Molecular Decompositions Research Articles

Quantum Tunneling Instability in Pericyclic Reactions: The Cheletropic, Coarctate, and Ene Cases.

Some retro-pericyclic reactions, as a result of their high exothermicity and short trajectories, are the perfect ground for heavy atom tunneling molecular decompositions, also known as "quantum tunneling instability" (QTI). Considering this effect, in our first installment [Frenklach, A.; Amlani, H.; Kozuch, S. Quantum Tunneling Instability in Pericyclic Reactions. J. Am. Chem. Soc. 2024, 146 (17), 11823-11834, DOI: 10.1021/jacs.4c00608], we computed several retro-Diels-Alder reactions, predicting that many studied reactants cannot be isolated. Herein, we will explore the QTI of retro-cheletropic, coarctate, and ene exemplars, where again we hypothesize the impossibility to detect their reactants.

Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

We obtain new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel–Lizorkin spaces and use such tools to prove boundedness properties of Hermite pseudo-multipliers on those spaces. The notion of molecule we develop leads to boundedness of pseudo-multipliers associated to symbols of Hormander-type adapted to the Hermite setting on spaces for which the smoothness allowed includes non-positive values; in particular, we obtain continuity results for such operators on Lebesgue and Hermite local Hardy spaces. As a byproduct of our results on boundedness properties of pseudo-multipliers, we show that Hermite Besov spaces and Hermite Triebel–Lizorkin spaces are closed under non-linearities.

The [formula omitted] type Morrey–Triebel–Lizorkin spaces with variable smoothness and integrability

In this paper the authors introduce the Bωu type Morrey–Triebel–Lizorkin spaces with variable smoothness and integrability and give atomic and molecular decompositions of these new spaces. The lifting property of these spaces is also included.

The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents

Abstract In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝ n {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.

Variable Exponent Besov–Morrey Spaces

In this paper we introduce Besov-Morrey spaces with all indices variable and study some fundamental properties. This includes a description in terms of Peetre maximal functions and atomic and molecular decompositions. This new scale of non-standard function spaces requires the introduction of variable exponent mixed Morrey-sequence spaces, which in turn are defined within the framework of semimodular spaces. In particular, we obtain a convolution inequality involving special radial kernels, which proves to be a key tool in this work.

Atomic and Molecular Decomposition of Homogeneous Spaces of Distributions Associated to Non-negative Self-Adjoint Operators

We deal with homogeneous Besov and Triebel–Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel–Lizorkin spaces. Spectral multipliers for these spaces are established as well.

Weak Triebel–Lizorkin Spaces with Variable Integrability, Summability and Smoothness

Weak Triebel–Lizorkin spaces with variable integrability, summability and smoothness are first introduced. Then we establish a vector estimate for weak Lebesgue spaces with variable exponent. As an application we give equivalent quasi-norms in these new spaces by means of Peetre's maximal functions. Finally, we obtain the boundedness of the \varphi transform on these new spaces and their atomic and molecular decompositions.

Molecular decomposition and Fourier multipliers for holomorphic Besov and Triebel–Lizorkin spaces

Smooth molecular decompositions for holomorphic Besov and Triebel–Lizorkin spaces on the unit disk of the complex plane are constructed. The decompositions are used to obtain a boundedness result for Fourier multipliers. As further applications, we provide equivalent norms for the spaces under consideration, we consider the implications of the results on Hardy and Hardy–Sobolev spaces, and we study boundedness of coefficient multipliers.

Variable exponent Hardy spaces associated with discrete Laplacians on graphs

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.

Herz-Type Hardy Spaces Associated with Operators

Suppose L is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator L. Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator L. As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.

Anisotropic Herz-type Hardy spaces with variable exponent and their applications

A class of anisotropic Herz-type Hardy spaces with variable exponent associated with a non-isotropic dilation on $${\mathbb{R}^{n}}$$ are introduced, and characterizations of these spaces are established in terms of atomic and molecular decompositions. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of a linear operator on the anisotropic Herz-type Hardy spaces with variable exponent.

Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators

Anisotropic homogeneous mixed-norm Besov and Triebel–Lizorkin spaces are introduced and their properties are explored. A discrete adapted φ -transform decomposition is obtained. An associated class of almost diagonal operators is introduced and a boundedness result for such operators is obtained. Molecular decompositions for all the considered spaces are derived with the help of the algebra of almost diagonal operators. As an application, we obtain boundedness results on the considered spaces for Fourier multipliers and for pseudodifferential operators with suitable adapted homogeneous symbols using the molecular decomposition theory.

Bilinear operators with homogeneous symbols, smooth molecules, and Kato-Ponce inequalities

We present a unifying approach to establish mapping properties for bilinear pseudodifferential operators with homogeneous symbols in the settings of function spaces that admit a discrete transform and molecular decompositions in the sense of Frazier and Jawerth. As an application, we obtain related Kato-Ponce inequalities.

Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates

Let $(X, d, \mu)$ be a space of homogeneous type equipped with a distance $d$ and a measure $\mu$. Assume that $L$ is a closed linear operator which generates an analytic semigroup $e^{-tL}, t > 0$. Also assume that $L$ has a bounded $H_\infty$-calculus on $L^2(X)$ and satisfies the $L^p-L^q$ semigroup estimates (which is weaker than the pointwise Gaussian or Poisson heat kernel bounds). The aim of this paper is to establish a theory of inhomogeneous Besov spaces associated to such an operator $L$. We prove the molecular decompositions for the new Besov spaces and obtain the boundedness of the fractional powers $(I+L)^{-\gamma}, \gamma > 0$ on these Besov spaces. Finally, we carry out a comparison between our new Besov spaces and the classical Besov spaces.

Can a Photosensitive Oxide Catalyze Decomposition of Energetic Materials?

Organic–inorganic interfaces provide both intrigues and opportunities for designing systems that possess properties and functionalities inaccessible by each individual component. In particular, the electronic, catalytic, and defect properties of inorganic surfaces can significantly affect the adsorption, decomposition, and photoresponse of organic molecules. Here, we choose the formulation of TiO2 and trinitrotoluene (TNT), a highly catalytic oxide and a prominent explosive, as a prototypical example to explore the effect of a catalytic oxide additive on the photosensitivity of energetic materials. We show that whether or not a catalytic oxide additive can help molecular decompositions under light illumination depends largely on the band alignment between the oxide surface and the energetic molecule. For the composite of TiO2 and TNT, the lowest unoccupied molecular orbitals (LUMOs) of TNT merge within the conduction band (CB) of TiO2. As such, no optical transition corresponding to available laser energies is observed. However, oxygen vacancy can lead to electron density transfer from the surface to the energetic molecules, causing an enhancement of the bonding between molecules and surface and a reduction of the molecular decomposition activation barriers. Therefore, when other (than optical) forms of energy (shock, heat, etc.) flow into molecules, molecular decompositions may be triggered more easily.

Anisotropic Hardy-Lorentz spaces and their applications

Let $p\in(0,1]$, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. The authors introduce the anisotropic Hardy-Lorentz space $H^{p,q}_A(\mathbb{R}^n)$ associated with $A$ via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic or the molecular decompositions, the radial or the non-tangential maximal functions, or the finite atomic decompositions. All these characterizations except the $\infty$-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on $\mathbb{R}^n$. As applications, the authors first prove that $H^{p,q}_A(\mathbb{R}^n)$ is an intermediate space between $H^{p_1,q_1}_A(\mathbb{R}^n)$ and $H^{p_2,q_2}_A(\mathbb{R}^n)$ with $0<p_1<p<p_2<\infty$ and $q_1,\,q,\,q_2\in(0,\infty]$, and also between $H^{p,q_1}_A(\mathbb{R}^n)$ and $H^{p,q_2}_A(\mathbb{R}^n)$ with $p\in(0,\infty)$ and $0<q_1<q<q_2\leq\infty$ in the real method of interpolation. The authors then establish a criterion on the boundedness of sublinear operators from $H^{p,q}_A(\mathbb{R}^n)$ into a quasi-Banach space; moreover, the authors obtain the boundedness of $\delta$-type Calder\'{o}n-Zygmund operators from $H^p_A(\mathbb{R}^n)$ to the weak Lebesgue space $L^{p,\infty}(\mathbb{R}^n)$ (or $H^{p,\infty}_A(\mathbb{R}^n)$) in the critical case, from $H_A^{p,q}(\mathbb{R}^n)$ to $L^{p,q}(\mathbb{R}^n)$ (or $H_A^{p,q}(\mathbb{R}^n)$) with $\delta\in(0,\frac{\ln\lambda_-}{\ln b}]$, $p\in(\frac1{1+\delta},1]$ and $q\in(0,\infty]$, as well as the boundedness of some Calder\'{o}n-Zygmund operators from $H_A^{p,q}(\mathbb{R}^n)$ to $L^{p,\infty}(\mathbb{R}^n)$, where $b:=|\det A|$, $\lambda_-:=\min\{|\lambda|:\ \lambda\in\sigma(A)\}$ and $\sigma(A)$ denotes the set of all eigenvalues of $A$.

Intrinsic atomic and molecular decompositions of Hardy–Musielak–Orlicz spaces

We introduce the Hardy type space for Musielak–Orlicz spaces. It includes several existing Hardy type spaces such as the Hardy–Orlicz spaces and the Hardy spaces with variable exponents. Furthermore, we develop an atomic decomposition such that the size condition just relies on the norms of Musielak–Orlicz spaces. This gives us a nature extension of the molecular decompositions to the Hardy type space for Musielak–Orlicz spaces.

Atomic and molecular decompositions in variable exponent 2-microlocal spaces and applications

In this article we study atomic and molecular decompositions in 2-microlocal Besov and Triebel–Lizorkin spaces with variable integrability. We show that, in most cases, the convergence implied in such decompositions holds not only in the distributions sense, but also in the function spaces themselves. As an application, we give a simple proof for the denseness of the Schwartz class in such spaces. Some other properties, like Sobolev embeddings, are also obtained via atomic representations.

Smooth Decompositions of Triebel-Lizorkin and Besov Spaces on Product Spaces of Homogeneous Type

We introduce Triebel-Lizorkin and Besov spaces by Calderón’s reproducing formula on product spaces of homogeneous type. We also obtain smooth atomic and molecular decompositions for these spaces.

Erratum to: Atomic and molecular decompositions of anisotropic Besov spaces

We give a corrected proof of Lemma 3.1 in [1]. While the statement of [1, Lemma 3.1] is true, its proof is incorrect. The argument contains a serious defect which can not be easily corrected. The inequality that appears in [1] before (3.5) is not true. If this inequality was true, then we could conclude that, even for a non doubling measure μ, (3.5) was also true. But there exist some non doubling measures for which (3.5) is not true. Since this result plays a fundamental role in the rest of the paper, it becomes compulsory to provide the correct proof of [1, Lemma 3.1].