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Let L = - Δ ℍ n + V be a Schrödinger operator on Heisenberg groups ℍ n , where Δ ℍ n is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 풬 / 2 . Here 풬 is the homogeneous dimension of ℍ n . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 , α > 0 , associated with L. By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ (⋅ , ⋅) , respectively. As an application, we characterize the space BMO L γ ⁢ (ℍ n) via { e - t ⁢ L α } t > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded, compact, and order bounded sums of weighted composition-differentiation operators from Bergman-type spaces to weighted Banach spaces of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the frequent hypercyclicity of a scalar multiple of the tensor product of two linear fractional composition operators $ \lambda C_{\varphi }\otimes C_{\psi } $ λ C φ ⊗ C ψ on $ S_{\nu }\otimes S_{\nu } $ S ν ⊗ S ν , where $ S_{\nu } $ S ν is the weighted Dirichlet space. We will restrict the discussion to the situations that both φ and ψ are hyperbolic non automorphisms, hyperbolic automorphisms and parabolic automorphisms, respectively. Meanwhile, we give the equivalent conditions of $ \lambda C_{\varphi }\otimes C_{\psi } $ λ C φ ⊗ C ψ to be frequently hypercyclic, hypercyclic, mixing and chaotic. [ABSTRACT FROM AUTHOR]

Let μ be a positive finite Borel measure on [0, 1). Cesàro-type operators C μ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that C μ is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of C μ on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments (μ n) n ∈ N 0 . The continuity, compactness and spectrum of C μ acting on Fréchet and (LB) Korenblum type spaces are also considered. [ABSTRACT FROM AUTHOR]

Motivated by certain results known from the literature, in this paper we give several new dynamic inequalities for B-monotone functions with respect to time scales nabla calculus. If the time scale represents the set of real numbers, our results reduce to integral inequalities known from the literature. On the other hand, in the setting of positive integers, we obtain new discrete inequalities for B-monotone sequences. [ABSTRACT FROM AUTHOR]

We utilize the $ (A^p_\alpha,q) $ (A α p , q) -Carleson measure to investigate the bounded small Hankel operator $ h_f^\alpha $ h f α acting between the Bergman-type space $ A^p_\alpha $ A α p and $ \overline {A_\alpha ^q} $ A α q ¯ on the complex unit ball $ \mathbb {B}_n $ B n for some $ \alpha \le -1 $ α ≤ − 1. For $ n\ge 2 $ n ≥ 2 , and $ f\in H(\mathbb {B}_n) $ f ∈ H (B n) , we have the conclusions: Suppose $ \alpha \le -1 $ α ≤ − 1 , $ \max \{1,-2\alpha -2\} max { 1 , − 2 α − 2 } < p < ∞ and $ pN+\alpha >-1 $ pN + α > − 1. Then $ h_f^\alpha : A^p_\alpha \to \bar { A^p_\alpha } $ h f α : A α p → A α p ¯ is bounded if and only if $ |R^N f(z)|^p\mathrm {d} v_{pN+\alpha }(z) $ | R N f (z) | p d v pN + α (z) is an $ (A^p_\alpha,p) $ (A α p , p) -Carleson measure. Suppose $ -n-1 − n − 1 < α ≤ − 1 , $ \max \{1,-2\alpha -2\} max { 1 , − 2 α − 2 } < p < q < ∞ and $$\begin{align*} pN>\max\left\{ \frac {q-p}p (n+1+\alpha)-2(\alpha+1),\frac{-p(\alpha+1)}{p-1}\right\}. \end{align*} $$ pN > max { q − p p (n + 1 + α) − 2 (α + 1) , − p (α + 1) p − 1 }. Then $ h_f^\alpha : A^p_\alpha \to \overline {A_\alpha ^q} $ h f α : A α p → A α q ¯ is bounded if and only if $ |R^N f(z)|^q\mathrm {d} v_{qN+\alpha }(z) $ | R N f (z) | q d v qN + α (z) is an $ (A^p_\alpha,q) $ (A α p , q) -Carleson measure. Suppose $ 1 1 < q < p < ∞ , $ \alpha \le -1 $ α ≤ − 1 , N be a positive integer such that $ qN+\alpha >-1 $ qN + α > − 1. If $ |R^N f(z)|^q\mathrm {d} v_{qN+\alpha }(z) $ | R N f (z) | q d v qN + α (z) is an $ (A^p_\alpha,q) $ (A α p , q) -Carleson measure, then $ h_f^\alpha : A^p_\alpha \to \overline {A_{\alpha }^q} $ h f α : A α p → A α q ¯ is bounded. Let $ \kappa =\frac {pq}{p-q} $ κ = pq p − q and $ \kappa '=\frac {\kappa }{\kappa -1} $ κ ′ = κ κ − 1 . If $ \alpha +(p-1)\kappa 'N>-1 $ α + (p − 1) κ ′ N > − 1 and $ h_f^\alpha : A^p_\alpha \to \overline {A_{\alpha }^q} $ h f α : A α p → A α q ¯ is bounded, then $ f\in A_{\alpha }^{\kappa } $ f ∈ A α κ with $ \|f\|_{A_{\alpha }^{\kappa }}\lesssim \|h_{f}^\alpha \| $ ‖ f ‖ A α κ ≲ ‖ h f α ‖. If $ |R^N f(z)|^q\mathrm {d} v_{qN+\alpha }(z) $ | R N f (z) | q d v qN + α (z) is an $ (A^p_\alpha,q) $ (A α p , q) -Carleson measure, then $ f\in A_\alpha ^\kappa $ f ∈ A α κ . But, there is an $ f\in A_\alpha ^\kappa $ f ∈ A α κ such that $ |R^N f(z)|^q\mathrm {d} v_{qN+\alpha } (z) $ | R N f (z) | q d v qN + α (z) is not an $ (A^p_\alpha,q) $ (A α p , q) -Carleson measure. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions for the boundedness of Hausdorff operator on power-weighted Morrey–Herz spaces that are associated with the Opdam–Cherednik transform. Moreover, we provide some Lipschitz estimates for its commutator on such spaces. [ABSTRACT FROM AUTHOR]

Our main focus in this paper is to explore the mapping properties for the n-dimensional fractional Hausdorff operator H Φ , β from L p ( R n , | x | α) to L q ( R n , | x | γ) , where p , q < 1 (p , q ≠ 0) , α , γ ∈ R , 0 ≤ β < n and Φ is a nonnegative measurable function on R n . For p , q < 1 (p , q ≠ 0) satisfying some additional assumptions, we give sufficient conditions for the validity of the reverse fractional Hausdorff inequality H Φ , β f L q ( R n , | x | γ) ≥ C ‖ f ‖ L p ( R n , | x | α) for some positive constant C and all nonnegative functions f ∈ L p ( R n , | x | α) . For the particular case 0 < p = q < 1 , we obtain the sharp reverse fractional Hausdorff inequality. As applications, we establish the sharp reverse inequalities for the n-dimensional fractional Hardy operator and its adjoint operator, and also the n-dimensional fractional Hardy–Littlewood–Pólya operator on power-weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In a self contained presentation we discuss the Gamma functions spaces with variable exponent. As an application we investigate the boundedness, compactness and closed range of the weighted composition operator on Gamma spaces with variable exponent. [ABSTRACT FROM AUTHOR]

We investigate the boundedness and compactness properties of integral operators, Volterra operators, and generalized Volterra operators between the Dales–Davie algebras. Additionally, we study the analogous properties of these operators when applied to the Lipschitz versions of Dales–Davie algebras. [ABSTRACT FROM AUTHOR]

We discuss the Weyl multipliers for pairs of Lebesgue spaces, and identify some sub classes of B (L 2 (R n)) which form Weyl multipliers for (L p , L q) , 1 ≤ p , q ≤ ∞ . Similar results are also obtained for Lorentz spaces. [ABSTRACT FROM AUTHOR]

Given a bounded operator Q on a Hilbert space H , a pair of bounded operators (T 1 , T 2) on H is said to be Q-commuting if one of the following holds: T 1 T 2 = Q T 2 T 1 or T 1 T 2 = T 2 Q T 1 or T 1 T 2 = T 2 T 1 Q. We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88–90, 1963) for pairs of commuting contractions. In particular, for Q = q I H , where q is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of q-commuting contractions, which are well studied. There is an extended notion of q-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an n-tuple of q-commuting contractions, where n ≥ 3 . Generalizing the class of commuting contractions considered by Brehmer (Acta Sci Math (Szeged) 22:106–111, 1961), we construct a class of n-tuples of q-commuting contractions and find isometric dilations explicitly for the class. [ABSTRACT FROM AUTHOR]

In this article, we will demonstrate some Hardy’s inequalities by utilizing Hölder inequality, integration by parts, and chain rule of the conformable fractional calculus. When α=1\alpha =1, we can obtain some of classical Copson’s and Hardy’s inequalities. In the end, we will obtain an application of Hardy’s inequality utilizing the conformable fractional calculus.

In this paper, we characterize integro substitution operators that maintain closed ranges within L 2 (X , Σ , μ) , where (X , Σ , μ) represents a σ -finite measure space. Next, we give an explicit formula for the pseudo-inverse denoted as (T u φ) † of a integro substitution operators T u φ on this space using conditional expectation operators. [ABSTRACT FROM AUTHOR]

In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

For $ 1 1 < p < ∞ and a nondecreasing function K on $ [0,\infty) $ [ 0 , ∞) , we give an explicit derivative-free characterization of $ \mathcal {Q}_K $ Q K type space, denoted by $ \mathcal {Q}_K^p $ Q K p and characterize the boundary version of $ \mathcal {Q}_K^p $ Q K p . By using these results, we prove the corona theorems both for $ \mathcal {Q}_K^p\cap H^\infty $ Q K p ∩ H ∞ and for the multiplier algebra of $ \mathcal {Q}_K^p $ Q K p under some assumptions of the weight function K. [ABSTRACT FROM AUTHOR]

Let ω be a doubling weight and 0 < p ≤ q < ∞ . The essential norm of Riemann–Stieltjes operator T g from the weighted Bergman space A ω p to A ω q was investigated in the unit ball of ℂ n . [ABSTRACT FROM AUTHOR]

In this paper, we study composition and weighted composition operators that are close to isometries on H 2 but not necessarily isometric. We also obtain a Wold type decomposition for such operators. [ABSTRACT FROM AUTHOR]

We introduce the Hadamard–Bergman convolution on the half-plane. We show that it exists in terms of the Hadamard product and it is commutative on the Bergman space (more appropriately called the Bergman–Jerbashian space) in the half-plane. Further, we explore mapping properties of the generalized Bergman-type operators with exponential weights in weighted Bergman spaces in the half-plane. Finally, we deduce sharp inclusions for weighted Bergman spaces, from corresponding Sobolev-type inequalities. [ABSTRACT FROM AUTHOR]

Let S n − 1 denote unit sphere in R n equipped with the normalized Lebesgue measure. Let Φ ∈ L s (S n − 1) be a homogeneous function of degree zero such that ∫ S n − 1 Φ (y ′) d σ (y ′) = 0 , where y ′ = y / | y | for any y ≠ 0 . The commutators of variable Marcinkiewicz fractional integral operator is defined as [ b , μ Φ ] β m (f) (x) = (∫ 0 ∞ | ∫ | x − y | ≤ s Φ (x − y) [ b (x) − b (y) ] m | x − y | n − 1 − β (x) f (y) d y | 2 d s s 3) 1 2. In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces K ˙ p (⋅) α (⋅) , q) , θ (R n) . [ABSTRACT FROM AUTHOR]

In this paper, the structure of the nearly invariant subspaces for discrete semigroups generated by several (even infinitely many) automorphisms of the unit disc is described. As part of this work, the near S ∗ -invariance property of the image space C φ (ker T) is explored for composition operators C φ , induced by inner functions φ , and Toeplitz operators T. After that, the analysis of nearly invariant subspaces for strongly continuous multiplication semigroups of isometries is developed with a study of cyclic subspaces generated by a single Hardy class function. These are characterised in terms of model spaces in all cases when the outer factor is a product of an invertible function and a rational (not necessarily invertible) function. Techniques used include the theory of Toeplitz kernels and reproducing kernels. [ABSTRACT FROM AUTHOR]

This note provides an effective bound in the Gauss-Kuzmin-Lévy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval I 0 = 0 , 1 2 or I 0 = - 1 2 , 1 2 . We prove asymptotic formulas λ (T - n I) = μ (I) (λ (I 0) + O (q n)) for such transformations T, where λ is the Lebesgue measure on R , μ the normalized T-invariant Lebesgue absolutely continuous measure, I subinterval in I 0 , and q = 0.288 is smaller than the Wirsing constant q W ≈ 0.3036 . [ABSTRACT FROM AUTHOR]

On the Fock–Sobolev spaces, we study the range of Volterra inner derivations and composition inner derivations. The Volterra inner derivation ranges in the ideal of compact operators if and only if the induced function g is a linear polynomial. The composition inner derivation ranges in the ideal of compact operators if and only if the induced function φ is either identity or a contractive linear self-mapping of C . Moreover, we describe the compact intertwining relations for composition operators and Volterra operators between different Fock–Sobolev spaces. In this paper, our results are complement and in a sense extend some aspects of Calkin’s result (Ann Math 42:839–873, 1941) to the algebras of bounded linear operators on Fock–Sobolev spaces. [ABSTRACT FROM AUTHOR]

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range. [ABSTRACT FROM AUTHOR]

Consider the differential operator Δ a , b = ( d 2 d t 2 + 4 π i a b t d dt - 4 π 2 a 2 t 2 b 2 + 2 π i a b I) , t > 0 , a , b ∈ R , where I is the identity operator. The operator Δ a , b is known as affine Laplacian. We consider the heat equation associated to the operator Δ a , b with initial condition f from L 2 (R n) . Its solution is denoted by e t Δ a , b f . The transform f ↦ e t Δ a , b f is called affine heat kernel transform (or A-heat kernel transform). In this article, we consider (analytically extended) affine heat kernel transform and characterize the image of L 2 (R) under it as a weighted Bergman space of analytic functions on C with nonnegative weight. Consequently, we study L p -boundedness of affine heat kernel transform, L p -boundedness of affine Bargmann projection and related duality results. Moreover, we define affine Weyl translations and characterize the maximal and minimal spaces of analytic functions on C which are invariant under the affine Weyl translations. [ABSTRACT FROM AUTHOR]

In this paper, we completely describe the closed range of bounded weighted composition operator on the quaternionic Fock space F 2 (H) . This result further facilitates the characterizations for C f , φ or C f , φ ∗ preserving frames, Riesz bases or tight frames on F 2 (H) . Especially, we creatively obtain a useful formula of the adjoint operator C f , φ ∗ on F 2 (H) , which is crucial for showing the equivalence of the unitary, isometry and co-isometry of the weighted composition operator on F 2 (H) . This result is new even on the ordinary complex Fock space. Moreover, a series of new equivalent characterizations for quaternionic composition operators on F 2 (H) follow as a special case. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the d × d matrix-valued weights W on the complex plane C such that the Fock projection P α is bounded on the vector-valued spaces L α , W 2 (C d) induced by W. It is proved that P α is bounded on L α , W 2 (C d) if and only if W satisfies a restricted A 2 -condition. Then we establish some function-theoretic and operator-theoretic properties for the Fock spaces F α , W 2 (C d) induced by the d × d matrix-valued weights W satisfying the restricted A 2 -condition: we show that the C d -valued polynomials are dense in F α , W 2 (C d) ; a Littlewood–Paley formula for F α , W 2 (C d) is established; the bounded differentiation and integration operators D (n) : F α , W 2 (C d) → L 2 (Φ d A ; C d) are characterized, where Φ is a nonnegative matrix-valued function; we also investigate the boundedness of the Volterra type integration operator T G acting on F α , W 2 (C d) , where G is a matrix-valued entire function. In particular, it is shown that for d ≥ 2 , T G may be unbounded on F α , W 2 (C d) when G is a linear polynomial, and T G may be compact on F α , W 2 (C d) when G is a polynomial of degree greater than 3. These phenomena are in sharp contrast with the case d = 1 , where T G is bounded (resp. compact) if and only if G is a polynomial of degree not more than 2 (resp. a linear polynomial). [ABSTRACT FROM AUTHOR]

We study composition operators between variable exponent Lebesgue spaces and characterize boundedness and compactness of the composition operators on a variable exponent Lebesgue space. We also derive a sufficient condition for composition operator to have a closed range and explain some properties which these operators share with the case of Lebesgue spaces. [ABSTRACT FROM AUTHOR]

This study deals with the one-parameter family {Dq}q∈[0,1]{\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit qq-Durrmeyer operators. The continuity of this family with respect to the parameter qq is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that {Dq}q∈[0,1]{\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q∈[0,1]q\in \left[0,1]. When it comes to the uniform operator topology, the continuity is preserved solely at q=0q=0 and fails at all q∈(0,1].q\in \left(0,1]. In addition, a few estimates for the distance between two limit qq-Durrmeyer operators have been derived in the operator norm on C[0,1]C\left[0,1].

In this paper, we establish some reversed dynamic inequalities of Hilbert type on time scales nabla calculus by applying reversed Hölder's inequality, chain rule on time scales, and the mean inequality. As particular cases of our results (when T = N and T = R ), we get the reversed form of discrete and continuous inequalities proved by Chang-Jian, Lian-Ying and Cheung (Math. Slovaca 61(1):15–28, 2011). [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the boundedness of commutators of Hardy operators with rough kernels on grand variable Herz spaces K ˙ q ⁢ (⋅) a ⁢ (⋅) , u , θ ⁢ (ℝ n) by applying some properties of variable exponent. Moreover, by using the idea of grand variable Herz–Morrey spaces, we will prove the boundedness of Hardy operators on these spaces. [ABSTRACT FROM AUTHOR]

We completely characterize the boundedness of area operators from the Bergman spaces A α p (B n) to the Lebesgue spaces L q (S n) for all 0 < p,q < ∞. For the case n = 1, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math., 54, 2487–2500 (2011)]. Especially, in the case q < p and q < s, we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to n-complex dimension. [ABSTRACT FROM AUTHOR]

The Cesàro means of Taylor polynomials σ n , n ≥ 0 , are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polynomials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces D ζ , they serve as a proper LPAS. The primary objective of this note is to accurately determine the norm of σ n when it is considered as an operator on D ζ. There exist several practical methods to impose a norm on D ζ , and each norm results in a distinct operator norm for σ n. In this context, we explore three different norms on D ζ and, for each norm, precisely compute the value of ‖ σ n ‖ D ζ → D ζ . Furthermore, in all instances, we identify the maximizing functions and demonstrate their uniqueness. [ABSTRACT FROM AUTHOR]

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space H 2 . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly S ∗ -invariant subspace of H 2 , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study the existence of periodic solutions for the p-Laplacian systems with delay - (| z ′ (t) | p - 2 z ′ (t)) ′ = f (t , z (t + τ) , z (t) , z (t - τ)) , z (τ) - z (- τ) = z ′ (τ) - z ′ (- τ) = 0. Using the saddle point theorem and the linking theorem, some new existence theorems are obtained for second-order p-Laplacian systems with delay. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved. [ABSTRACT FROM AUTHOR]

We introduce grand complementary Morrey spaces, in a general form, on open sets Ω in R n . In case of unbounded sets Ω , this requires the notion of aggrandizer. We find natural conditions on the aggrandizer under which the grand space is larger than the initial space. We provide a counterexample showing that this embedding is strict in general. In the case Ω = R n , we study weighted Hardy operators with radial weights and find conditions for their boundedness in grand complementary Morrey spaces. In case of quasi-monotone weights, we also show that the conditions on the weight in terms of Matuszewska–Orlicz indices sufficient for the boundedness of Hardy operators in complementary Morrey spaces also remain to be sufficient for grand complementary Morrey spaces. [ABSTRACT FROM AUTHOR]

We characterize the L v p (R n) - L u q (R n) boundedness of the Dunkl-Hausdorff operator for radially decreasing functions. Both the cases 1 < p ≤ q < ∞ as well as 1 < q < p < ∞ have been discussed. For p = q , the boundedness has also been characterized in case the weights are radially monotone. [ABSTRACT FROM AUTHOR]

Let S (R n) be the Schwartz space and S ′ (R n) be the space of tempered distributions on R n . In this article, we prove that if H ⊆ S ′ (R n) is a non-zero Hilbert space of tempered distributions which is translation and modulation invariant such that | (f , g) | ≤ C ‖ f ‖ H for some C > 0 and for all f ∈ H , then H = L 2 (R n) , where g (x) = e - x 2 for all x ∈ R n and (· , ·) denotes the standard duality pairing between S ′ (R n) and S (R n) with respect to which (S (R n)) ∗ = S ′ (R n) . [ABSTRACT FROM AUTHOR]

This paper characterizes boundedness and compactness of the identity operator I d : F (p , p - 2 , s) → T t , m q (μ) . Applying this result, we obtain the characterization of the boundedness of the Volterra integral operator T g from F (p , p - 2 , s) spaces to LF (q , q - 2 , t) spaces. Furthermore, the compactness and essential norm of T g : F (p , p - 2 , s) → LF (q , q - 2 , t) are also investigated. [ABSTRACT FROM AUTHOR]

In this work, we provide a complete characterization of the boundedness of two classes of multiparameter Forelli–Rudin type operators from one mixed-norm Lebesgue space L p → to another space L q → , when 1 ≤ p → ≤ q → < ∞ , equipped with possibly different weights. Using these characterizations, we establish the necessary and sufficient conditions for both L p → - L q → boundedness of the weighted multiparameter Berezin transform and L p → - A q → boundedness of the weighted multiparameter Bergman projection, where A q → denotes the mixed-norm Bergman space. Our approach presents several novelties. Firstly, we conduct refined integral estimates of holomorphic functions on the unit ball in C n . Secondly, we adapt the classical Schur’s test to different weighted mixed-norm Lebesgue spaces. These improvements are crucial in our proofs and allow us to establish the desired characterization and sharp conditions. [ABSTRACT FROM AUTHOR]

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over N 0 k. As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system. [ABSTRACT FROM AUTHOR]

In this paper, we study Hankel operators on the Fock–Sobolev type spaces for all possible 1 ≤ p , q < ∞ and α ∈ R . We introduce a function space called integrable mean oscillation on C n . Then we characterize those symbols f for which the Hankel operators H f α and H f ¯ α are simultaneously bounded (or compact) from Fock–Sobolev type space F α p to Lebesgue space L α q . [ABSTRACT FROM AUTHOR]

This article continues the study of the Hadamard–Bergman operators in the unit disc of the complex plane. These operators arose as a natural generalization of the orthogonal projector and represent an integral realization of multiplier operators. Here we consider mainly the further development of the theory of such operators in the context of operators of variable order, that is, with kernels depending on an external variable. We give some important examples illustrating the relevance of such research, including the integrodifferentiation operator of variable order and certain Volterra type operator studied in complex analysis earlier. It is worth noting that it is the Hadamard–Bergman operators that seem to lay claim to the most direct complex analogue of operators with homogeneous kernels in the unit disc. This opens up excellent opportunities for further development of the study of operators of the Hadamard–Bergman type. [ABSTRACT FROM AUTHOR]