In this paper, under natural regularity assumptions on the exponent function, we prove some boundedness results for the functions of Littlewood-Paley, Lusin and Marcinkiewicz on a new class of generalized Herz-Morrey ...In this paper, under natural regularity assumptions on the exponent function, we prove some boundedness results for the functions of Littlewood-Paley, Lusin and Marcinkiewicz on a new class of generalized Herz-Morrey spaces with weight and variable exponent, which essentially extend some known results.展开更多
Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper,...Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator D^γ on the weighted Morrey spaces.展开更多
In this paper,we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions.Then we obtain decomposition characterizations of these spaces by atom,molecule and wavele...In this paper,we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions.Then we obtain decomposition characterizations of these spaces by atom,molecule and wavelet.As an application,we obtain the boundedness of the pseudo-differential operators on these spaces.展开更多
We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.
In [4], a new family W（L^p（x）, Lm^q） of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L^p（x） （R） and the global...In [4], a new family W（L^p（x）, Lm^q） of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L^p（x） （R） and the global component is a weighted Lebesgue space Lm^q （R）. This present paper is a sequel to our work [4]. In Section 2, we discuss necessary and sufficient conditions for the equality W （L^p（x）, Lm^q） = L^q （R）. Later we give some characterization of Wiener amalgam space W （L^p（x）, Lm^q）.In Section 3 we define the Wiener amalgam space W （FL^p（x）, Lm^q） and investigate some properties of this space, where FL^p（x） is the image of L^p（x） under the Fourier transform. In Section 4, we discuss boundedness of the Hardy- Littlewood maximal operator between some Wiener amalgam spaces.展开更多
In this paper,the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized w...In this paper,the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces.And the corresponding commutators generated by BMO function are also considered.展开更多
In this paper,the authors obtain the boundedness of vector valued bilinear Calderón-Zygmund operators on products of weighted Herz-Morrey spaces with variable exponents.
After studying in a previous work the smoothness of the space where dr- measF0 〉 O, with p（.） E C（~） and p（x） 〉 1 for all x E , the authors study in this paper the strict and uniform convexity as well as some ...After studying in a previous work the smoothness of the space where dr- measF0 〉 O, with p（.） E C（~） and p（x） 〉 1 for all x E , the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space. The results obtained in this direction are used for proving existence results for operator equations having the form Ju = Niu, where J is a duality mapping on Uro corresponding to the gauge function ~, and Nf is the Nemytskij operator generated by a Caratheodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from Ur0 into its dual.展开更多
Using the theory of weighted Sobolev spaces with variable exponent and the <em>L</em><sup>1</sup>-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Di...Using the theory of weighted Sobolev spaces with variable exponent and the <em>L</em><sup>1</sup>-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.展开更多
We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the lo...We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.展开更多
In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebe...In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander＇s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p（x）-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander＇s condition, but they also hold for Grushin vector fields as well with obvious modifications.展开更多
In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H...In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].展开更多
Let Ω be a domain in RN. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space Wl,P（ ）（Ω）, where p（.） ： Ω → [1, ∞[ is a variable exponent. This inequality is...Let Ω be a domain in RN. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space Wl,P（ ）（Ω）, where p（.） ： Ω → [1, ∞[ is a variable exponent. This inequality is itself a corollary to a more general result about equivalent norms over such cones. The approach in this paper avoids the difficulty arising from the possible lack of density of the space ;D（Ω） in the space {v ∈ Wl,P（ ）（Ω）; tr v = 0 on δΩ}. Two applications are also discussed.展开更多
文摘In this paper, under natural regularity assumptions on the exponent function, we prove some boundedness results for the functions of Littlewood-Paley, Lusin and Marcinkiewicz on a new class of generalized Herz-Morrey spaces with weight and variable exponent, which essentially extend some known results.
基金supported by NSF of China (Grant No. 11471033)NCET of China (Grant No. NCET-11-0574)the Fundamental Research Funds for the Central Universities (FRF-TP-12-006B)
文摘Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator D^γ on the weighted Morrey spaces.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12161022 and 12061030)Hainan Provincial Natural Science Foundation of China(Grant No.122RC652)the Science and Technology Project of Guangxi(Grant No.Guike AD23023002)。
文摘In this paper,we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions.Then we obtain decomposition characterizations of these spaces by atom,molecule and wavelet.As an application,we obtain the boundedness of the pseudo-differential operators on these spaces.
文摘We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.
文摘In [4], a new family W（L^p（x）, Lm^q） of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L^p（x） （R） and the global component is a weighted Lebesgue space Lm^q （R）. This present paper is a sequel to our work [4]. In Section 2, we discuss necessary and sufficient conditions for the equality W （L^p（x）, Lm^q） = L^q （R）. Later we give some characterization of Wiener amalgam space W （L^p（x）, Lm^q）.In Section 3 we define the Wiener amalgam space W （FL^p（x）, Lm^q） and investigate some properties of this space, where FL^p（x） is the image of L^p（x） under the Fourier transform. In Section 4, we discuss boundedness of the Hardy- Littlewood maximal operator between some Wiener amalgam spaces.
基金supported by the National Natural Science Foundation of China(No.11561062)Natural Science Foundation of Gansu Province(21JR1RM337).
文摘In this paper,the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces.And the corresponding commutators generated by BMO function are also considered.
基金supported by the National Natural Science Foundation of China(Nos.11761026)Guangxi Natural Science Foundation(No.2020GXNSFAA159085)。
文摘In this paper,the authors obtain the boundedness of vector valued bilinear Calderón-Zygmund operators on products of weighted Herz-Morrey spaces with variable exponents.
文摘After studying in a previous work the smoothness of the space where dr- measF0 〉 O, with p（.） E C（~） and p（x） 〉 1 for all x E , the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space. The results obtained in this direction are used for proving existence results for operator equations having the form Ju = Niu, where J is a duality mapping on Uro corresponding to the gauge function ~, and Nf is the Nemytskij operator generated by a Caratheodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from Ur0 into its dual.
文摘Using the theory of weighted Sobolev spaces with variable exponent and the <em>L</em><sup>1</sup>-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.
文摘We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.
基金supported by NSFC(Grant No.11371056)supported by a US NSF grant
文摘In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander＇s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p（x）-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander＇s condition, but they also hold for Grushin vector fields as well with obvious modifications.
基金Natural Science Foundation of China(11771166)Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University#IRT17R46.
文摘In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].
文摘Let Ω be a domain in RN. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space Wl,P（ ）（Ω）, where p（.） ： Ω → [1, ∞[ is a variable exponent. This inequality is itself a corollary to a more general result about equivalent norms over such cones. The approach in this paper avoids the difficulty arising from the possible lack of density of the space ;D（Ω） in the space {v ∈ Wl,P（ ）（Ω）; tr v = 0 on δΩ}. Two applications are also discussed.
