The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity.

We introduce and study Dirichlet-type spaces $\mathcal D(\mu _1, \mu _2)$ of the unit bidisc $\mathbb D^2,$ where $\mu _1, \mu _2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(\mu _1, \mu _2)$ and the complex polynomials are dense in $\mathcal D(\mu _1, \mu _2).$ Further, we obtain the division property and solve Gleason’s problem for $\mathcal D(\mu _1, \mu _2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr M_z$ of the multiplication operators $\mathscr M_{z_1}, \mathscr M_{z_2}$ on $\mathcal D(\mu _1, \mu _2)$ defines a cyclic toral $2$-isometry and $\mathscr M^*_z$ belongs to the Cowen–Douglas class $\mathbf {B}_1(\mathbb D^2_r)$ for some $r>0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter’s representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair T with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(\mu _1, \mu _2)$ for some $\mu _1,\mu _2$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

Let $\sigma \in (0,\,2)$, $\chi ^{(\sigma )}(y):={\mathbf 1}_{\sigma \in (1,2)}+{\mathbf 1}_{\sigma =1} {\mathbf 1}_{y\in B(\mathbf {0},\,1)}$, where $\mathbf {0}$ denotes the origin of $\mathbb {R}^n$, and $a$ be a non-negative and bounded measurable function on $\mathbb {R}^n$. In this paper, we obtain the boundedness of the non-local elliptic operator

\[ Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{{\rm d}y}{|y|^{n+\sigma}} \]

$\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$\lambda \in (0,\,\infty )$

$Lu-\lambda u=f$

$\mathbb {R}^n$

$f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$H^1(\mathbb {R}^n)$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$L^p(\mathbb {R}^n)$

$p\in (1,\,\infty )$

$p=1$

$p=\infty$

In this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.

In this note, we mainly study operator-theoretic properties on the Besov space $B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.

In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$ , as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

For a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $ , we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$ , denoted by $\mathcal {D}^s_K$ . Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$ , we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.

Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by

\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]

We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.

In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.

For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$

$$ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$$

$T_\alpha $

$L^p$

$1\leq p\leq +\infty $

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $K_{i}$ be a locally compact (Hausdorff) topological space and let ${\mathcal{H}}_{i}$ be a closed subspace of ${\mathcal{C}}_{0}(K_{i},\mathbb{F})$ such that each point of the Choquet boundary $\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$ of ${\mathcal{H}}_{i}$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$ with $\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$, then $\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$ is homeomorphic to $\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$. We then provide a one-sided version of this result. Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided ${\mathcal{H}}_{1}$ is isomorphic to ${\mathcal{H}}_{2}$.

We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$, where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spaces ${\dot{B}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ and weighted Triebel–Lizorkin spaces ${\dot{F}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ associated with the operator $L$ for the full range $0<p,q\leqslant \infty$, $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ and $w$ being in the Muckenhoupt weight class $A_{\infty }$. Under rather weak assumptions on $L$ as stated above, we prove that our new spaces satisfy important features such as continuous characterizations in terms of square functions, atomic decompositions and the identifications with some well-known function spaces such as Hardy-type spaces and Sobolev-type spaces. One of the highlights of our result is the characterization of these spaces via noncompactly supported functional calculus. An important by-product of this characterization is the characterization via the heat kernel for the full range of indices. Moreover, with extra assumptions on the operator $L$, we prove that the new function spaces associated with $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$, the spectral multiplier of $L$ in our new function spaces and the dispersive estimates of wave equations.

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.

We use the Carleson measure-embedding theorem for weighted Bergman spaces to characterize the positive Borel measures $\unicode[STIX]{x1D707}$ on the unit disc such that certain analytic function spaces of Dirichlet type are embedded (compactly embedded) in certain tent spaces associated with a measure $\unicode[STIX]{x1D707}$. We apply these results to study Volterra operators and multipliers acting on the mentioned spaces of Dirichlet type.

The aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator

$${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$

${\rm {\cal H}}_a^p ({\open C}_ + )$

$p\in [1,\infty ]$

Let $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$-capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$), $f$ vanishes on $E$, and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$.