Related papers: Characterizations of $A_2$ Matrix Power Weights

We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover,…

In the one-parameter family of power-law maps of the form $f_a(x)=-|x|^{\alpha}+a,$ $\alpha >1,$ we give examples of mutually related dynamically determined quantities, depending on the parameter $a$, such that one is a Pick function of the…

We construct rational all-pass matrix functions with real-valued coefficients for mirroring pairs of complex-conjugated determinantal roots of a rational matrix. This problem appears, for example, when proving the spectral factorization…

In this paper, we give necessary and sufficient conditions for weighted $L^2$ estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form: \[ \| T(\mathbf{W} f)\|_{L^2(\mathbf{V})} \le…

If $A$ is an $n\times n$ matrix whose $n$ eigenvalues are ordered in terms of decreasing modules, $|\lambda_1 | \geq |\lambda_2| \geq ... |\lambda_n|$, it is often of interest to estimate $\frac{|\lambda_2|}{|\lambda_1|}$. If $A$ is a row…

In this paper, we exhibit explicitly a sequence of $2\times2$ matrix valued orthogonal polynomials with respect to a weight $W_{p,n}$, for any pair of real numbers $p$ and $n$ such that $0<p<n$. The entries of these polynomiales are…

We define the vector-valued, matrix-weighted function spaces $\dot{F}^{\alpha q}_p(W)$ (homogeneous) and $F^{\alpha q}_p(W)$ (inhomogeneous) on $\mathbb{R}^n$, for $\alpha \in \mathbb{R}$, $0<p<\infty$, $0<q \leq \infty$, with the matrix…

In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The results we obtain…

In this work we use matrix models to study the problem of strength distributions. This is motivated by noticing near exponential fall offs of strengths in calculated magnetic dipole excitations. We emphasize that the quality of the…

Let $ Tf =\sum_{ I} \varepsilon_I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L…

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…

The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight function $| x_{1}^{2}-x_{2}^{2}|^{2k_{0}}| x_{1}x_{2}|^{2k_{1}}e^{-(x_{1}^{2}+x_{2}^{2})/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the…

The aim of this paper is to show some examples of matrix-valued orthogonal functions on the real line which are simultaneously eigenfunctions of a second-order differential operator of Schr\"{o}dinger type and an integral operator of…

We compute various correlation functions at the planar level in a simple supersymmetric matrix model, whose scalar potential is in shape of a double-well. The model has infinitely degenerate vacua parametrized by filling fractions \nu_\pm…

We prove some properties of completely monotonic functions and apply them to obtain results on gamma and $q$-gamma functions.

We continue the analysis in [H. Osaka and J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear and its Applications 431(2009), 1825 - 1832] in which the followings three assertions at each…

We show how to calculate correlation functions of two matrix models. Our method consists in making full use of the integrable hierarchies and their reductions, which were shown in previous papers to naturally appear in multi--matrix models.…

We show that the large-charge formalism can be successfully applied to models that go beyond the vector models discussed so far in the literature. We study the explicit example of a conformal $SU(3)$ matrix model in 2+1 space-time…

We study the matrix factorization problem associated with an SO(2) spinning top by using the algebro-geometric approach. We derive the explicit expressions in terms of Riemann theta functions and discus some related problems including a…

Special functions are often defined as a Fourier or Laplace transform of a positive measure, and the positivity of the measure manifests as positive definiteness of certain matrices. The purpose of this expository note is to give a sample…