with variable kernels are discussed.It is proved that if the kernel satisfies the Dini-condition,then the fractional integral operators with variable kernels are bounded from Hp(Rn) i...The fractional integral operators with variable kernels are discussed.It is proved that if the kernel satisfies the Dini-condition,then the fractional integral operators with variable kernels are bounded from Hp(Rn) into Lq(Rn) when 0<p≤1 and 1/q=1/p-α/n.The results in this paper improve the results obtained by Ding,Chen and Fan in 2002.展开更多
In this paper,the authors introduce the central bounded oscillation space CBMO q (R n),let [b,T,α ] be the commutator generated by fractional integral operators with variable kernels and CBMO function,we establish th...In this paper,the authors introduce the central bounded oscillation space CBMO q (R n),let [b,T,α ] be the commutator generated by fractional integral operators with variable kernels and CBMO function,we establish the boundedness of [b,T,α ] on homogeneous Morrey-Herz spaces.展开更多
Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-t...Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.展开更多
For 0 〈 α 〈 mn and nonnegative integers n ≥ 2, m≥ 1, the multilinear fractional integral is defined bywhere →y= （y1, Y2,…, ym） and 7 denotes the m-tuple （f1, f2,…, fm）. In this note, the one- weighted and ...For 0 〈 α 〈 mn and nonnegative integers n ≥ 2, m≥ 1, the multilinear fractional integral is defined bywhere →y= （y1, Y2,…, ym） and 7 denotes the m-tuple （f1, f2,…, fm）. In this note, the one- weighted and two-weighted boundedness on Lp （JRn） space for multilinear fractional integral operator I（am） and the fractional multi-sublinear maximal operator Mα（m） are established re- spectively. The authors also obtain two-weighted weak type estimate for the operator Mα（m）.展开更多
Let μ be a Borel measure on R^d which may be non doubling. The only condition that μ must satisfy is μ（Q） ≤ col（Q）^n for any cube Q belong to R^d with sides parallel to the coordinate axes and for some fixed ...Let μ be a Borel measure on R^d which may be non doubling. The only condition that μ must satisfy is μ（Q） ≤ col（Q）^n for any cube Q belong to R^d with sides parallel to the coordinate axes and for some fixed n with 0 〈 n ≤ d. The purpose of this paper is to obtain a boundedness property of fractional integrals in Hardy spaces H^1（μ）.展开更多
In this paper, we will discuss the behavior of a class of rough fractional integral operators on variable exponent Lebesgue spaces,and establish their boundedness from Lp1 （＇） （Rn） to Lp2（） （Rn）.
In this paper, we will establish the boundedness of the commutator generated by fractional integral operator and RBMO（μ） function on generalized Morrey space in the non-homogeneous space.
In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boun...In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.展开更多
In this paper, the authors establish the(L^p(μ), L^q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 < p < 1/(α + β...In this paper, the authors establish the(L^p(μ), L^q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 < p < 1/(α + β) and 1/q = 1/p-(α + β), and obtain their weak(L^1(μ), L^(1/(1-α-β))(μ))-type. Moreover, the authors also consider the boundedness in the case that 1/(α+β) < p < 1/α,1/α≤ p ≤∞ and the endpoint cases, namely, p = 1/(α + β).展开更多
Considering a class of operators which include fractional integrals related to operators with Gaussian kernel bounds, the fractional integral operators with rough kernels and fractional maximal operators with rough ke...Considering a class of operators which include fractional integrals related to operators with Gaussian kernel bounds, the fractional integral operators with rough kernels and fractional maximal operators with rough kernels as special cases, we prove that if these operators are bounded on weighted Lebesgue spaces and satisfy some local pointwise control, then these operators and the commutators of these operators with a BMO functions are also bounded on generalized weighted Morrey spaces.展开更多
In this paper, we study the boundedness of higher order commutators of gen- eralized fractional integral operators on weighted Lp spaces and Herz-type Hardy spaces.
文摘This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.
基金Supported by the National Natural Science Foundation of China(11271330)
文摘In this paper, we obtain the boundedness of the fractional integral operators, the bilineax fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.
基金This work was supported by the Bulgarian National Science Fund under Project KP-06-M32/2"Advanced Stochastic and Deterministic Approaches for Large-Scale Problems of Computational Mathematics"。
文摘In the present paper,we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral.We use the expansion formula to obtain approximations for the fractional integral of ordersα,1+α,2+α,3+αand 4+α.The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.
基金Supported by the National Natural Sciences Foundation of China (10771110)the Natural Science Founda- tion of Ningbo City (2006A610090)
文摘In this paper the boundedness for the multilinear fractional integral operator Iα^（m） on the product of Herz spaces and Herz-Morrey spaces are founded, which improves the Hardy- Littlewood-Sobolev inequality for classical fractional integral Iα. The method given in the note is useful for more general multilinear integral operators.
基金Supported by the National Natural Science Foundation of China(11271330,11226104,11226108)the Jiangxi Natural Science Foundation of China(20114BAB211007)the Science Foundation of Jiangxi Education Department(GJJ13703)
文摘In this paper, the authors get the Coifman type weighted estimates and weak weighted LlogL estimates for vector-valued generalized commutators of multilinear fractional integral with w ∈ A∞. Furthermore, both the boundedness of vector-valued multilinear frac- tional integral and the weak weighted LlogL estimates for vector-valued multilinear fractional integral are also obtained.
文摘Let L be the infinitesimal generator of an analytic semigroup on L^2 （R^n） with Gaussian kernel bound, and let L^-α/2 be the fractional integrals of L for 0 〈 α 〈 n. In this paper, we will obtain some boundedness properties of commutators [b, L^-α/2] on weighted Morrey spaces L^p,k（w） when the symbol b belongs to BMO（Rn） or the homogeneous Lipschitz space.
文摘Let ∠= -△Hn＋ V be the Schrdinger operator on the Heisenberg groups Hn,where V is a nonnegative function satisfying the reverse Hlder inequality. In this article, the author obtains the BMO_∠ and BLO_∠ estimates of the fractional integrals associated to ∠.
文摘In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.
基金Supported by the973Project( G1 9990 75 1 0 5 ) and the National Natural Science Foundation of China( 1 0 2 71 0 1 6)
文摘The fractional integral operators with variable kernels are discussed.It is proved that if the kernel satisfies the Dini-condition,then the fractional integral operators with variable kernels are bounded from Hp(Rn) into Lq(Rn) when 0<p≤1 and 1/q=1/p-α/n.The results in this paper improve the results obtained by Ding,Chen and Fan in 2002.
基金Supported by the Anhui Polytechnic University Foundation for Recruiting Talent(2011YQQ004)Supported by the Provincial Natural Science Research Project of Anhui Colleges(KJ2011A032)+1 种基金Supported by the Young Teachers Program of Anhui Province(2006jql042)Supported by the Grant for Younth of Anhui Polytechnic University (2010YQ047)
文摘In this paper,the authors introduce the central bounded oscillation space CBMO q (R n),let [b,T,α ] be the commutator generated by fractional integral operators with variable kernels and CBMO function,we establish the boundedness of [b,T,α ] on homogeneous Morrey-Herz spaces.
基金Supported Partially by NSF of China (10371087) Education Committee of Anhui Province (2003kj034zd).
文摘Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.
基金the NNSF of China under Grant#10771110NSF of Ningbo City under Grant#2006A610090
文摘For 0 〈 α 〈 mn and nonnegative integers n ≥ 2, m≥ 1, the multilinear fractional integral is defined bywhere →y= （y1, Y2,…, ym） and 7 denotes the m-tuple （f1, f2,…, fm）. In this note, the one- weighted and two-weighted boundedness on Lp （JRn） space for multilinear fractional integral operator I（am） and the fractional multi-sublinear maximal operator Mα（m） are established re- spectively. The authors also obtain two-weighted weak type estimate for the operator Mα（m）.
文摘Let μ be a Borel measure on R^d which may be non doubling. The only condition that μ must satisfy is μ（Q） ≤ col（Q）^n for any cube Q belong to R^d with sides parallel to the coordinate axes and for some fixed n with 0 〈 n ≤ d. The purpose of this paper is to obtain a boundedness property of fractional integrals in Hardy spaces H^1（μ）.
基金Supported by the NSF of Zhejiang Province (Y6090681)the Education Dept.of Zhejiang Province(Y201120509)
文摘In this paper, we will discuss the behavior of a class of rough fractional integral operators on variable exponent Lebesgue spaces,and establish their boundedness from Lp1 （＇） （Rn） to Lp2（） （Rn）.
基金Supported by the NSF of Education Committee of Anhui Province (KJ2011A138)
文摘In this paper, we will establish the boundedness of the commutator generated by fractional integral operator and RBMO（μ） function on generalized Morrey space in the non-homogeneous space.
文摘In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.
基金Supported by the National Natural Science Foundation of China(Grant No.11661075).
文摘In this paper, the authors establish the(L^p(μ), L^q(μ))-type estimate for fractional commutator generated by fractional integral operators Tα with Lipschitz functions(b ∈ Lipβ(μ)),where 1 < p < 1/(α + β) and 1/q = 1/p-(α + β), and obtain their weak(L^1(μ), L^(1/(1-α-β))(μ))-type. Moreover, the authors also consider the boundedness in the case that 1/(α+β) < p < 1/α,1/α≤ p ≤∞ and the endpoint cases, namely, p = 1/(α + β).
文摘Considering a class of operators which include fractional integrals related to operators with Gaussian kernel bounds, the fractional integral operators with rough kernels and fractional maximal operators with rough kernels as special cases, we prove that if these operators are bounded on weighted Lebesgue spaces and satisfy some local pointwise control, then these operators and the commutators of these operators with a BMO functions are also bounded on generalized weighted Morrey spaces.
基金Supported by the NSF of China (10371087)NSF of Anhui Province (07021019)+2 种基金Education Committee ofAnhui Province (KJ2007A009Kj2008B244)the Grant for Younth of Anhui Normal University (2009xqn58)
文摘In this paper, we study the boundedness of higher order commutators of gen- eralized fractional integral operators on weighted Lp spaces and Herz-type Hardy spaces.
