[en] In this paper we pursue the study of the best approximation operator extended from L^Φ to L^φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calderón–Zygmund class t_m^p (x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.

[en] The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel on the upper half space $${\int <!--\; ???\; -->}_{{\mathbb{R}}_{+}^n}{{\int <!--\; ???\; -->}_{\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n}f\left(\mathrm{\xi <!--\; ??\; -->}\right)P\left(x,\mathrm{\xi <!--\; ??\; -->},\mathrm{\alpha <!--\; ??\; -->}\right)g\left(x\right)d\mathrm{\xi <!--\; ??\; -->}dx\le <!--\; ???\; -->C_{n,\mathrm{\alpha <!--\; ??\; -->},p,q^{\prime }}\parallel <!--\; ???\; -->g\parallel <!--\; ???\; -->}_{L{q}^{\prime }\left({\mathbb{R}}_{+}^n\right)}\parallel <!--\; ???\; -->f{\parallel <!--\; ???\; -->}_{L{q}^{\prime }\left(\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n\right),}$$ where $f\in <!--\; ???\; -->L^p\left(\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n\right),g\in <!--\; ???\; -->L^{q^{\prime }}\left({\mathbb{R}}_{+}^n\right)\text{and}p,q^{\prime }\in <!--\; ???\; -->\left(1+\mathrm{\infty <!--\; ???\; -->}\right),2\le <!--\; ???\; -->\mathrm{\alpha <!--\; ??\; -->}<n\text{satisfying}\frac{n-<!--\; ???\; -->1}{np}+\frac{1}{q^{\prime }}+\frac{2-<!--\; ???\; -->\mathrm{\alpha <!--\; ??\; -->}}{n}=1.$Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_n,α,p,q′. Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore, in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler-Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ${\mathrm{\xi <!--\; ??\; -->}}_0\in <!--\; ???\; -->\mathrm{\partial <!--\; ???\; -->}{\mathbb{R}}_{+}^n.$ Our results proved in this paper play a crucial role in establishing the Stein-Weiss inequalities with the Poisson kernel in our subsequent paper.

[en] A set E ⊂ ℝ^d whose indicator function 1_E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1_E has nearly maximal Gowers norm then E nearly coincides with some ellipsoid, in the sense that their symmetric difference has small Lebesgue measure.

[en] Let G be a graph with vertex set V(G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …, |E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to $\frac{1+\left|E(G)\right|}{2}\cdot <!--\; ???\; -->d(v)$, where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V(G), |{e : e ∈ E_v, f(e) ≤ Δ/2}| = |{e : e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.

[en] In this paper, the weak (1,1) boundedness of oscillatory singular integral with variable phase P(x)γ(y) for any x, y ∈ ℝ, $$Tf(x):=\mathrm{p}.\mathrm{v}.{\int <!--\; ???\; -->}_{-<!--\; ???\; -->\mathrm{\infty <!--\; ???\; -->}}^{\mathrm{\infty <!--\; ???\; -->}}{\mathrm{e}}^{\mathrm{i}P(x)\mathrm{\gamma <!--\; ??\; -->}(y)}f(x-<!--\; ???\; -->y)\frac{dy}{y}$$ is studied, where P is a real monic polynomial on ℝ.

[en] In this paper, we obtain the quantitative bound of the exponential convergence rates of Markov chains under a weaken minorization condition, using the coupling method and the analytic approach. And also, we obtain the convergence rates for continuous time Markov processes.

[en] The minimax concave penalty (MCP) has been demonstrated theoretically and practically to be effective in nonconvex penalization for variable selection and parameter estimation. In this paper, we develop an efficient alternating direction method of multipliers (ADMM) with continuation algorithm for solving the MCP-penalized least squares problem in high dimensions. Under some mild conditions, we study the convergence properties and the Karush–Kuhn–Tucker (KKT) optimality conditions of the proposed method. A high-dimensional BIC is developed to select the optimal tuning parameters. Simulations and a real data example are presented to illustrate the efficiency and accuracy of the proposed method.

[en] We introduce a new function space that is much finer than the classical Lebesgue spaces, and investigate the continuity and the discontinuity of the Calderón–Zygmund operators on the new weighted spaces. In order to identify the continuity of singular integrals, we prove an interpolation theorem on the new spaces and propose a concept of the spectrum of base functions.

[en] Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β. Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation (MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well.

[en] In this paper, by making use of Abel’s theorem on power series, the reflection formula and the function equation for Hurwitz zeta function, we establish several expressions of Dirichlet L-function at positive integers by means of some finite sums of different types. Some special cases as well as immediate consequences of the results presented here are also considered.