In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,Ф along a surface Г = {x = Ф(|y|)y/|y|)} on the Triebel-Lizorkin space Fq,q^α(R^n ) for Ω belonging to H1 (Sn-...In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,Ф along a surface Г = {x = Ф(|y|)y/|y|)} on the Triebel-Lizorkin space Fq,q^α(R^n ) for Ω belonging to H1 (Sn-1) and some class WFα(S^n-1), which relates to Grafakos-Stefanov class. Some previous results are extended and improved.展开更多
In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan’s blowup and Laza’s GIT construction. More precisely, we obtain the Betti numbers of Kirwan’s...In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan’s blowup and Laza’s GIT construction. More precisely, we obtain the Betti numbers of Kirwan’s resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.展开更多
We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R^n) for ? ∈ H^1(^(Sn-1)) and ? ∈ Llog^+L(S^(n-1)) ∪_1<q<...We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R^n) for ? ∈ H^1(^(Sn-1)) and ? ∈ Llog^+L(S^(n-1)) ∪_1<q<∞(B^((0,0))_q(S^(n-1))), respectively.展开更多
Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e^(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)...Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e^(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ^(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i^2=1ω_i^(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^(p1)(ω_1) × L^(p2)(ω_2) to L^p(ν_ω) if p0 < p1, p2 < ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 > 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^(p1)(ω_1) × L^(p2)(ω_2) to L^(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.展开更多
Let Ω be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let Th,Ω,φ be the classical singular Radon transform, and let Th^ε,...Let Ω be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let Th,Ω,φ be the classical singular Radon transform, and let Th^ε,Ω,φ be its truncated operator with rough kernels associated to polynomial mapping which is defined by Th^ε,Ω,φf(x)=|f|y|>εf(x-φ(y))h(|y|)Ω(y)|y|^-ndy|.In this paper, we show that for any a ∈(-∞,∞) and (p, q) satisfying certain index condition, the operator Th^ε,Ω,φ enjoys the following convergence properties lim ε→0||Th^ε,Ω,φf-Th,Ω,φf||Fα^p,q(R^d)= 0 and limε→0||Th^ε,Ω,φf-Th,Ω,φf||Bα^p,q(R^d)=0, provided that Ω∈L(log+ L)β(S^n-1) for some β∈(0,1], or Ω∈H^1(S^n-1), or Ω∈(U1<q<∞Bq^(0,0)(S^n-1)).展开更多
In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and contin...In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.展开更多
基金partially supported by Grant-in-Aid for Scientific Research(C)(No.23540228),Japan Society for the Promotion of Science
文摘In the present paper, we consider the boundedness of Marcinkiewicz integral operator μΩ,h,Ф along a surface Г = {x = Ф(|y|)y/|y|)} on the Triebel-Lizorkin space Fq,q^α(R^n ) for Ω belonging to H1 (Sn-1) and some class WFα(S^n-1), which relates to Grafakos-Stefanov class. Some previous results are extended and improved.
基金Supported by NSFC grants for General Program (Grant No. 11771086)Key Program (Grant No. 11731004)National Key Research and Development Program of China (Grant No. 2020YFA0713200)。
文摘In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan’s blowup and Laza’s GIT construction. More precisely, we obtain the Betti numbers of Kirwan’s resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371057, 11471033 and 11571160)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20130003110003)+2 种基金the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10)Grant-in-Aid for Scientific Research (C) (Grant No. 23540228)Japan Society for the Promotion of Science
文摘We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R^n) for ? ∈ H^1(^(Sn-1)) and ? ∈ Llog^+L(S^(n-1)) ∪_1<q<∞(B^((0,0))_q(S^(n-1))), respectively.
基金supported by National Natural Science Foundation of China (Grant Nos. 11401175, 11501169 and 11471041)the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10)+2 种基金Program for New Century Excellent Talents in University (Grant No. NCET-13-0065)Grantin-Aid for Scientific Research (C) (Grant No. 15K04942)Japan Society for the Promotion of Science
文摘Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e^(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ^(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i^2=1ω_i^(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^(p1)(ω_1) × L^(p2)(ω_2) to L^p(ν_ω) if p0 < p1, p2 < ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 > 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^(p1)(ω_1) × L^(p2)(ω_2) to L^(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.
文摘Let Ω be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let Th,Ω,φ be the classical singular Radon transform, and let Th^ε,Ω,φ be its truncated operator with rough kernels associated to polynomial mapping which is defined by Th^ε,Ω,φf(x)=|f|y|>εf(x-φ(y))h(|y|)Ω(y)|y|^-ndy|.In this paper, we show that for any a ∈(-∞,∞) and (p, q) satisfying certain index condition, the operator Th^ε,Ω,φ enjoys the following convergence properties lim ε→0||Th^ε,Ω,φf-Th,Ω,φf||Fα^p,q(R^d)= 0 and limε→0||Th^ε,Ω,φf-Th,Ω,φf||Bα^p,q(R^d)=0, provided that Ω∈L(log+ L)β(S^n-1) for some β∈(0,1], or Ω∈H^1(S^n-1), or Ω∈(U1<q<∞Bq^(0,0)(S^n-1)).
基金supported by National Natural Science Foundation of China(Grant No.11701333)Support Program for Outstanding Young Scientific and Technological Top-Notch Talents of College of Mathematics and Systems Science(Grant No.Sxy2016K01)+3 种基金supported by National Natural Science Foundation of China(Grant Nos.11471041 and 11671039)National Natural Science Foundation of China-Deutsche Forschungsgemeinschaft(Grant No.11761131002)supported by Grant-in-Aid for Scientific Research(C)(Grant No.15K04942)Japan Society for the Promotion of Science。
文摘In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov class.The boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.