Recent zbMATH articles in MSC 30Hhttps://zbmath.org/atom/cc/30H2021-06-15T18:09:00+00:00WerkzeugHarmonic extension of \(Q_{\mathcal{K}}\)-type spaces via regular wavelets.https://zbmath.org/1460.420362021-06-15T18:09:00+00:00"Han, Fang"https://zbmath.org/authors/?q=ai:han.fang"Li, Pengtao"https://zbmath.org/authors/?q=ai:li.pengtaoSummary: In this paper, we use regular wavelets to investigate the harmonic extension of a class of \(Q_{\mathcal{K}}\)-type spaces \(Q_{\mathcal{K},\gamma}(\mathbb{R})\). For a locally integrable function \(f\), we apply regular wavelets to decompose and estimate the Poisson integral of \(f\). Then, by the aid of a reproducing formula, we characterize the harmonic extension of \(Q_{\mathcal{K},\gamma}(\mathbb{R})\).Univariate tight wavelet frames of minimal support.https://zbmath.org/1460.420452021-06-15T18:09:00+00:00"Gómez-Cubillo, F."https://zbmath.org/authors/?q=ai:gomez-cubillo.fernando"Villullas, S."https://zbmath.org/authors/?q=ai:villullas.sergioSummary: This work characterizes (dyadic homogeneous) wavelet frames for \(L^2(\mathbb{R})\) by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated with the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis, and extension principles. Spectral formulas are used to determine all the tight wavelet frames for \(L^2(\mathbb{R})\) with a fixed finite number of generators of minimal support. The method associates wavelet frames of this type with certain inner operator-valued functions in Hardy spaces. The cases with one and two generators are completely solved.Bohr's inequality for harmonic mappings and beyond.https://zbmath.org/1460.300032021-06-15T18:09:00+00:00"Kayumova, Anna"https://zbmath.org/authors/?q=ai:kayumova.anna"Kayumov, Ilgiz R."https://zbmath.org/authors/?q=ai:kayumov.ilgiz-rifatovich"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: There has been a number of problems closely connected with the classical Bohr inequality for bounded analytic functions defined on the unit disk centered at the origin. Several extensions, generalizations and modifications of it are established by many researchers and they can be found in the literature, for example, in the multidimensional setting and in the case of the Dirichlet series, functional series, function spaces, etc. In this survey article, we mainly focus on the recent developments on this topic and in particular, we discuss new and sharp improvements on the classical Bohr inequality and on the Bohr inequality for harmonic functions.
For the entire collection see [Zbl 1411.65006].Dirichlet series and holomorphic functions in high dimensions.https://zbmath.org/1460.300042021-06-15T18:09:00+00:00"Defant, Andreas"https://zbmath.org/authors/?q=ai:defant.andreas"García, Domingo"https://zbmath.org/authors/?q=ai:garcia.domingo"Maestre, Manuel"https://zbmath.org/authors/?q=ai:maestre.manuel"Sevilla-Peris, Pablo"https://zbmath.org/authors/?q=ai:sevilla-peris.pabloPublisher's description: Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twenty years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.Cyclicity in Dirichlet type spaces.https://zbmath.org/1460.470062021-06-15T18:09:00+00:00"Kellay, K."https://zbmath.org/authors/?q=ai:kellay.karim"Le Manach, F."https://zbmath.org/authors/?q=ai:le-manach.florian"Zarrabi, M."https://zbmath.org/authors/?q=ai:zarrabi.mohamedSummary: We study cyclicity in the Dirichlet type spaces for outer functions whose zero set is countable.
For the entire collection see [Zbl 1444.30001].Two weight commutators on spaces of homogeneous type and applications.https://zbmath.org/1460.420312021-06-15T18:09:00+00:00"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinh"Gong, Ruming"https://zbmath.org/authors/?q=ai:gong.ruming"Kuffner, Marie-Jose S."https://zbmath.org/authors/?q=ai:kuffner.marie-jose-s"Li, Ji"https://zbmath.org/authors/?q=ai:li.ji.1"Wick, Brett D."https://zbmath.org/authors/?q=ai:wick.brett-d"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyongSummary: In this paper, we establish the two weight commutator theorem of Calderón-Zygmund operators in the sense of Coifman-Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for \(A_2\) weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderón-Zygmund operators: Cauchy integral operator on \(\mathbb{R}\), Cauchy-Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).A new proof of the ultrametric Hermite-Lindermann theorem [A new proof of the ultrametric Hermite-Lindemann theorem].https://zbmath.org/1460.111022021-06-15T18:09:00+00:00"Escassut, A."https://zbmath.org/authors/?q=ai:escassut.alainSummary: We propose a new proof of the Hermite-Lindemann Theorem in an ultrametric field by using classical properties of analytic functions. The proof remains valid in zero residue characteristic.Dynamical C*-algebras and kinetic perturbations.https://zbmath.org/1460.810462021-06-15T18:09:00+00:00"Buchholz, Detlev"https://zbmath.org/authors/?q=ai:buchholz.detlev"Fredenhagen, Klaus"https://zbmath.org/authors/?q=ai:fredenhagen.klausSummary: The framework of dynamical \(\mathrm{C^*}\)-algebras for scalar fields in Minkowski space, based on local scattering operators, is extended to theories with locally perturbed kinetic terms. These terms encode information about the underlying spacetime metric, so the causality relations between the scattering operators have to be adjusted accordingly. It is shown that the extended algebra describes scalar quantum fields, propagating in locally deformed Minkowski spaces. Concrete representations of the abstract scattering operators, inducing this motion, are known to exist on Fock space. The proof that these representers also satisfy the generalized causality relations requires, however, novel arguments of a cohomological nature. They imply that Fock space representations of the extended dynamical \(\mathrm{C^*}\)-algebra exist, involving linear as well as kinetic and pointlike quadratic perturbations of the field.Sharp Riesz-Fejér inequality for harmonic Hardy spaces.https://zbmath.org/1460.310052021-06-15T18:09:00+00:00"Melentijević, Petar"https://zbmath.org/authors/?q=ai:melentijevic.petar"Božin, Vladimir"https://zbmath.org/authors/?q=ai:bozin.vladimirSummary: We prove sharp version of Riesz-Fejér inequality for functions in harmonic Hardy space \(h^p(\mathbb{D})\) on the unit disk \(\mathbb{D}\), for \(p > 1\), thus extending the result from \textit{I. R. Kayumov} et al. [Potential Anal. 52, No. 1, 105--113 (2020; Zbl 1439.31001)] and resolving the posed conjecture.Weak Hardy-type spaces associated with ball quasi-Banach function spaces. II: Littlewood-Paley characterizations and real interpolation.https://zbmath.org/1460.420332021-06-15T18:09:00+00:00"Wang, Songbai"https://zbmath.org/authors/?q=ai:wang.songbai"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen"Zhang, Yangyang"https://zbmath.org/authors/?q=ai:zhang.yangyangSummary: Let \(X\) be a ball quasi-Banach function space on \(\mathbb{R}^n\). In this article, assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on \(X\) as well as it is bounded on both the weak ball quasi-Banach function space \(WX\) and the associated space, the authors establish various Littlewood-Paley function characterizations of \(WH_X(\mathbb{R}^n)\) under some weak assumptions on the Littlewood-Paley functions. The authors also prove that the real interpolation intermediate space \((H_X(\mathbb{R}^n),L^\infty(\mathbb{R}^n))_{\theta,\infty}\), between the Hardy space associated with \(X, H_X(\mathbb{R}^n)\), and the Lebesgue space \(L^\infty(\mathbb{R}^n)\), is \(WH_{X^{1/(1-\theta)}}(\mathbb{R}^n)\), where \(\theta\in (0,1)\). All these results are of wide applications. Particularly, when \(X:=M_q^p(\mathbb{R}^n)\) (the Morrey space), \(X:=L^{\vec{p}}(\mathbb{R}^n)\) (the mixed-norm Lebesgue space) and \(X:=(E_\Phi^q)_t(\mathbb{R}^n)\) (the Orlicz-slice space), all these results are even new; when \(X:=L_\omega^\Phi(\mathbb{R}^n)\) (the weighted Orlicz space), the result on the real interpolation is new and, when \(X:=L^{p(\cdot)}(\mathbb{R}^n)\) (the variable Lebesgue space) and \(X:=L_\omega^\Phi (\mathbb{R}^n)\), the Littlewood-Paley function characterizations of \(WH_X(\mathbb{R}^n)\) obtained in this article improves the existing results via weakening the assumptions on the Littlewood-Paley functions.A \(q\)-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application.https://zbmath.org/1460.420382021-06-15T18:09:00+00:00"Song, Liang"https://zbmath.org/authors/?q=ai:song.liang"Wu, Liangchuan"https://zbmath.org/authors/?q=ai:wu.liangchuanSummary: The theory of tent spaces on \(\mathbb{R}^n\) was introduced by \textit{R. R. Coifman} et al. [Lect. Notes Math. 992, 1--15 (1983; Zbl 0523.42016), ``Some new functions and their applications to harmonic analysis'', J. Funct. Anal. 62, 304--335 (1985)], including atomic decomposition, duality theory and so on. \textit{E. Russ} [``The atomic decomposition for tent spaces on spaces of homogeneous type'' (2006), \url{https://maths.anu.edu.au/files/CMAProc42-r.pdf}] generalized the atomic decomposition for tent spaces to the case of spaces of homogeneous type \((X,d,\mu)\). The main purpose of this paper is to extend the results of Coifman et al. [loc. cit.] and Russ [loc. cit.] to weighted version. More precisely, we obtain a \(q\)-atomic decomposition for the weighted tent spaces \(T^p_{2,w}(X)\), where \(0<p\leq 1\), \(1<q<\infty\), and \(w\in A_\infty\). As an application, we give an atomic decomposition for weighted Hardy spaces associated to non-negative self-adjoint operators on \(X\).A variant of Yano's extrapolation theorem on Hardy spaces.https://zbmath.org/1460.300182021-06-15T18:09:00+00:00"Bakas, Odysseas"https://zbmath.org/authors/?q=ai:bakas.odysseasLet \((X,\mu)\) and \((Y,\nu)\) be two finite measure spaces, and \(T \colon (X,\mu) \rightarrow (Y,\nu)\) a sublinear operator, i.e,
\[ ||T(f+g)|| \le ||T(f)||+||T(g)||\quad\text{and} \quad||T(\alpha f)|| \le |\alpha|||T(f)||\]
such that there exist constants \(C_0, r > 0\) s.t.
\[\sup\limits_{||g||_{{\mathcal{L}}^p(X)} = 1}||T(g)||_{{\mathcal{L}}^p(Y)} \le C_0(p-1)^{-r}\] for every \(1 < p \le 2\). Then a classical theorem of \textit{Y. Yano} [J. Math. Soc. Japan 3, 296--305 (1951; Zbl 0045.17901)] asserts that \[ ||T(f)|| _{{\mathcal{L}}^1(Y)} \le A+B \int_X |f(x)||\log^r(1+|f(x)|)|d\mu(x)\] for all measurable functions \(f\) on \(X\), where \(A\),\(B > 0\) are constants depending only on \(C_0\), \(r\), \(\mu(X)\) and \(\nu(Y)\). Motivated by some classical results in [\textit{S. K. Pichorides}, Proc. Am. Math. Soc. 114, No. 3, 787--789 (1992; Zbl 0744.42005); \textit{A. Zygmund}, Fundam. Math. 30, 170--196 (1938; Zbl 0019.01602)], the paper under review proves a version of the aforementioned extrapolation theorem of Yano for sublinear operators acting on analytic Hardy spaces over \(\mathbb T\). For \(1 \le p \le \infty\).
The main result of the paper (Theorem 1) reads: For a sublinear operator \(T\) acting on the spaces of measurable functions on \(\mathbb{T}\), if there exist constants \(C_0, r > 0\) s.t \[\sup\limits_{g \in H^p(\mathbb{T}),||g||_{{\mathcal{L}}^p(\mathbb{T})} = 1}||T(g)||_{{\mathcal{L}}^p(\mathbb{T})} \le C_0(p-1)^{-r}\]
for every \(1 < p \le 2\), then there exists a constant \(D > 0\), depending only on \(C_0, r\) s.t. \( ||T(g)|| _{{\mathcal{L}}^1(\mathbb{T})} \le D||f||_{L\log^r({\mathcal{L}}(\mathbb{T})}\) for every analytic trigonometric polynomial \(f\) on \(\mathbb{T}\).
It is interesting to note that, for sublinear operators satisfying the assumptions of the main theorem (Theorem 1), the main theorem improves the exponent
\(s=r+1\) in \(L\log^sL(\mathbb{T})\) to the optimal one \(s=r\). The author also proves that Theorem 1 is sharp. The paper concludes with discussions on extensions of the main result (Theorem 1) to Hardy-Orlicz spaces and their higher-dimensional variants.
Reviewer: Abebaw Tadesse (Langston)Higher-order Riesz transforms of Hermite operators on new Besov and Triebel-Lizorkin spaces.https://zbmath.org/1460.420342021-06-15T18:09:00+00:00"Bui, The Anh"https://zbmath.org/authors/?q=ai:bui.the-anh"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinhSummary: Consider the Hermite operator \(H=-\Delta +|x|^2\) on the Euclidean space \(\mathbb{R}^n\). The aim of this article is to prove the boundedness of higher-order Riesz transforms on appropriate Besov and Triebel-Lizorkin spaces. As an application, we prove certain regularity estimates of second-order elliptic equations in divergence form with the oscillator perturbations.