Orthogonal polynomials on the circle for the weight w satisfying conditions: w,1/w in BMO(T)

Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators

Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, then $$ \| [b_k,\cdots[b_2, [b_1, T]]\cdots]\|_{L^p(μ) \to L^p(λ)} \lesssim_{[μ]_{A_p}, [λ]_{A_p}} \prod_{i=1}^k\|b_i\|_{\operatorname{bmo}(ν^{θ_i})} , $$ where $p \in (1,\infty)$, $θ_i \in [0,1]$, $\sum_{i=1}^kθ_i=1$, $μ, λ\in A_p$, $ν:= μ^{1/p}λ^{-1/p}$. Here $A_p$ stan…

About Nodal systems for Lagrange interpolation on the circle

We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by the n roots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, some consequences for the Lagrange interpolation on [-1,1] and the Lagrange trigonometric interpolation are obtained.

Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potent…

$R_{II}$ type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle

We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple and lie on the real line. It turns out that the polynomial $P_n$, for any $n \geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized eigenvalue problem. It is shown that with this $R_{II}$ type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by $P_n$ with respect to this measure is also obtained. Finally, examples are given to justify the results.

Equivalent Characterizations for Boundedness of Maximal Singular Integrals on $ax+b$\,--Groups

Let $(S, d, ρ)$ be the affine group $\mathrm{R}^n \ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $ρ$, which is of exponential growth at infinity. In this paper, for any linear operator $T$ on $(S, d, ρ)$ associated with a kernel $K$ satisfying certain integral size condition and Hörmander's condition, the authors prove that the following four statements regarding the corresponding maximal singular integral $T^\ast$ are equivalent: $T^\ast$ is bounded from $L_c^\infty$ to $\mathrm{BMO}$, $T^\ast$ is bounded on $L^p$ for all $p\in(1, \infty…

The Steklov problem and Remainder Estimates for Krein Systems generated by a Muckenhoupt weight

We show that solutions to Krein systems, the continuous frequency analogue of orthogonal polynomials on the unit circle, generated by an $A_2 (\mathbb{R})$ weight $w$ satisfying $w-1 \in L^1 (\mathbb{R}) + L^2 (\mathbb{R})$, are uniformly bounded in $L^p_{\mathrm{loc}} (w, \mathbb{R})$ for $p$ sufficiently close to $2$. This provides a positive answer to the Steklov problem for Krein systems. Furthermore, we define a "remainder" which measures the difference between the solution to a Krein system and a polynomial-like approximant, and we estimate these remainders in $L^p_w (\mathbb{R})$ for $w…

Best rational approximation of functions with logarithmic singularities

We consider functions $ω$ on the unit circle $\mathbb T$ with a finite number of logarithmic singularities. We study the approximation of $ω$ by rational functions and find an asymptotic formula for the distance in the BMO-norm between $ω$ and the set of rational functions of degree $n$ as $n\to\infty$. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.

单位圆上的 Nikishin 系统

Nikishin systems on the unit circle

本文定义单位圆上的 $r$ 个概率测度 Nikishin 系统，证明其在特定同奇偶、有序多指标下满足 AT 性质与正规性，并推广到 $r=2$ 及实轴情形。

Szegő映射与Hermite–Padé多项式用于单位圆上的多重正交性

Szegő Mapping and Hermite--Padé Polynomials for Multiple Orthogonality on the Unit Circle

本文研究了单位圆上满足多重正交条件的多项式，通过Szegő映射和Geronimus关系将其与实线上多重正交多项式联系起来，推导了多项种递推关系和行列式表示。

关于象征属于$S^{m}_{0,δ}$类的傅里叶积分算子正则性的注记

Notes on Regularity of Fourier integral operators with symbol in $S^{m}_{0,δ}$

本文针对象征属$S^{m}_{0,δ}$类、位相满足强非退化条件的傅里叶积分算子，推导其$L^p$有界性与$L^\infty$到$BMO$的映射性质，证明所得阶数界的紧性。

Boundedness of Commutators on Hardy Spaces over Metric Measure Spaces of Non-homogeneous Type

Let $(\mathcal{X},d,μ)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $b\in\rm{\widetilde{RBMO}(μ)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(μ)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(μ)$. The b…

Geometry of limits of zeros of polynomial sequences of type $(1,2)$

We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of an arc, a circle, an interval or its exterior under the real line, the union of at most two different shapes of the above cases expect the union of an arc and a circle, and some degenerate forms.

Asymptotics of greedy energy sequences on the unit circle and the sphere

For a parameter $λ>0$, we investigate greedy $λ$-energy sequences $(a_{n})_{n=0}^{\infty}$ on the unit sphere $S^{d}\subset\mathbb{R}^{d+1}$, $d\geq 1$, satisfying the defining property that each $a_{n}$, $n\geq 1$, is a point where the potential $\sum_{k=0}^{n-1}|x-a_{k}|^λ$ attains its maximum value on $S^{d}$. We show that these sequences satisfy the symmetry property $a_{2k+1}=-a_{2k}$ for every $k\geq 0$. The asymptotic distribution of the sequence undergoes a sharp transition at the value $λ=2$, from uniform distribution ($λ<2$) to concentration on two antipodal points ($λ>2$). We invest…

On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Let $Ω\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (i.e., quantitatively open and path-connected) satisfiying the capacity density condition. Let $L_0 u=-\mathrm{div}(A_0 \nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real uniformly elliptic operators in $Ω$, with $ω_{L_0}, ω_L$ the associated elliptic measures. We establish the equivalence between the following properties: (i) $ω_L \in A_{\infty}(ω_{L_0})$, (ii) $L$ is $L^p(ω_{L_0})$-solvable for some $p\in (1,\infty)$, (iii) bounded null solutions of $L$ satisfy Carleson measure estimates with respec…