Acta Mathematica Sinica-English Series最新文献

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces.

Let (mathbb{B}) be a unit ball in ℝ², (W_{0}^{1,2}(mathbb{B})) be the standard Sobolev space. For any ϵ > 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely,

can be attained by some radially symmetric function (u_{epsilon}in W_{0}^{1,2}(mathbb{B})) with (int_{mathbb{B}}vertnabla u_{epsilon}vert^{2}dx=1). In this note, we concern the compactness of the function family {u_ϵ}_ϵ>0 and prove that up to a subsequence u_ϵ converges to some function u₀ in (C^{1}(overline{mathbb{B}})) as ϵ → 0. Furthermore, u₀ is an extremal function of the supremum

Let us explain the result in geometry. Denote (omega_{0}=dx_{1}^{2}+dx_{2}^{2}) be the standard Euclidean metric. Define a conical metric (omega_{epsilon}=vert xvert^{2epsilon}omega_{0}) for (xinmathbb{B}). Then the extremal family {u_ϵ}_ϵ>0 of the following Trudinger-Moser functionals

under the constraint (uin W_{0}^{1,2}(mathbb{B})) and (int_{mathbb{B}}vertnabla_{omega_{epsilon}u}vert^{2}dv_{omega_{epsilon}}leq 1) is compact as ϵ → 0.

Let L/F be a finite Galois extension of number fields of degree n and let p be a prime which does not divide n. We shall study the p^j-rank of (K_{2i}(mathcal{O}_{L})) via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano. Along the way, we generalize previous observations of Browkin, Wu and Zhou on K₂-groups to higher even K-groups. We also give examples to illustrate our results. Finally, we apply our discussion to refine a result of Kitajima pertaining to the p-rank of even K-groups in the cyclotomic ℤ_l-extension, where l ≠ p.

This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(X), Φ(X)^⊥) and (^⊥Ψ(Y), Ψ(Y)) in Rep(Q, A)—the category of all A-valued representations of a quiver Q—are complete whenever (X, Y) is a complete cotorsion pair in an abelian category A satisfying some mild conditions. We give an affirmative answer if the quiver Q is rooted. As an application, we show under certain mild conditions that if a subcategory L, which is not necessarily closed under direct summands, of A is special precovering (resp., preenveloping), then Φ(L)(resp., Ψ(L)) is special precovering (resp., preenveloping) in Rep(Q, A).

Let f be any arithmetic function and define ({S_{f}}(x):=sumnolimits_{{n le x}}f([x/n])). If the function f is small, namely, f(n) ≪ n^ε, then the error term E_f(x) in the asymptotic formula of S_f(x) has the form O(x^1/2+ε). In this paper, we shall study the mean square of E_f(x) and establish some new results of E_f(x) for some special functions.

Using methods from complex analysis in one variable, we define an integral operator that solves ({bar partial}) with supnorm estimates on product domains in ℂⁿ.

In this paper, we studied the metric mean dimension in Feldman–Katok (FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subsets. And we established two variational principles.

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact.

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.

In this paper, we consider the reducibility of three-dimensional skew symmetric systems. We obtain a reducibility result if the base frequency is high-dimensional weak Liouvillean and the parameter is sufficiently small. The proof is based on a modified KAM theory for 3-dimensional skew symmetric systems.