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Let q be a positive integer. Recently, Niu and Liu proved that, if , then the product is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and , the product is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer such that, for any positive integer , the product is not a powerful number. 相似文献
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Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on R~n. Let H_A~(p,q )(R~n) be the anisotropic Hardy-Lorentz spaces associated with A defined via the nontangential grand maximal function. In this article, the authors characterize H_A~(p,q )(R~n) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley g~*_λ-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space L_(p,q)(R~n). All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on R~n. Moreover, the range of λ in the g~*_λ-function characterization of H_A~(p,q )(R~n) coincides with the best known one in the classical Hardy space H~p(R~n) or in the anisotropic Hardy space H_A~p (R~n). 相似文献
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In this article, we obtain the sharp bounds from LP to the space wLP for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from LP to the space LPI are obtained. 相似文献
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Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let be a consistent theory. Then there is a complete theory extending T such that T is an open core of , but every model of interprets a model of . If is NIP, can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion. 相似文献
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Quốc Anh Ngô 《Comptes Rendus Mathematique》2017,355(5):526-532
In this note, we mainly study the relation between the sign of and in with and for . Given the differential inequality , first we provide several sufficient conditions so that holds. Then we provide conditions such that for all , which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to and with in . 相似文献
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If for a vector space V of dimension g over a characteristic zero field we denote by its alternating powers, and by its linear dual, then there are natural Poincaré isomorphisms: We describe an analogous result for objects in rigid pseudo-abelian -linear ACU tensor categories. 相似文献
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Gianfranco Casnati Daniele Faenzi Francesco Malaspina 《Journal of Pure and Applied Algebra》2018,222(3):585-609
In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology modules on the image of the Segre embedding and its general hyperplane sections.Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers 1 and 3 and dimension at least 3. 相似文献
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Let . Let and be open bounded connected subsets of containing the origin. Let be such that contains the closure of for all . Then, for a fixed we consider a Dirichlet problem for the Laplace operator in the perforated domain . We denote by the corresponding solution. If and , then we know that under suitable regularity assumptions there exist and a real analytic operator from to such that for all . Thus it is natural to ask what happens to the equality for ? negative. We show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality for ? negative depends on the parity of the dimension n. 相似文献
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Pierre Deligne 《Comptes Rendus Mathematique》2018,356(7):717-719
Let G be a reductive group over a field k of characteristic . For and , let be deduced from G by the extension of scalars . If k is perfect, this keeps making sense for . We show that, if k is perfect, there exists such that the algebraic groups G and over k are isomorphic. The isomorphism class of , as a reductive group over k, then depends only on n modulo m. For k not necessarily perfect, we show that such a periodicity remains true for n large enough. 相似文献
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M. Burak Erdoğan Michael Goldberg William R. Green 《Journal of Functional Analysis》2018,274(7):2139-2161
Let be a Schrödinger operator on with real-valued potential V, and let . If V has sufficient pointwise decay, the wave operators are known to be bounded on for all if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on for . This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents p. 相似文献
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Let and Ω be a bounded Lipschitz domain in . Assume that and the function is non-negative, where ?Ω denotes the boundary of Ω. Denote by ν the outward unit normal to ?Ω. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation in Ω with boundary data , respectively, in terms of a weak reverse Hölder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Ω with boundary data in (weighted) for any given . 相似文献
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M. Burak Erdoğan William R. Green Ebru Toprak 《Journal of Differential Equations》2018,264(9):5802-5837
We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the decay rate holds in the setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of is attained for large t, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies. 相似文献
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In this paper, we prove that if Ω is a bounded convex domain in , , and S is an affine complex hyperplane such that is not empty, then is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in , then is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to . 相似文献
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Douglas P. Hardin Michael C. Northington Alexander M. Powell 《Applied and Computational Harmonic Analysis》2018,44(2):294-311
A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators must satisfy , namely, . Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space ; our results provide an absolutely sharp improvement with . Our results are sharp in the sense that cannot be replaced by for any . 相似文献
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We study time decay estimates of the fourth-order Schrödinger operator in for and . We analyze the low energy and high energy behaviour of resolvent , and then derive the Jensen–Kato dispersion decay estimate and local decay estimate for under suitable spectrum assumptions of H. Based on Jensen–Kato type decay estimate and local decay estimate, we obtain the estimate of in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of for . Furthermore, using the local decay estimate and the Georgescu–Larenas–Soffer conjugate operator method, we prove the Jensen–Kato type decay estimates for some functions of H. 相似文献