Acta Mathematica Sinica-English Series最新文献

In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.

In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on ℝ^∞ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.

We prove that the index is bounded from below by a linear function of its first Betti number for any compact free boundary f-minimal hypersurface in certain positively curved weighted manifolds.

In clinical studies, it is often that the medical treatments take a period of time before having an effect on patients and the delayed time may vary from person to person. Even though there exists a rich literature developing methods to estimate the time-lag period and treatment effects after lag time, most of these existing studies assume a fixed lag time. In this paper, we propose a hazard model incorporating a random treatment time-lag effect to describe the heterogeneous treatment effect among subjects. The EM algorithm is used to obtain the maximum likelihood estimator. We give the asymptotic properties of the proposed estimator and evaluate its performance via simulation studies. An application of the proposed method to real data is provided.

Let ({cal C}) be a Krull–Schmidt n-exangulated category and ({cal A}) be an n-extension closed subcategory of ({cal C}). Then ({cal A}) inherits the n-exangulated structure from the given n-exangulated category in a natural way. This construction gives n-exangulated categories which are neither n-exact categories in the sense of Jasso nor (n + 2)-angulated categories in the sense of Geiss–Keller–Oppermann in general. Furthermore, we also give a sufficient condition on when an n-exangulated category ({cal A}) is an n-exact category. These results generalize work by Klapproth and Zhou.

Let X and Y be two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: If f:X → Y is a standard coarse Lipschitz embedding, then for each x* ∈ Lip₀(X) there exist α, γ > 0 depending only on f and Q_x* ∈ Lip₀(Y) with ({Vert{{Q_{{x^*}}}}Vert_{{rm{Lip}}}} le alpha {Vert {{x^*}}Vert_{{rm{Lip}}}}) such that

Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space.

Let (overrightarrow{b}=(b_{1},b_{2},ldots,b_{m})) be a collection of locally integrable functions and (T_{Sigmaoverrightarrow{b}}) the commutator of multilinear singular integral operator T. Denote by (mathbb{L}(delta)) and (mathbb{L}(delta(cdot))) the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator (T_{Sigmaoverrightarrow{b}}) in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for b_j (j = 1, 2, …, m) to be (mathbb{L}(delta)) or (mathbb{L}(delta(cdot))) via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.