Hilbert–Schmidt Hankel Operators with Anti-holomorphic Symbols on Complex Ellipsoids

On a complex ellipsoid in ${\mathbb{C}^n}$ C n , we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert–Schmidt.     

Hilbert–Schmidt Hankel Operators over Complete Reinhardt Domains

Let ${\mathcal{R}}$ be an arbitrary bounded complete Reinhardt domain in ${\mathbb{C}^n}$ . We show that for ${n \geq 2}$ , if a Hankel operator with an anti-holomorphic symbol is Hilbert–Schmidt on the Bergman space ${A^2(\mathcal{R})}$ , then it must equal zero. This fact has previously been proved only for strongly pseudoconvex domains and for a certain class of pseudoconvex domains.     

An Index Formula in Connection with Meromorphic Approximation

Let $\Phi $ be a continuous $n\times n$ matrix-valued function on the unit circle $\mathbb T $ such that the $(k-1)$ st singular value of the Hankel operator with symbol $\Phi $ is greater than the $k$ th singular value. In this case, it is well-known that $\Phi $ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty }_{(k)}$ ; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb D $ (in the sense that the McMillan degree of $Q$ in $\mathbb D $ is at most $k$ ) and $Q$ minimizes the essential suprema of singular values $s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0$ , with respect to the lexicographic ordering. For each $j\ge 0$ , the essential supremum of $s_{j}\left((\Phi -Q)(\zeta )\right)$ is called the $j$ th superoptimal singular value of degree $k$ of $\Phi $ . We prove that if $\Phi $ has $n$ non-zero superoptimal singular values of degree $k$ , then the Toeplitz operator $T_{\Phi -Q}$ with symbol $\Phi -Q$ is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where $\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}$ and $H_{\Phi }$ denotes the Hankel operator with symbol $\Phi $ . This result can in fact be extended from continuous matrix-valued functions to the wider class of $k$ -admissible matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi $ on $\mathbb T $ for which the essential norm of the Hankel operator $H_{\Phi }$ is strictly less than the smallest non-zero superoptimal singular value of degree $k$ of $\Phi $ .     

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ? ⁿ . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L ² estimates for the $\bar{\partial}$ operator are key ingredients in the proof.     

Estimates for the Fourier-Bessel transforms of multivariate functions

Two estimates useful in applications are proved for the Fourier-Bessel (or Hankel) transform in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + ^2 } \right)$ for some classes of two-variable functions characterized by a generalized modulus of continuity.     

Erratum to: Algèbres de lie quasi-réductives

In Corollary 12(ii) and Theorem 13(v) of [1] we omitted the hypothesis dim $ \mathfrak{z}\leq 1 $ . Moreover, in some places the symbol $ \mathbb{K} $ must be replaced by the symbol $ {{\mathbb{K}}^{\times }} $ .     

A Note on Recent Papers by Grafakos and Teschl,and Estrada

We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in \(\mathbb R^n\) and the Fourier transform of the same function in \(\mathbb R^{n+2}\) and \(\mathbb R^{n+1}\) , respectively, are located within known results on transplantation for Hankel transforms.     

Estimates for Solutions of the \bar{\partial}-Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type

In the last ten years, the resolution of the equation \(\bar{\partial}u=f\) with sharp estimates has been intensively studied for convex domains of finite type in \(\mathbb{C}^{n}\) by many authors. Generally, they used kernels constructed with holomorphic support function satisfying “good” global estimates. In this paper, we consider the case of lineally convex domains. Unfortunately, the method used to obtain global estimates for the support function cannot be carried out in that case. Then we use a kernel that does not directly give a solution of the \(\bar{\partial}\) -equation, but only a representation formula which allows us to end the resolution of the equation using Kohn’s L ² theory. As an application, we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.     

Sharp geometric rigidity of isometries on Heisenberg groups

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon )$ -quasi-isometry on a John domain of the Heisenberg group $\mathbb H ^n, n>1,$ is close to some isometry up to proximity order $\sqrt{\varepsilon }+\varepsilon $ in the uniform norm, and up to proximity order $\varepsilon $ in the $L_p^1$ -norm. We give examples showing the asymptotic sharpness of our results.     

Root n estimates of vectors of integrated density partial derivative functionals

Based on a random sample of size \(n\) from an unknown \(d\) -dimensional density \(f\) , the nonparametric estimations of a single integrated density partial derivative functional as well as a vector of such functionals are considered. These single and vector functionals are important in a number of contexts. The purpose of this paper is to derive the information bounds for such estimations and propose estimates that are asymptotically optimal. The proposed estimates are constructed in the frequency domain using the sample characteristic function. For every \(d\) and sufficiently smooth \(f\) , it is shown that the proposed estimates are asymptotically normal, attain the optimal \(O_p(n^{-1/2})\) convergence rate and achieve the (conjectured) information bounds. In simulation studies the superior performances of the proposed estimates are clearly demonstrated.     

Logarithmic Mean Oscillation on the Polydisc,Multi-Parameter Paraproducts and Iterated Commutators

We introduce another notion of bounded logarithmic mean oscillation in the \(N\) -torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\) , \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\) . We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\) .     

A T(1)-Theorem for non-integral operators

Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$ . We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies–Gaffney estimates. Associated with $L$ are certain approximations of the identity. We call an operator $T$ a non-integral operator if compositions involving $T$ and these approximations satisfy certain weighted norm estimates. The Davies–Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on $T$ in Calderón–Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood–Paley–Stein square function associated with $L$ is bounded on $L^2(X)$ , that a non-integral operator $T$ is bounded on $L^2(X)$ if and only if $T(1) \in BMO_L(X)$ and $T^{*}(1) \in BMO_{L^{*}}(X)$ . Here, $BMO_L(X)$ and $BMO_{L^{*}}(X)$ denote the recently defined $BMO(X)$ spaces associated with $L$ that generalize the space $BMO(X)$ of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a $T(1)$ -Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove $L^2(X)$ -boundedness of a paraproduct operator associated with $L$ . We moreover study criterions for a $T(b)$ -Theorem to be valid.     

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [0,\infty [

In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $\alpha ,\beta \rightarrow \infty $ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces ${H}_d(\mathbb F )$ with growing dimensions $d$ over the fields $\mathbb F =\mathbb R ,\ \mathbb C $ or the quaternions $\mathbb H $ .     

Hankel Operators on Holomorphic Hardy–Orlicz Spaces

We characterize the symbols of Hankel operators that extend into bounded operators from the Hardy–Orlicz ${\mathcal H^{\Phi_1}(\mathbb B^n)}$ into ${\mathcal H^{\Phi_2}(\mathbb B^n)}$ in the unit ball of ${\mathbb C^n}$ , in the case where the growth functions ${\Phi_1}$ and ${\Phi_2}$ are either concave or convex. The case where the growth functions are both concave has been studied by Bonami and Sehba. We also obtain several weak factorization theorems for functions in ${\mathcal H^{\Phi}(\mathbb B^n)}$ , with concave growth function, in terms of products of Hardy–Orlicz functions with convex growth functions.     

Determination of optimal convergence-control parameter value in homotopy analysis method

In the framework of the Homotopy Analysis Method (HAM) the so-called convergence-control parameter $c_{0}$ (Liao (Int J Non-Linear Mech 32:815–822, 1997) originally used the symbol $\hbar $ to denote the auxiliary parameter. But, $\hbar $ is well-known as Planck’s constant in quantum mechanics. To avoid misunderstanding, Liao (Commun Nonlinear Sci Numer Simulat 15:2003–2016, 2010) suggest to use the symbol $c_0$ to denote the basic convergence-control parameter.) has a key role in convergence of obtained series solution of solving non-linear equations. In this paper a modified approach in the determining of the convergence-control parameter value $c_{0}$ is proposed. The purpose of this paper is to find a proper convergence-control parameter $c_0$ to get a convergent series solution, or a faster convergent one. This modified approach minimizes the norm of a discrete residual function, systematically, in which seeks to find an optimal value of the convergence-control parameter $c_0$ at each order of HAM approximation, instead of the so-called $c_0$ -curve process. The proved theorems and numerical results demonstrate the validity, efficiency, and performance of the proposed approach.     

Comparing Invariants of SK1

In this text, we compare an invariant of the reduced Whitehead group SK ₁ of a central simple algebra recently introduced by Kahn (2010) to other invariants of SK ₁. Doing so, we prove the non-triviality of Kahn’s invariant using the non-triviality of an invariant introduced by Suslin (1991) which is non-trivial for Platonov’s examples of non-trivial SK ₁ (Platonov, Math USSR Izv 10(2):211–243, 1976). We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras (Merkurjev 1995).