In this paper, we prove the $(L^p, L^q)$-boundedness of (fractional) Hausdorff operators with power weight on Euclidean spaces. As special cases, we can obtain some well known results about Hardy operators.

In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for $L^2(\mathbb{R}^d)$ by replacing some mother framelets.

The aim of this paper is to give a simple proof of the restriction theorem for the maximal operators on the $d$-dimensional Euclidean space $\mathbb{R}^d$, whose theorem was proved by Carro-Rodriguez in 2012. Moreover, we shall give some remarks of the restriction theorem for the linear and the multilinear operators by Carro-Rodriguez and Rodriguez, too.

Let $L = −∆+V$ be a Schrödinger operator acting on $L^2(\mathbb{R}^n)$, $n ≥ 1$, where $V \not\equiv 0$ is a nonnegative locally integrable function on $\mathbb{R}^n$. In this article, we will introduce weighted Hardy spaces $H^p_L(w)$ associated with $L$ by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform $∇L^{−1/2}$ associated with $L$ is bounded from our new space $H^p_L (w)$ to the classical weighted Hardy space $H^p(w)$ when $n/(n+1)< p<1$ and $w ∈ A_1∩RH_{(2/p)′}$.

In this paper, we study optimal recovery (reconstruction) of functions on the sphere in the average case setting. We obtain the asymptotic orders of average sampling numbers of a Sobolev space on the sphere with a Gaussian measure in the $L_q({\mathbb{S}^{d-1}})$ metric for $1\le q\le \infty$, and show that some worst-case asymptotically optimal algorithms are also asymptotically optimal in the average case setting in the $L_q(\mathbb{S}^{d-1})$ metric for $1\le q\le \infty$.

A class $\Phi$ of 5-dimensional functions was introduced and an existence and uniqueness of common fixed points for a family of non-self mappings satisfying a $\phi_i$-quasi-contractive condition and a certain boundary condition was given on complete metrically convex metric spaces, and from which, more general unique common fixed point theorems were obtained. Our main results generalize and improve many same type common fixed point theorems in references.

In this paper, we establish the $L^p({\Bbb R}^{n+1} )$ boundedness for the commutators of singular integrals associated to surfaces of revolution, $\{(t,\phi(|t|)):t\in {\Bbb R}^{n}\}$, with rough kernels $\Omega\in L(\log L)^2({\Bbb S}^{n-1})$, if $\phi(|t|)=|t|$.

Let $L = L_0+V$ be the higher order Schrödinger type operator where $L_0$ is a homogeneous elliptic operator of order $2m$ in divergence form with bounded coefficients and $V$ is a real measurable function as multiplication operator (e.g., including $(−∆) ^m+V (m∈\mathbb{N})$ as special examples). In this paper, assume that $V$ satisfies a strongly subcritical form condition associated with $L_0$, the authors attempt to establish a theory of Hardy space $H^p_L(\mathbb{R}^n) (0 < p ≤ 1)$ associated with the higher order Schrödinger type operator $L$. Specifically, we first define the molecular Hardy space $H^p_L(\mathbb{R}^n)$ by the so-called $(p,q,ε,M)$ molecule associated to $L$ and then establish its characterizations by the area integral defined by the heat semigroup $e^{−tL}$.

In this paper, we deal with the complex Baskakov-Szász-Durrmeyer mixed operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth in $\mathbb{D}_R=\{z∈\mathbb{C};|z|<R\}$. Also, the exact order of approximation is found. The method used allows to construct complex Szász-type and Baskakov-type approximation operators without involving the values on $[0,∞)$.