We study the periodic boundary value problems for nonlinear integro-differential equa- tions of Volterra type with Carathéodory functions. For two situations relative to lower and upper solutions α and β:αβ ...We study the periodic boundary value problems for nonlinear integro-differential equa- tions of Volterra type with Carathéodory functions. For two situations relative to lower and upper solutions α and β:αβ or β α, the existence of solutions and the monotone iterative method for establishing extreme solutions are considered.展开更多
In this paper we study a certain directional Hilbert transform and the boundedness on some mixed norm spaces. As one of applications, we prove the Lp-boundedness of the Littlewood-Paley operators with variable kernels...In this paper we study a certain directional Hilbert transform and the boundedness on some mixed norm spaces. As one of applications, we prove the Lp-boundedness of the Littlewood-Paley operators with variable kernels. Our results are extensions of some known theorems.展开更多
Abstract In this paper, we give a similarity classification for the multiplication operator Mg on the Sobolev disk algebra R（D） with g analytic on the closure of the unit disk D.
Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ）...Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.展开更多
In this paper the regularity of the Lagrangians f（x,ζ）=|ζ|<sup>α（x）</sup> （1【α<sub>1</sub> α（x） α<sub>2</sub>【 +∞） is studied. Our main result: If α（x） is Holder...In this paper the regularity of the Lagrangians f（x,ζ）=|ζ|<sup>α（x）</sup> （1【α<sub>1</sub> α（x） α<sub>2</sub>【 +∞） is studied. Our main result: If α（x） is Holder continuous, then the Lagrangian f（x,ζ）=|ζ|<sup>α（x）</sup> is regular. This result gives a negative answer to a conjecture of V. Zhikov.展开更多
In this paper, we introduce and study the Sobolev spaces of exponential type associated with the Weinstein operator, via some elements of harmonic analysis related to this operator. In particular, some properties, inc...In this paper, we introduce and study the Sobolev spaces of exponential type associated with the Weinstein operator, via some elements of harmonic analysis related to this operator. In particular, some properties, including completeness and imbedding theorem, are proved. Finally, using the theory of reproducing kernels, some applications are given for these spaces.展开更多
We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（...We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.展开更多
Civen two doubling measures μ and v in a metric apace (S.p)of homogeneous type. let B_0 S be a given ball. It has been a well-known result bv now (see)[1 4])theat the validity of an L^1→L^1 Poincaré inequality ...Civen two doubling measures μ and v in a metric apace (S.p)of homogeneous type. let B_0 S be a given ball. It has been a well-known result bv now (see)[1 4])theat the validity of an L^1→L^1 Poincaré inequality of the following form: f_B|f-f_B|dv≤cr(B)f_Bgdμ. for all metric balls B B_0 S, implies a variant of representation formula of fractonal integral type: |f(x)-f_(B(11))|≤C integral from n=B_(11) g(y)p(x, y)/μ(B(x, p(x, y)))dμ(y)+C(r(B_0))/(μ(B_0))integral from n=B_0 g(y)dμ(y). One of the main results of this paper shows that an L^1 to L^q Poincaré inequality for some 01, i.e.. (f_B|f-f_B|~q dv)^(1/q)≤cr(B) f_B gdμ, for all metric balls B B_0. will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition. sup_(λ>0)(λv({x ∈ B:|f(x)-f_B|>λ}))/v(B)≤Gr (B)f_B gdμ. also implies the same formula. Analogous theorems related to high-order Poincaréinequalities and Sobolev spaces in metric spaces are also proved.展开更多
In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general ...In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general results,we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G.Given any set E(?)G,B_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)G of Hausdorff dimension Q-d,where 0<d<Q,B_(α,p)(E)=0 holds if and only if αp(?)d. A version of O.Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4).展开更多
The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S-(R) and some subspaces called energy spaces first defined in [2] and [3]. The main...The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S-(R) and some subspaces called energy spaces first defined in [2] and [3]. The main idea is to look for solutions of a given linear PDE in those subspaces. Here, this work extends previous developments in S-(Rm)?(m∈Z+) using the theory of Sobolev spaces. Furthermore, we also define the concept of Energy Parallax, which is the inclusion of additional solutions when varying the energy of a predefined system locally by taking into account additional smaller quantities. We show that it is equivalent to take into account solutions in other energy subspaces. To illustrate the theory, one of our examples is based on the variation of Electro Magnetic (EM) energy density within the skin depth of a conductive material, leading to take into account derivatives of EM evanescent waves, particular solutions of the wave equation. The last example is the derivation of the Woodward effect [4] with the variations of the EM energy density under strict assumptions in general relativity. It finally leads to a theoretical definition of an electromagnetic and gravitational (EMG) coupling.展开更多
基金Project supported by the National Natural Science Foundation of China
文摘We study the periodic boundary value problems for nonlinear integro-differential equa- tions of Volterra type with Carathéodory functions. For two situations relative to lower and upper solutions α and β:αβ or β α, the existence of solutions and the monotone iterative method for establishing extreme solutions are considered.
基金supported by the National Natural Science Foundation of China (61071189)Innovation Scientists and Technicians Troop Construction of Henan Province of China (084100510012)the Natural Science Foundation for the Education Department of Henan Province of China (2008B510001)
文摘In this paper, a characterization of orthonormal wavelet families in Sobolev spaces H s （R） is established.
基金supported by the fund of the 973 Project,the National Natural Science Foundation of China(Grant Nos.10571156,10571015&10371043)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20050027025).
文摘In this paper we study a certain directional Hilbert transform and the boundedness on some mixed norm spaces. As one of applications, we prove the Lp-boundedness of the Littlewood-Paley operators with variable kernels. Our results are extensions of some known theorems.
基金Supported by National Natural Science Foundation of China(Grant No.10901046)Foundation for the Author of National Excellent Doctoral Dissertation of China(Grant No.201116)
文摘Abstract In this paper, we give a similarity classification for the multiplication operator Mg on the Sobolev disk algebra R（D） with g analytic on the closure of the unit disk D.
基金The author is partially supported by NNSF(10271015) RFDP(20020027004)of China
文摘Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.
基金Supported by the National Natural Science Foundation of China.
文摘In this paper the regularity of the Lagrangians f（x,ζ）=|ζ|<sup>α（x）</sup> （1【α<sub>1</sub> α（x） α<sub>2</sub>【 +∞） is studied. Our main result: If α（x） is Holder continuous, then the Lagrangian f（x,ζ）=|ζ|<sup>α（x）</sup> is regular. This result gives a negative answer to a conjecture of V. Zhikov.
文摘In this paper, we introduce and study the Sobolev spaces of exponential type associated with the Weinstein operator, via some elements of harmonic analysis related to this operator. In particular, some properties, including completeness and imbedding theorem, are proved. Finally, using the theory of reproducing kernels, some applications are given for these spaces.
文摘We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.
基金The first author is supported partly by the U.S. National Science Foundation Grant Nos. DMS96-22996 and DMS99-70352.supported partly by DGICYT Grant PB940192. Spainsupported partly by NATO Collaborative Research G
文摘Civen two doubling measures μ and v in a metric apace (S.p)of homogeneous type. let B_0 S be a given ball. It has been a well-known result bv now (see)[1 4])theat the validity of an L^1→L^1 Poincaré inequality of the following form: f_B|f-f_B|dv≤cr(B)f_Bgdμ. for all metric balls B B_0 S, implies a variant of representation formula of fractonal integral type: |f(x)-f_(B(11))|≤C integral from n=B_(11) g(y)p(x, y)/μ(B(x, p(x, y)))dμ(y)+C(r(B_0))/(μ(B_0))integral from n=B_0 g(y)dμ(y). One of the main results of this paper shows that an L^1 to L^q Poincaré inequality for some 01, i.e.. (f_B|f-f_B|~q dv)^(1/q)≤cr(B) f_B gdμ, for all metric balls B B_0. will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition. sup_(λ>0)(λv({x ∈ B:|f(x)-f_B|>λ}))/v(B)≤Gr (B)f_B gdμ. also implies the same formula. Analogous theorems related to high-order Poincaréinequalities and Sobolev spaces in metric spaces are also proved.
基金Research supportde partly by the U.S.National Science Foundation Grant No.DMS99-70352
文摘In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general results,we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G.Given any set E(?)G,B_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)G of Hausdorff dimension Q-d,where 0<d<Q,B_(α,p)(E)=0 holds if and only if αp(?)d. A version of O.Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4).
文摘The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S-(R) and some subspaces called energy spaces first defined in [2] and [3]. The main idea is to look for solutions of a given linear PDE in those subspaces. Here, this work extends previous developments in S-(Rm)?(m∈Z+) using the theory of Sobolev spaces. Furthermore, we also define the concept of Energy Parallax, which is the inclusion of additional solutions when varying the energy of a predefined system locally by taking into account additional smaller quantities. We show that it is equivalent to take into account solutions in other energy subspaces. To illustrate the theory, one of our examples is based on the variation of Electro Magnetic (EM) energy density within the skin depth of a conductive material, leading to take into account derivatives of EM evanescent waves, particular solutions of the wave equation. The last example is the derivation of the Woodward effect [4] with the variations of the EM energy density under strict assumptions in general relativity. It finally leads to a theoretical definition of an electromagnetic and gravitational (EMG) coupling.
