The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

Toeplitz operators between large Fock spaces in several complex variables

<abstract><p>Let $ \omega $ belong to the weight class $ \mathcal{W} $, the large Fock space $ \mathcal{F}_{\omega}^{p} $ consists of all holomorphic functions $ f $ on $ \mathbb{C}^{n} $ such that the function $ f(\cdot)\omega(\cdot)^{1/2} $ is in $ L^p(\mathbb{C}^{n}, dv) $. In this paper, given a positive Borel measure $ \mu $ on $ {\mathbb C}^n $, we characterize the boundedness and compactness of Toeplitz operator $ T_\mu $ between two large Fock spaces $ F^{p}_\omega $ and $ F^{q}_\omega $ for all possible $ 0 &lt; p, q &lt; \infty $.</p></abstract>

The Range of the Cesàro Operator Acting on

In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

Embeddings of Müntz Spaces in

In this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$.

Charges solve the truncated complex moment problem

Let [Formula: see text], with [Formula: see text] and [Formula: see text], be a given complex-valued sequence. The complex moment problem (respectively, the general complex moment problem) associated with [Formula: see text] consists in determining necessary and sufficient conditions for the existence of a positive Borel measure (respectively, a charge) [Formula: see text] on [Formula: see text] such that [Formula: see text] In this paper, we investigate the notion of recursiveness in the two variable case. We obtain several useful results that we use to deduce new necessary and sufficient conditions for the truncated complex moment problem to admit a solution. In particular, we show that the general complex moment problem always has a solution. A concrete construction of the solution and an illustrating example are also given.

Flows of measures generated by vector fields

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

Application of localization to the multivariate moment problem II

The paper is a sequel to the paper [5], Math. Scand. 115 (2014), 269--286, by the same author. A new criterion is presented for a PSD linear map $L \colon \mathbb{R}[\underline{x}] \to \mathbb{R}$ to correspond to a positive Borel measure on $\mathbb{R}^n$. The criterion is stronger than Nussbaum's criterion (Ark. Math. 6 (1965), 171--191) and is similar in nature to Schmüdgen's criterion in Marshall [5] and Schmüdgen, Ark. Math. 29 (1991), 277--284. It is also explained how the criterion allows one to understand the support of the associated measure in terms of the non-negativity of $L$ on a quadratic module of $\mathbb{R}[\underline{x}]$. This latter result extends a result of Lasserre, Trans. Amer. Math. Soc. 365 (2013), 2489--2504. The techniques employed are the same localization techniques employed already in Marshall (Cand. Math. Bull. 46 (2003), 400--418, and [5]), specifically one works in the localization of $\mathbb{R}[\underline{x}]$ at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$.

TOEPLITZ OPERATORS BETWEEN FOCK SPACES

Given a positive Borel measure ${\it\mu}$ on the $n$-dimensional Euclidean space $\mathbb{C}^{n}$, we characterise the boundedness (and compactness) of Toeplitz operators $T_{{\it\mu}}$ between Fock spaces $F^{\infty }({\it\varphi})$ and $F^{p}({\it\varphi})$ with $0<p\leq \infty$ in terms of $t$-Berezin transforms and averaging functions of ${\it\mu}$. Our result extends recent work of Mengestie [‘On Toeplitz operators between Fock spaces’, Integral Equations Operator Theory78 (2014), 213–224] and others.

Addendum to “Limit Sets of Typical Homeomorphisms”

Given an integer n ≥ 3, a metrizable compact topological n-manifold X with boundary, and a finite positive Borel measure μ on X, we prove that for the typical homeomorphism ƒ : X → X, it is true that for μ-almost every point x in X the restriction of ƒ (respectively of f-1) to the omega limit set ω( ƒ ; x) (respectively to the alpha limit set α( ƒ ; x)) is topologically conjugate to the universal odometer.