In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein.

We study the image of the model subspace $K_\theta$ under the composition operator $C_\varphi$, where $\varphi$ and $\theta$ are inner functions, and find the smallest model subspace which contains the linear manifold $C_\varphi K_\theta$. Then we characterize the case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to the study of the inner functions $\varphi$ and $\psi$ such that the composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of inner functions.

Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.

We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen-Douglas operators given by C. L. Jiang.

Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's theorem to manifolds of different dimensions.

We define the higher order Riesz transforms and the Littlewood--Paley $g$-function associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta f'(\theta)+\l^2f(\theta)$. We prove that these operators are Calder\'{o}n--Zygmund operators in the homogeneous type space $((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted, $H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs.

Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.

We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.

We give optimal upper and lower bounds for the function $H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ for $x\geq 0$ and $s>1$. These bounds improve the standard inequalities with integrals. We deduce from them inequalities about Riemann's $\zeta$ function, and we give a conjecture about the monotonicity of the function $s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. Some applications concern the convexity of functions related to Euler's $\Gamma$ function and optimal majorization of elementary functions of Baskakov's operators. Then, the result proved for the function $x\mapsto x^{-s}$ is extended to completely monotonic functions. This leads to easy evaluation of the order of the generating series of some arithmetical functions when $z$ tends to 1. The last part is concerned with the class of non negative decreasing convex functions on $]0,+\infty[$, integrable at infinity. Nous prouvons un encadrement optimal pour la quantit\'e $H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ pour $x\geq 0$ et $s>1$, qui am\'eliore l'encadrement standard par des int\'egrales. Cet encadrement entra{\^\i}ne des in\'egalit\'es sur la fonction $\zeta$ de Riemann, et am\`ene \`a conjecturer la monotonie de la fonction $s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. On donne des applications \`a l'\'etude de la convexit\'e de fonctions li\'ees \`a la fonction $\Gamma$ d'Euler et \`a la majoration optimale des fonctions \'el\'ementaires intervenant dans les op\'erateurs de Baskakov. Puis, nous \'etendons aux fonctions compl\`etement monotones sur $]0,+\infty[$ les r\'esultats \'etablis pour la fonction $x\mapsto x^{-s}$, et nous en d\'eduisons des preuves \'el\'ementaires du comportement, quand $z$ tend vers $1$, des s\'eries g\'en\'eratrices de certaines fonctions arithm\'etiques. Enfin, nous prouvons qu'une partie du r\'esultat se g\'en\'eralise \`a une classe de fonctions convexes positives d\'ecroissantes.

We study when the spaces of Lorentz multipliers from $L^{p,t} \rightarrow L^{p,s}$ are distinct. Our main interest is the case when $s
Keywords:multipliers, convolution operators, Lorentz spaces, Lorentz-improving multipliers
Categories:43A22, 42A45, 46E30