Abstract In this paper, we give counterexamples to show that variation and oscillation operators related to rough truncations of the Hilbert transform are not bounded from H1 to L1, and prove variation, oscillation an...Abstract In this paper, we give counterexamples to show that variation and oscillation operators related to rough truncations of the Hilbert transform are not bounded from H1 to L1, and prove variation, oscillation and A-jump operators related to smooth truncations are bounded from H1 to L1.展开更多
Let O（PτL be the oscillation of the Possion semigroup associated with the parabolic Hermite operator = t-△＋｜x｜2. We show that O（PτL） is bounded from LP （Rn＋1） into itself for 1 〈 p 〈 ∞, bounded from L1...Let O（PτL be the oscillation of the Possion semigroup associated with the parabolic Hermite operator = t-△＋｜x｜2. We show that O（PτL） is bounded from LP （Rn＋1） into itself for 1 〈 p 〈 ∞, bounded from L1（Rn＋l） into weak-L1（Rn＋1） and bounded from Lc∞（Rn＋1） into BMO（Rn＋1）. In the case p = ∞ we show that the range of the image of the operator O（PτL） is strictly smaller than the range of a general singular operator.展开更多
To shed some light on the John-Nirenberg space,the authors of this article introduce the John-Nirenberg-Q space via congruent cubes,JNQp,qα(Rn),which,when p=∞and q=2,coincides with the space Qα(Rn)introduced by Ess...To shed some light on the John-Nirenberg space,the authors of this article introduce the John-Nirenberg-Q space via congruent cubes,JNQp,qα(Rn),which,when p=∞and q=2,coincides with the space Qα(Rn)introduced by Essen,Janson,Peng and Xiao in[Indiana Univ Math J,2000,49(2):575-615].Moreover,the authors show that,for some particular indices,JNQp,qα(Rn)coincides with the congruent John-Nirenberg space,or that the(fractional)Sobolev space is continuously embedded into JNQp,qα(Rn).Furthermore,the authors characterize JNQp,qα(Rn)via mean oscillations,and then use this characterization to study the dyadic counterparts.Also,the authors obtain some properties of composition operators on such spaces.The main novelties of this article are twofold:establishing a general equivalence principle for a kind of’almost increasing’set function that is here introduced,and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.展开更多
Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.
In this paper, some sufficient and necessary conditions are established for the oscillatory of solutions for nonlinear functional difference equations, which extend and improve some corresponding results obtained and ...In this paper, some sufficient and necessary conditions are established for the oscillatory of solutions for nonlinear functional difference equations, which extend and improve some corresponding results obtained and are discrete analogues of the corresponding results for the continuous version.展开更多
In this paper,we first introduce the weak Lipschitz spaces WLip_(q,α),1<q<∞,0<α<1 which are the analog of weak Lebesgue spaces L^(q,∞)in the setting of Lipschitz space.We obtain the equivalence between...In this paper,we first introduce the weak Lipschitz spaces WLip_(q,α),1<q<∞,0<α<1 which are the analog of weak Lebesgue spaces L^(q,∞)in the setting of Lipschitz space.We obtain the equivalence between the norm ||·||_(Lipα)and ||·||_(()WLip_(q,α)).As an application,we show that the commutator M_(β)~b is bounded from L~p to L^(q,∞) for some p ∈(1,∞) and 1/p-1/q=(α+β)/n if and only if b is in Lip_(α).We also introduce the weak central bounded mean oscillation space WCBMO_(q,α) and give a characterization of WCBMO_(q,α) via the boundedness of the commutators of Hardy type operators.展开更多
Letλ〉0,and let the Bessel operator△λ:=-d2/dx2-2λ/x d/dx defined on R+:=(0,∞).We show that the oscillation andρ-variation operators of the Riesz transform R△λassociated with△λare bounded on BMO(R+,dmλ),wher...Letλ〉0,and let the Bessel operator△λ:=-d2/dx2-2λ/x d/dx defined on R+:=(0,∞).We show that the oscillation andρ-variation operators of the Riesz transform R△λassociated with△λare bounded on BMO(R+,dmλ),whereρ>2 and dmλ=x2λdx.Moreover,we construct a(1,∞)△λ-atom as a counterexample to show that the oscillation andρ-variation operators of R△λare not bounded from to L1(R+,dmλ).Finally,we prove that the oscillation and theρ-variation operators for the smooth truncations associated with Bessel operators R~△λare bounded from H1(R+:dmλ)to L1(R+,dmλ).展开更多
基金Supported by NSF of China(Grant Nos.11501169,11371057)
文摘Abstract In this paper, we give counterexamples to show that variation and oscillation operators related to rough truncations of the Hilbert transform are not bounded from H1 to L1, and prove variation, oscillation and A-jump operators related to smooth truncations are bounded from H1 to L1.
基金supported by National Natural Science Foundation of China(11471251 and 11671308)
文摘Let O（PτL be the oscillation of the Possion semigroup associated with the parabolic Hermite operator = t-△＋｜x｜2. We show that O（PτL） is bounded from LP （Rn＋1） into itself for 1 〈 p 〈 ∞, bounded from L1（Rn＋l） into weak-L1（Rn＋1） and bounded from Lc∞（Rn＋1） into BMO（Rn＋1）. In the case p = ∞ we show that the range of the image of the operator O（PτL） is strictly smaller than the range of a general singular operator.
基金partially supported by the National Natural Science Foundation of China(12122102 and 11871100)the National Key Research and Development Program of China(2020YFA0712900)。
文摘To shed some light on the John-Nirenberg space,the authors of this article introduce the John-Nirenberg-Q space via congruent cubes,JNQp,qα(Rn),which,when p=∞and q=2,coincides with the space Qα(Rn)introduced by Essen,Janson,Peng and Xiao in[Indiana Univ Math J,2000,49(2):575-615].Moreover,the authors show that,for some particular indices,JNQp,qα(Rn)coincides with the congruent John-Nirenberg space,or that the(fractional)Sobolev space is continuously embedded into JNQp,qα(Rn).Furthermore,the authors characterize JNQp,qα(Rn)via mean oscillations,and then use this characterization to study the dyadic counterparts.Also,the authors obtain some properties of composition operators on such spaces.The main novelties of this article are twofold:establishing a general equivalence principle for a kind of’almost increasing’set function that is here introduced,and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.
文摘Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.
基金Supported by the Nature Science Foundation of Jining(JB10)
文摘In this paper, some sufficient and necessary conditions are established for the oscillatory of solutions for nonlinear functional difference equations, which extend and improve some corresponding results obtained and are discrete analogues of the corresponding results for the continuous version.
基金Supported by the National Natural Science Foundation of China (Grant No.11871452)the Natural Science Foundation of Henan Province (Grant No.202300410338)the Nanhu Scholar Program for Young Scholars of Xinyang Normal University。
文摘In this paper,we first introduce the weak Lipschitz spaces WLip_(q,α),1<q<∞,0<α<1 which are the analog of weak Lebesgue spaces L^(q,∞)in the setting of Lipschitz space.We obtain the equivalence between the norm ||·||_(Lipα)and ||·||_(()WLip_(q,α)).As an application,we show that the commutator M_(β)~b is bounded from L~p to L^(q,∞) for some p ∈(1,∞) and 1/p-1/q=(α+β)/n if and only if b is in Lip_(α).We also introduce the weak central bounded mean oscillation space WCBMO_(q,α) and give a characterization of WCBMO_(q,α) via the boundedness of the commutators of Hardy type operators.
基金This work was supported in part by the Doctoral Scientific Research of Yili Normal University(No.2017YSBS09).
文摘Letλ〉0,and let the Bessel operator△λ:=-d2/dx2-2λ/x d/dx defined on R+:=(0,∞).We show that the oscillation andρ-variation operators of the Riesz transform R△λassociated with△λare bounded on BMO(R+,dmλ),whereρ>2 and dmλ=x2λdx.Moreover,we construct a(1,∞)△λ-atom as a counterexample to show that the oscillation andρ-variation operators of R△λare not bounded from to L1(R+,dmλ).Finally,we prove that the oscillation and theρ-variation operators for the smooth truncations associated with Bessel operators R~△λare bounded from H1(R+:dmλ)to L1(R+,dmλ).
