Recent zbMATH articles in MSC 42B https://zbmath.org/atom/cc/42B 2022-06-24T15:10:38.853281Z Werkzeug The reverse Hölder inequality for an elementary function https://zbmath.org/1485.26025 2022-06-24T15:10:38.853281Z "Korenovskii, A. O." https://zbmath.org/authors/?q=ai:korenovskii.a-o Summary: For a positive function $$f$$ on the interval $$[0,1]$$, the power mean of order $$p\in\mathbb{R}$$ is defined by $\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\quad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$ Assume that $$0<A<B$$, $$0<\theta<1$$ and consider the step function $$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$$, where $$\chi_E$$ is the characteristic function of the set $$E$$. Let $$-\infty<p<q<+\infty$$. The main result of this work consists in finding the term $C_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$ For fixed $$p<q$$, we study the behaviour of $$C_{p<q,A<B}$$ and $$\theta_{p<q,A<B}$$ with respect to $$\beta=B/A\in(1,+\infty)$$. The cases $$p=0$$ or $$q=0$$ are considered separately. The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse Hölder inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in [the author, Ann. Mat. Pura Appl. (4) 195, No. 2, 659--680 (2016; Zbl 1342.26044)]. Holomorphic function spaces on homogeneous Siegel domains https://zbmath.org/1485.32003 2022-06-24T15:10:38.853281Z "Calzi, Mattia" https://zbmath.org/authors/?q=ai:calzi.mattia "Peloso, Marco M." https://zbmath.org/authors/?q=ai:peloso.marco-maria The authors study holomorphic functions on homogeneous Siegel domains. They concentrate mainly on weighted mixed norm Bergman spaces. The problems considered include: sampling, atomic decomposition, duality, boundary values, and boundedness of the Bergman projectors. The work consists of an introduction and five chapters, it also contains an appendix devoted to mixed norm spaces. We shall briefly describe the contents of the consecutive chapters. In the introduction, the authors motivate their study by discussing the simplest example of a Siegel domain: the upper half-plane $$\mathbb{C}_{+}:=\mathbb{R}+i\mathbb{R}_{+}^{*}$$. They also present known results concerning function theory on Siegel domains. Let $$E$$ be a vector space over $$\mathbb{C}$$ of finite dimension $$n$$ and $$F$$ be a vector space over $$\mathbb{R}$$ of finite dimension $$m>0$$. Let $$\Omega$$ be an open convex cone with vertex $$0$$ in $$F$$ and not containing any affine lines. Also, let $$\Phi\colon E\times E\rightarrow F_{\mathbb{C}}$$, $$F_{\mathbb{C}}$$ is the complexification of $$F$$, i.e., $$F_{\mathbb{C}}=F\otimes_{\mathbb{R}}\mathbb{C}$$, be a positive non-degenerate hermitian mapping, that is: \begin{itemize} \item[(i)] $$\Phi$$ is linear in the first argument; \item[(ii)] $$\Phi(\zeta,\zeta^{'})=\overline{\Phi(\zeta^{'},\zeta)}$$ for all $$\zeta,\zeta^{'}\in E$$; \item[(iii)] $$\Phi$$ is non-degenerate; \item[(iv)] $$\Phi(\zeta):=\Phi(\zeta,\zeta)\in \overline{\Omega}$$. \end{itemize} {\em The Siegel domain of type II} associated with the cone $$\Omega$$ and the mapping $$\Phi$$ is defined the following way: $D:=\{(\zeta,z)\in E\times F_{\mathbb{C}}\colon \Im z-\Phi(\zeta)\in \Omega\}.$ Chapter one contains, beside the above definition and basic examples, definitions of the Fourier transform, the Bergman and the Hardy spaces, the formulation of the corresponding Paley-Wiener theorems and a discussion of the Kohn Laplacian. In Chapter two the authors introduce various objects related to homogeneous Siegel domains of type II. This includes a discussion of $$T$$-algebras and the associated homogeneous cones, the generalized power functions $$\Delta_{\Omega}^{s}$$, $$\Delta_{\Omega^{'}}^{s}$$, $$\Omega^{'}$$ the dual cone and the associated gamma and beta functions. They also introduce the corresponding Bergman metric. Chapter three is devoted to the study of the weighted Bergman spaces $$A_{s}^{p,q}(D)$$. These spaces are defined for $$\mathbf{s}\in\mathbb{R}^{r}$$ and $$p,q\in (0,\infty]$$ as \begin{align*} A_{\mathbf{s}}^{p,q}(D)&:=\mathrm{Hol}(D)\cap L_{\mathbf{s}}^{p,q}(D)\\ A_{\mathbf{s},0}^{p,q}(D)&:=\mathrm{Hol}(D)\cap L_{\mathbf{s},0}^{p,q}(D), \end{align*} where $$L_{\mathbf{s}}^{p,q}(D)$$ is the Hausdorff space associated with the space $\{f\colon D\rightarrow \mathbb{C}\colon f \text{\:measurable\:} \int_{\Omega}(\Delta_{\Omega}^{\mathbf{s}}\|f_{h}\|_{L^{p}(\mathcal{N}})^{q}d\nu_{\Omega}(h)<\infty\}$ and $$L_{\mathbf{s},0}^{p,q}(D)$$ is the closure of $$C_{c}(D)$$ in $$L_{\mathbf{s}}^{p,q}(D)$$. The measure $$\nu_{\Omega}$$ is $$\Delta_{\Omega}^{\mathbf{d}}\cdot \mathcal{H}^{m}$$ for an appropriate $$\mathbf{d}$$ and $$\mathcal{H}^{m}$$ the Hausdorff measure. The symbol $$\mathcal{N}$$ stands for $$E\times F$$ endowed with the group structure $(\zeta,x)(\zeta^{'},x^{'}):=(\zeta+\zeta^{'},x+x^{'}+2\Im \Phi(\zeta,\zeta^{'})$ and $f_{h}\colon \mathcal{N}\ni (\zeta,x)\mapsto f(\zeta,x+i\Phi(\zeta)+ih)\in \mathbb{C}.$ The values $$p,q,\mathbf{s}$$ for which $$A_{\mathbf{s}}^{p,q}(D)$$ is non-trivial are characterized (Proposition 3.5). Some sampling results are obtained (Theorems 3.22 and 3.23). In Section 3.4 the authors deal with atomic decomposition for the spaces $$A_{\mathbf{s}}^{p,q}(D)$$. In Section 3.5 there is studied duality of these spaces. In Chapter 4 the authors introduce and study Besov-type spaces $$B_{p,q}^{\mathbf{s}}(\mathcal{N},\Omega)$$. The theory parallels the classical one defined on $$\mathbb{R}^{n}$$. It is modelled in relation to the boundary values of the spaces $$A_{\mathbf{s}}^{p,q}(D)$$. It should be noted that the group $$\mathcal{N}$$ is not commutative and the authors deal with the full range of exponents $$p,q\in (0,\infty]$$. The main results of Chapter 5 concern the boundedness of the Bergman projectors. Theorem 5.25 gives some conditions on $$p,q\in [1,\infty]$$ and $$\mathbf{s}, \mathbf{s}^{'}\in \mathbb{R}^{r}$$ such that the corresponding projector $$P_{\mathbf{s}^{'}}$$ (defined in Definition 5.19) is a continuous linear mapping of $$L_{\mathbf{s},0}^{p,q}(D)$$ into $$\tilde{A}_{\mathbf{s}}^{p,q}(D)$$ (this space is defined as the image of some extension operator on appropriate Besov spaces -- see Definition 5.3). The study of the Bergman projectors is related to atomic decompositions. The chapter opens with a discussion of boundary values $$f_{h}$$ as $$h\rightarrow 0$$, $$h\in \Omega$$ for functions $$f\in A_{\mathbf{s}}^{p,q}(D)$$. Reviewer: Michal Jasiczak (Poznań) A mean value formula for the iterated Dunkl-Helmoltz operator https://zbmath.org/1485.35012 2022-06-24T15:10:38.853281Z "González Vieli, F. J." https://zbmath.org/authors/?q=ai:gonzalez-vieli.francisco-javier Summary: We establish a spherical mean value formula for the iterated Dunkl-Helmoltz operator, thus generalizing a result of Mejjaoli and Trimèche. We then give an application to distributions with Dunkl transform supported by the unit sphere. Strong traces to degenerate parabolic equations https://zbmath.org/1485.35274 2022-06-24T15:10:38.853281Z "Erceg, Marko" https://zbmath.org/authors/?q=ai:erceg.marko "Mitrović, Darko" https://zbmath.org/authors/?q=ai:mitrovic.darko Global Fourier integral operators in the plane and the square function https://zbmath.org/1485.35447 2022-06-24T15:10:38.853281Z "Manna, Ramesh" https://zbmath.org/authors/?q=ai:manna.ramesh "Ratnakumar, P. K." https://zbmath.org/authors/?q=ai:ratnakumar.p-k Summary: We prove the local smoothing estimate for general Fourier integral operators with phase function of the form $$\phi(x, t, \xi)=x\cdot\xi + t\,q(\xi)$$, with $$q \in C^\infty(\mathbb{R}^2\setminus\{0\})$$, homogeneous of degree one, and amplitude functions in the symbol class of order $$m \le 0$$. The result is global in the space variable, and also improves our previous work in this direction [the authors, in: Advances in harmonic analysis and partial differential equations. Based on the 12th ISAAC congress, session Harmonic analysis and partial differential equations'', Aveiro, Portugal, July 29 -- August 2, 2019. Cham: Birkhäuser. 1--35 (2020; Zbl 1477.42005)]. The approach involves a reduction to operators with amplitude function depending only on the covariable, and a new estimate for square function based on angular decomposition. Approximation of function using generalized Zygmund class https://zbmath.org/1485.42001 2022-06-24T15:10:38.853281Z "Nigam, H. K." https://zbmath.org/authors/?q=ai:nigam.hare-krishna "Mursaleen, Mohammad" https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-ayman "Rani, Supriya" https://zbmath.org/authors/?q=ai:rani.supriya Summary: In this paper we review some of the previous work done by \textit{M. V. Singh} et al. [J. Inequal. Appl. 2017, Paper No. 101, 11 p. (2017; Zbl 1360.42005)], \textit{S. Lal} and \textit{Shireen} [Bull. Math. Anal. Appl. 5, No. 4, 1--13 (2013; Zbl 1314.42012)], etc., on error approximation of a function $$g$$ in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions $$g$$ and $$g^{\prime}$$, where $$g^{\prime}$$ is a derived function of a $$2 \pi$$-periodic function $$g$$, in the generalized Zygmund class $$X_z^{(\eta)}$$, $$z\geq 1$$, using matrix-Cesàro $$(TC^\delta)$$ means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of \textit{B. P. Dhakal} [Int. Math. Forum 5, No. 33--36, 1729--1735 (2010; Zbl 1210.42002); Approximation of a function $$f$$ belonging to Lip class by $$(N, p, q)C_1$$ means of its Fourier series'', Int. J. Eng. Technol. 2, No. 3, 1--15 (2013)], the first author [Surv. Math. Appl. 5, 113--122 (2010; Zbl 1399.42005); Commun. Appl. Anal. 14, No. 4, 607--614 (2010; Zbl 1214.42012)], the first author and \textit{A. Sharma} [Kyungpook Math. J. 50, No. 4, 545--556 (2010; Zbl 1227.42008); Degree of approximation of a function belonging to $$\mathrm{Lip}(\xi,(t),r)$$ class by $$(E,1)(C,1)$$ product means'', Int. J. Pure Appl. Math. 70, No. 6, 775--784 (2011)], \textit{J. K. Kushwaha} and \textit{B. P. Dhakal} [Approximation of a function belonging to $$\mathrm{Lip} (\alpha, r)$$ class by $$N_{p,q}$$: $$C_1$$ summability method of its Fourier series'', Nepal J. Sci. Technol. 14, No. 2, 117--122 (2013; \url{doi:10.3126/njst.v14i2.10424})], \textit{U. K. Shrivastava} et al. [Approximation of function belonging to the $$\mathrm{Lip}(\Psi (t), p)$$ class by matrix-Cesáro summability method'', IOSR J. Math. 10, No. 1, 39--41 (2014)] become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1. Approximation of functions in generalized Zygmund class by double Hausdorff matrix https://zbmath.org/1485.42002 2022-06-24T15:10:38.853281Z "Nigam, H. K." https://zbmath.org/authors/?q=ai:nigam.hare-krishna "Mursaleen, M." https://zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammad "Rani, Supriya" https://zbmath.org/authors/?q=ai:rani.supriya Summary: In the present work, we emphasize, for the first time, the error estimation of a two-variable function $$g(y,z)$$ in the generalized Zygmund class $$Y_r^{(\xi)}$$ ($$r\geq 1$$) using the double Hausdorff matrix means of its double Fourier series. In fact, in this work, we establish two theorems on error estimation of a two-variable function of $$g$$ in the generalized Zygmund class. Approximation properties of the double Fourier sphere method https://zbmath.org/1485.42011 2022-06-24T15:10:38.853281Z "Mildenberger, Sophie" https://zbmath.org/authors/?q=ai:mildenberger.sophie "Quellmalz, Michael" https://zbmath.org/authors/?q=ai:quellmalz.michael Summary: We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical Hölder spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence. Trigonometric approximation of functions $$f(x,y)$$ of generalized Lipschitz class by double Hausdorff matrix summability method https://zbmath.org/1485.42012 2022-06-24T15:10:38.853281Z "Mishra, Abhishek" https://zbmath.org/authors/?q=ai:mishra.abhishek-c "Mishra, Vishnu Narayan" https://zbmath.org/authors/?q=ai:mishra.vishnu-narayan "Mursaleen, M." https://zbmath.org/authors/?q=ai:mursaleen.mohammad Summary: In this paper, we establish a new estimate for the degree of approximation of functions $$f(x,y)$$ belonging to the generalized Lipschitz class $$\mathrm{Lip} ((\xi_1, \xi_2);r)$$, $$r \geq 1$$, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $$\mathrm{Lip} ((\alpha,\beta);r)$$ and $$\mathrm{Lip}(\alpha,\beta)$$ in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $$(C, \gamma, \delta)$$ means. Quantum Fourier analysis https://zbmath.org/1485.42013 2022-06-24T15:10:38.853281Z "Jaffe, Arthur" https://zbmath.org/authors/?q=ai:jaffe.arthur-m "Jiang, Chunlan" https://zbmath.org/authors/?q=ai:jiang.chunlan "Wu, Jinsong" https://zbmath.org/authors/?q=ai:wu.jinsong Summary: Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform, as a map between suitably defined $$L^p$$ spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems. On the regularity of distributions via the convergence of the continuous shearlet transform in two dimensions https://zbmath.org/1485.42014 2022-06-24T15:10:38.853281Z "Navarro, Jaime" https://zbmath.org/authors/?q=ai:navarro.jaime "Elizarraraz, David" https://zbmath.org/authors/?q=ai:elizarraraz.david Fourier restriction above rectangles https://zbmath.org/1485.42015 2022-06-24T15:10:38.853281Z "Schwend, Jeremy" https://zbmath.org/authors/?q=ai:schwend.jeremy "Stovall, Betsy" https://zbmath.org/authors/?q=ai:stovall.betsy Summary: In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure. The Schrödinger equation in $$L^p$$ spaces for operators with heat kernel satisfying Poisson type bounds https://zbmath.org/1485.42016 2022-06-24T15:10:38.853281Z "Chen, Peng" https://zbmath.org/authors/?q=ai:chen.peng "Duong, Xuan Thinh" https://zbmath.org/authors/?q=ai:duong.xuan-thinh "Fan, Zhijie" https://zbmath.org/authors/?q=ai:fan.zhijie "Li, Ji" https://zbmath.org/authors/?q=ai:li.ji.1 "Yan, Lixin" https://zbmath.org/authors/?q=ai:yan.lixin The operator $$e^{it\Delta}$$ is $$L^p(\mathbb{R}^n) \mapsto L^p(\mathbb{R}^n)$$ bounded only when $$p = 2$$, but the $$p$$ range can be widened prior regularization. This type of phenomenon was already investigated, for example, in [\textit{T. A. Bui} et al., Rev. Mat. Iberoam. 36, No. 2, 455--484 (2020; Zbl 1448.35352)]. In the article under review, the authors consider mapping properties of $$e^{itL}$$ for non-negative self-adjoint operators $$L$$ in a metric space $$(X, d, \mu)$$ with doubling measure $$\mu$$. Their main result is the inequality $\lVert e^{itL}(I + L)^{-\sigma_pn}f \rVert_{L^p(X)} \le C(I + \lvert t \rvert)^{\sigma_pn}\lVert f \rVert_{L^p(X)}, \qquad \text{for } \sigma_p := \bigg\lvert \frac{1}{2} - \frac{1}{p} \bigg\rvert, \tag{1}$ where $$p \in (p_0, p_0^\prime)$$ with $$1\le p_0 < 2$$, and $$e^{-tL}$$ satisfies $\lVert \1_{B(x,t^{1/m})}e^{-tL}V_{t^{1/m}}^{\sigma_{p_0}} \1_{B(y,t^{1/m})} f \rVert_{L^2(X)} \le C \Big(1 + \frac{d(x,y)}{t^{1/m}}\Big)^{-n - \kappa} \lVert f \rVert_{L^{p_0}(X)}, \tag{2}$ where $$\kappa > [n/2] + 1$$ and $$V_r(x) := \mu(B(x,r))$$. The hypothesis (2) covers a wide class of operators. The authors develop a theory of Hardy spaces $$H_L^q(X)$$ adapted to $$L$$ and show that, by duality and interpolation, it suffices to prove (1) with $$H_L^q(X)$$, for $$q < 1$$, instead of $$L^p(X)$$. To show that the operator is bounded in $$H^q_L(X)$$, they prove new estimates for oscillatory multipliers to control off-diagonal terms. The proof uses several techniques, for example: $$L^2$$ based estimates; a dyadic-like decomposition (amalgam blocks); and commutator estimates. \par Editorial note: The reviewer found some mistakes in the original paper [see \url{arXiv:2007.01469}] and contacted the authors. They uploaded an amended version of the paper in [\url{arXiv:2007.01469}]. The review above relates to the amended version. Reviewer: Felipe Ponce-Vanegas (Bilbao) Multiplier conditions for boundedness into Hardy spaces https://zbmath.org/1485.42017 2022-06-24T15:10:38.853281Z "Grafakos, Loukas" https://zbmath.org/authors/?q=ai:grafakos.loukas "Nakamura, Shohei" https://zbmath.org/authors/?q=ai:nakamura.shohei "Nguyen, Hanh Van" https://zbmath.org/authors/?q=ai:van-nguyen.hanh "Sawano, Yoshihiro" https://zbmath.org/authors/?q=ai:sawano.yoshihiro Summary: In the present work we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $$\sigma (\xi_1,\dots,\xi_m)$$ and their derivatives vanish on the hyperplane $$\xi_1+\cdots +\xi_m=0$$. The multilinear Hörmander multiplier theorem with a Lorentz-Sobolev condition https://zbmath.org/1485.42018 2022-06-24T15:10:38.853281Z "Grafakos, Loukas" https://zbmath.org/authors/?q=ai:grafakos.loukas "Park, Bae Jun" https://zbmath.org/authors/?q=ai:park.bae-jun Summary: In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz-Sobolev space condition. The work is motivated by the recent result of \textit{L. Grafakos} and \textit{L. Slavíková} [Int. Math. Res. Not. 2019, No. 15, 4764--4783 (2019; Zbl 1459.42013)] where an analogous version of classical Hörmander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if $$mn/2<s<mn$$, then \begin{aligned} \Vert T_{\sigma }(f_1,\dots ,f_m) \Vert_{L^p(\mathbb{R}^n)}\lesssim \sup_{k\in \mathbb{Z}} \Vert \sigma (2^k\ \cdot)\widehat{\Psi^{(m)}} \Vert_{L_s^{mn/s,1}(\mathbb{R}^{mn})}\Vert f_1\Vert_{L^{p_1}(\mathbb{R}^n)}\cdots \Vert f_m\Vert_{L^{p_m}(\mathbb{R}^n)} \end{aligned} for certain $$p,p_1,\dots ,p_m$$ with $$1/p=1/p_1+\dots +1/p_m$$. We also show that the above estimate is sharp, in the sense that the Lorentz-Sobolev space $$L_s^{mn/s,1}$$ cannot be replaced by $$L_s^{r,q}$$ for $$r<mn/s, 0<q\le \infty$$, or by $$L_s^{mn/s,q}$$ for $$q>1$$. Riesz means on locally symmetric spaces https://zbmath.org/1485.42019 2022-06-24T15:10:38.853281Z "Papageorgiou, Effie" https://zbmath.org/authors/?q=ai:papageorgiou.effie-g Summary: We prove that for a certain class of $$n$$ dimensional rank one locally symmetric spaces, if $$f \in L^p$$, $$1\le p \le 2$$, then the Riesz means of order $$z$$ of $$f$$ converge to $$f$$ almost everywhere, for $$\Re z > (n-1)(1/p-1/2)$$. Fourier multipliers on a vector-valued function space https://zbmath.org/1485.42020 2022-06-24T15:10:38.853281Z "Park, Bae Jun" https://zbmath.org/authors/?q=ai:park.bae-jun Summary: We study multiplier theorems on a vector-valued function space, which is a generalization of the results of \textit{A. P. Calderon} and \textit{A. Torchinsky} [Adv. Math. 24, 101--171 (1977; Zbl 0355.46021)], and \textit{L. Grafakos} et al. [Ill. J. Math. 61, No. 1--2, 25--35 (2017; Zbl 1395.42025)], and an improvement of the result of \textit{H. Triebel} [J. Approx. Theory 28, 317--328 (1980; Zbl 0446.46018)]. For $$0<p<\infty$$ and $$0<q\le \infty$$ we obtain that if $$r>\frac{d}{s-(d/\min{(1,p,q)}-d)}$$, then $\big \Vert \big \{\big(m_k \widehat{f_k}\big)^{\vee}\big\}_{k\in{\mathbb{Z}}}\big \Vert_{L^p(\ell^q)}\lesssim_{p,q} \sup_{l\in{\mathbb{Z}}}{\big \Vert m_l(2^l\cdot)\big \Vert_{L_s^r({\mathbb{R}}^d)}} \big \Vert \big \{f_k\big\}_{k\in{\mathbb{Z}}}\big \Vert_{L^p(\ell^q)}, \quad f_k\in{\mathcal{E}}(A2^k),$ under the condition $$\max{(|d/p-d/2|,|d/q-d/2|)}<s<d/\min{(1,p,q)}$$. An extension to $$p=\infty$$ will be additionally considered in the scale of Triebel-Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by Sobolev spaces $$L_s^r$$ with $$r\le \frac{d}{s-(d/\min{(1,p,q)}-d)}$$. Bochner-Riesz operators between Morrey spaces https://zbmath.org/1485.42021 2022-06-24T15:10:38.853281Z "Wang, Hua" https://zbmath.org/authors/?q=ai:wang.hua.1|wang.hua|wang.hua.2 "Xiao, Jie" https://zbmath.org/authors/?q=ai:xiao.jie|xiao.jie.1|xiao.jie.2 "Xu, Shaozhen" https://zbmath.org/authors/?q=ai:xu.shaozhen Summary: This note concerns the boundedness of $$\mathscr{J}_\delta$$ (the Bochner-Riesz operator) mapping $$L^{p,\kappa}$$ (the $$(p,\kappa)$$-Morrey space) to $$L^{q,\lambda}$$ (the $$(q,\lambda)$$-Morrey space) or $$L^{q,\lambda;\ln}$$ (the $$(q,\lambda;\ln)$$-Morrey space), thereby showing $\|\mathscr{J}_\delta f\|_{L^{p,\lambda}}\lesssim\|f\|_{L^{p,\kappa}}\;\forall\;f\in L^{p,\kappa}\;\text{ under }\; \begin{cases} n\ge\kappa>\lambda>0;&\\ 1\le p<\infty;&\\ \delta\ge\frac{n-1}2+\frac{\lambda-\kappa}p, \end{cases}$ which may be regarded as the Morrey ($$\kappa>\lambda)$$-variant of the unsolved Bochner-Riesz conjecture (cf. [\textit{C. Benea} et al., Trans. Lond. Math. Soc. 4, No. 1, 110--128 (2017; Zbl 1395.42021)] or [\textit{E. M. Stein}, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton, NJ: Princeton University Press (1993; Zbl 0821.42001), p. 390]): $\|\mathscr{J}_\delta f\|_{L^p}\lesssim\|f\|_{L^p}\;\forall\; f\in L^p\;\text{ under }\; 2\ne p\in(1,\infty)\;\;\&\;\;\delta>n\left|\frac1p-\frac12\right|-\frac12.$ Quantitative weighted bounds for Calderón commutators with rough kernels https://zbmath.org/1485.42022 2022-06-24T15:10:38.853281Z "Chen, Yanping" https://zbmath.org/authors/?q=ai:chen.yanping.2|chen.yanping.3|chen.yanping.1 "Li, Ji" https://zbmath.org/authors/?q=ai:li.ji.1|li.ji.3|li.ji.2|li.ji.4 Summary: We obtain a quantitative weighted bound for the Calderón commutator $$\mathcal{C}_\Omega$$ which is a typical example of a non-convolution Calderón-Zygmund operator under the condition $$\Omega \in L^\infty (\mathbb{S}^{n-1})$$; this is the best known quantitative result for this class of rough operators. Upper and lower bounds for Littlewood-Paley square functions in the Dunkl Setting https://zbmath.org/1485.42023 2022-06-24T15:10:38.853281Z "Dziubański, Jacek" https://zbmath.org/authors/?q=ai:dziubanski.jacek "Hejna, Agnieszka" https://zbmath.org/authors/?q=ai:hejna.agnieszka In this paper the authors prove some integral bounds for Littlewood-Paley square functions in the Dunkl context. In $$\mathbb R^N$$ endowed with a normalized root system $$R$$ and a multiplicity function $$k\geq 0$$, the authors consider the classical gradient $$\nabla$$, the Dunkl gradient $$\nabla_k$$, the Dunkl Laplacian $$\Delta_k$$ and the carré du champ operator $$\Gamma$$ associated to $$\Delta_k$$. By means of two fixed functions $$\Phi$$ and $$\Psi$$ (not necessarily radial), defined on $$\mathbb R^N$$ and satisfying certain smoothness, integrability and decay conditions, the authors introduce some square functions, which are associated in a natural way to $$\nabla_k$$, $$\Delta_k$$, $$\Gamma$$. By adapting some tecniques from Calderón-Zygmund analysis to the Dunkl framework, they first prove an upper bound for the sum of the $$L^p$$-norms (with respect to a measure, suitably defined in terms of the root system and the multiplicity function) of the square functions associated to $$\nabla$$, $$\nabla_k$$, $$\Gamma$$. Lower bounds for the Dunkl square functions are also proved under an additional condition, stating essentially that the Dunkl transforms of $$\Phi$$ and $$\Psi$$ are not identically zero along any direction. Reviewer: Valentina Casarino (Vicenza) Hardy type estimates for Riesz transforms associated with Schrödinger operators on the Heisenberg group https://zbmath.org/1485.42024 2022-06-24T15:10:38.853281Z "Gao, Chunfang" https://zbmath.org/authors/?q=ai:gao.chunfang Summary: Let $$\mathbb{H}^n$$ be the Heisenberg group and $$Q=2n+2$$ be its homogeneous dimension. Let $$\mathcal{L}=-\Delta_{\mathbb{H}^n}+V$$ be the Schrödinger operator on $$\mathbb{H}^n$$, where $$\Delta_{\mathbb{H}^n}$$ is the sub-Laplacian and the nonnegative potential $$V$$ belongs to the reverse Hölder class $$B_{q_1}$$ for $$q_1\geq Q/2$$. Let $${H_{\mathcal{L}}^p(\mathbb{H}^n)}$$ be the Hardy space associated with the Schrödinger operator $$\mathcal{L}$$ for $$Q/(Q+\delta_0) < p \leq 1$$, where $$\delta_0 = \min\{1,2-Q/q_1\}$$. In this paper we consider the Hardy type estimates for the operator $$T_\alpha =V^\alpha (-\Delta_{\mathbb{H}^n} + V)^{-\alpha}$$, and the commutator $$[b,T_\alpha]$$, where $$0 < \alpha < Q/2$$. We prove that $$T_\alpha$$ is bounded from $$H_{\mathcal{L}}^p (\mathbb{H}^n)$$ into $$L^p(\mathbb{H}^n)$$. Suppose that $$b \in BMO_{\mathcal{L}}^\theta (\mathbb{H}^n)$$, which is larger than $$BMO(\mathbb{H}^n)$$. We show that the commutator $$[b,T_\alpha]$$ is bounded from $$H_\mathcal{L}^1 (\mathbb{H}^n)$$ into weak $$L^1(\mathbb{H}^n)$$. Boundedness and compactness of commutators associated with Lipschitz functions https://zbmath.org/1485.42025 2022-06-24T15:10:38.853281Z "Guo, Weichao" https://zbmath.org/authors/?q=ai:guo.weichao "He, Jianxun" https://zbmath.org/authors/?q=ai:he.jianxun|he.jianxun.1 "Wu, Huoxiong" https://zbmath.org/authors/?q=ai:wu.huoxiong "Yang, Dongyong" https://zbmath.org/authors/?q=ai:yang.dongyong Given $$\alpha\in[0,1]$$, $$f\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ and a cube $$Q$$, let $$f_Q=\frac1{|Q|}\int_Q f$$; $$Q$$ always stands for a cube with sides parallel to the coordinate axes. The space $$\mathrm{BMO}_\alpha (\mathbb{R}^n)$$ of functions with fractional mean oscillation consists of all $$f\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ such that $\|f\|_{\mathrm{BMO}_\alpha (\mathbb{R}^n)}:=\sup_Q \frac1{|Q|^{1+\frac {\alpha}{n}}}\int_Q|f-f_Q|<\infty.$ For $$\alpha=0$$ this is the usual BMO space on $$\mathbb{R}^n$$. For $$\beta\in[0,n)$$ and $$\Omega$$, a real-valued homogeneous function of degree 0 on $$\mathbb{R}^n$$, consider the operator defined for suitable functions $$f$$ by $T_{\Omega ,\beta} f(x)=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\beta}}f(y)\,dy;$ for $$\beta=0$$, when $$\int_{S^{n-1}}\Omega \,d\sigma=0$$ is additionally assumed, $$T_{\Omega ,\beta}$$ is a singular integral operator; if $$\beta\in(0,n)$$ and $$\Omega \equiv1$$, then $$T_{\Omega ,\beta}$$ is a fractional integral operator. For $$b\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$$ consider the commutator $$T^1_bf:=[b,T_{\Omega,\beta}]f=bT_{\Omega,\beta} f-T_{\Omega,\beta}(bf)$$ and the iterated commmutators $$T^m_b=[b,T^{m-1}_b]f$$, $$m\ge2$$ (in this case it is adittionally asumed that the symbol $$b$$ is real-valued). In the paper the authors study the $$(L^p,L^q)$$-compactness of the commutators $$T^m_b$$. An essential ingredient in this study consists in using $$\mathrm{CMO}$$ type spaces $$\mathrm{CMO}_\alpha(\mathbb{R}^n)$$. For a fixed $$\alpha\in[0,1]$$, $$\mathrm{CMO}_\alpha(\mathbb{R}^n)$$ is the closure of $$C^\infty_c(\mathbb R^n)$$ in $$\mathrm{BMO}_\alpha(\mathbb{R}^n)$$, and a useful characterization of functions from this space in terms of the mean value oscillation is obtained by the authors when $$\alpha\in[0,1)$$. In fact this extends an earlier result of \textit{A. Uchiyama} [Tohoku Math. J. (2) 30, 163--171 (1978; Zbl 0384.47023)] corresponding to $$\alpha=0$$, i.e. the classical BMO case. In one of the two main theorems of the paper it is proved that, under suitable assumptions on $$p$$, $$q$$, $$\alpha$$, $$\beta$$, $$m$$, $$\Omega$$ and a weight function $$\omega$$, the commutator $$T^m_b$$ is a compact operator from $$L^p(\omega^p)$$ to $$L^q(\omega^q)$$ if and only if $$b\in \mathrm{CMO}_\alpha(\mathbb{R}^n)$$. Reviewer: Krzysztof Stempak (Wrocław) Operator-free sparse domination https://zbmath.org/1485.42026 2022-06-24T15:10:38.853281Z "Lerner, Andrei K." https://zbmath.org/authors/?q=ai:lerner.andrei-k "Lorist, Emiel" https://zbmath.org/authors/?q=ai:lorist.emiel "Ombrosi, Sheldy" https://zbmath.org/authors/?q=ai:ombrosi.sheldy-j Summary: We obtain a sparse domination principle for an arbitrary family of functions $$f(x,Q)$$, where $$x\in\mathbb{R}^n$$ and $$Q$$ is a cube in $$\mathbb{R}^n$$. When applied to operators, this result recovers our recent works [\textit{A. K. Lerner} and \textit{S. Ombrosi}, J. Geom. Anal. 30, No. 1, 1011--1027 (2020; Zbl 1434.42020); \textit{E. Lorist}, J. Geom. Anal. 31, No. 9, 9366--9405 (2021; Zbl 1477.42016)]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré-Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [Lorist, loc. cit.], as we will demonstrate in an application to vector-valued square functions. $$L^p$$ boundedness of Carleson \& Hilbert transforms along plane curves with certain curvature constraints https://zbmath.org/1485.42027 2022-06-24T15:10:38.853281Z "Li, Junfeng" https://zbmath.org/authors/?q=ai:li.junfeng "Yu, Haixia" https://zbmath.org/authors/?q=ai:yu.haixia In the main result of this paper, which is Theorem 1.1, the authors show that if $$p\in(1,\infty)$$, $$u:\mathbb{R}\rightarrow\mathbb{R}$$ is a measurable function and $$\gamma$$ is a plane curve with certain constraints then the Carleson transform $\mathcal{C}_{u,\gamma}f(x)=p.v.\int_{-\infty}^{\infty}e^{iu(x)\gamma(t)}f(x-t)\frac{dt}{t}\qquad x\in\mathbb{R}$ and the Hilbert transform $H_{u,\gamma}f(x_{1},x_{2})=p.v.\int_{-\infty}^{\infty}f(x_{1}-t,x_{2}-u(x_{1})\gamma(t))\frac{dt}{t}\qquad(x_{1},x_{2})\in\mathbb{R}^{2}$ are bounded on $$L^{p}$$. Reviewer: Israel Pablo Rivera Ríos (Bahía Blanca) Rough singular integrals and maximal operator with radial-angular integrability https://zbmath.org/1485.42028 2022-06-24T15:10:38.853281Z "Liu, Ronghui" https://zbmath.org/authors/?q=ai:liu.ronghui "Wu, Huoxiong" https://zbmath.org/authors/?q=ai:wu.huoxiong This paper refers to the boundedness of the Calderón-Zygmund operator $T_\Omega f(x) = \text{p.v.} \int_{\mathbb{R}^n} f(x-y) \frac{\Omega(y')}{|y|^n} \, dy,$ where $$y^\prime = y/|y|$$ and the kernel $$\Omega$$ satisfies $$\int_{S^{n-1}} \Omega(y) d \sigma(y) = 0$$, $$S^{n-1}$$ being the unit sphere in $$\mathbb{R}^n$$, for functions $$f$$ in the mixed radial-angular space $$L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)$$, for any $$1\leq p, \, \tilde{p} \leq \infty$$. For these kind of funcions, the boundedness of $$T_\Omega$$ was previously obtained for $$\Omega \in \mathcal{C}^1 (S^{n-1})$$ and also for $$\Omega \in L \log L (S^{n-1})$$. Here one considers the case that $$\Omega \in H^1(S^{n-1})$$, the Hardy space on $$S^{n-1}$$. We recall that the space $$L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)$$ is formed by those funcions $$f$$ for which $\|f\|_{L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)} := \bigg( \int_0^\infty \|f(r \cdot)\|^p_{L^{\tilde{p}} (S^{n-1})} \, r^{n-1} \, dr \bigg)^{1/p} < \infty .$ In this paper one proves that for $$\Omega \in H^1(S^{n-1})$$ with $$\int_{S^{n-1}} \Omega(y) d \sigma(y) = 0$$ and $$1 < \tilde{p} \leq p < \tilde{p} n / (n-1)$$ or $$\tilde{p} n /(\tilde{p} +n-1) <p\leq \tilde{p} < \infty$$, the following inequality holds, $\|T_\Omega f\|_{L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)} \leq C_{p, \tilde{p}} \|\Omega\|_{H^1 (S^{n-1})} \|f\|_{L^p_{|x|} L^{\tilde{p}}_\theta (\mathbb{R}^n)}.$ An analogous inequality is stablished for the maximal singular integral operator, $$T^\ast_\Omega$$, given by $T^\ast_\Omega f(x) = \sup_{\varepsilon >0} \bigg| \int_{|y|\geq \varepsilon} f(x-y) \frac{\Omega(y^\prime)}{|y|^n} \, dy\bigg|,$ when $$1 < \tilde{p} \leq p < \tilde{p} n /(n-1)$$. Reviewer: Julià Cufí (Bellaterra) A nonlinear version of Roth's theorem on sets of fractional dimensions https://zbmath.org/1485.42029 2022-06-24T15:10:38.853281Z "Li, Xiang" https://zbmath.org/authors/?q=ai:li.xiang.2|li.xiang.3|li.xiang.4 "He, Qianjun" https://zbmath.org/authors/?q=ai:he.qianjun "Yan, Dunyan" https://zbmath.org/authors/?q=ai:yan.dunyan "Zhang, Xingsong" https://zbmath.org/authors/?q=ai:zhang.xingsong Summary: Let $$E\subset \mathbb{R}$$ be a closed set, which has nonzero Hausdorff dimension and some other properties. For some $$t>0$$, we proved the three- point patterns $$x$$, $$x+t$$, $$x+\gamma (t)$$ belong to $$E$$, where $$\gamma (t)$$ is a convex curve with some curvature constraints. Removable singularities for Lipschitz caloric functions in time varying domains https://zbmath.org/1485.42030 2022-06-24T15:10:38.853281Z "Mateu, Joan" https://zbmath.org/authors/?q=ai:mateu.joan "Prat, Laura" https://zbmath.org/authors/?q=ai:prat.laura "Tolsa, Xavier" https://zbmath.org/authors/?q=ai:tolsa.xavier Summary: In this paper we study removable singularities for regular $$(1,1/2)$$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $$L^2$$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation. Small cap decoupling inequalities: bilinear methods https://zbmath.org/1485.42031 2022-06-24T15:10:38.853281Z "Oh, Changkeun" https://zbmath.org/authors/?q=ai:oh.changkeun Summary: We obtain sharp small cap decoupling inequalities associated to the moment curve for certain range of exponents $$p$$. Our method is based on the bi-linearization argument due to \textit{J. Bourgain} and \textit{C. Demeter} [Ann. Math. (2) 182, No. 1, 351--389 (2015; Zbl 1322.42014)]. Our result generalizes theirs to all higher dimensions. Boundedness for commutators of rough $$p$$-adic Hardy operator on $$p$$-adic central Morrey spaces https://zbmath.org/1485.42032 2022-06-24T15:10:38.853281Z "Sarfraz, Naqash" https://zbmath.org/authors/?q=ai:sarfraz.naqash "Aslam, Muhammad" https://zbmath.org/authors/?q=ai:aslam.muhammad-saeed|aslam.muhammad-jamil|aslam.muhammad-kamran|aslam.muhammad-zubair|aslam.muhammad-nauman|aslam.muhammad-shamrooz "Jarad, Fahd" https://zbmath.org/authors/?q=ai:jarad.fahd Summary: In the present article we obtain the boundedness for commutators of rough $$p$$-adic Hardy operator on $$p$$-adic central Morrey spaces. Furthermore, we also acquire the boundedness of rough $$p$$-adic Hardy operator on Lebesgue spaces. Energy counterexamples in two weight Calderón-Zygmund theory https://zbmath.org/1485.42033 2022-06-24T15:10:38.853281Z "Sawyer, Eric T." https://zbmath.org/authors/?q=ai:sawyer.eric-t "Shen, Chun-Yen" https://zbmath.org/authors/?q=ai:shen.chun-yen "Uriarte-Tuero, Ignacio" https://zbmath.org/authors/?q=ai:uriarte-tuero.ignacio Summary: We show that the energy conditions are not necessary for boundedness of Riesz transforms in dimension $$n\geq 2$$. In dimension $$n=1$$, we construct an elliptic singular integral operator $$H_{\flat}$$ for which the energy conditions are not necessary for boundedness of $$H_{\flat}$$. The convolution kernel $$K_{\flat}(x)$$ of the operator $$H_{\flat}$$ is a smooth flattened version of the Hilbert transform kernel $$K(x)=\frac{1}{x}$$ that satisfies ellipticity $$\vert K_{\flat}(x)\vert\gtrsim\frac{1}{\vert x\vert}$$, but not gradient ellipticity $$\vert K^\prime_{\flat}(x)\vert\gtrsim\frac{1}{\vert x\vert^2}$$. Indeed the kernel has flat spots where $$K^\prime_{\flat}(x)=0$$ on a family of intervals, but $$K^\prime_{\flat}(x)$$ is otherwise negative on $$\mathbb{R}\setminus\{0\}$$. On the other hand, if a one-dimensional kernel $$K(x,y)$$ is both elliptic and gradient elliptic, then the energy conditions are necessary, and so by our theorem in [\textit{E. T. Sawyer} et al., in: Harmonic analysis, partial differential equations and applications. In honor of Richard L. Wheeden. Basel: Birkhäuser/Springer. 125--164 (2017; Zbl 1380.42015)], the $$T1$$ theorem holds for such kernels on the line. This paper includes results from [\textit{E. T. Sawyer}, Energy conditions and twisted localizations of operators'', Preprint, \url{arXiv:1801.03706v2}]. Sharp weak type estimates for a family of Soria bases https://zbmath.org/1485.42034 2022-06-24T15:10:38.853281Z "Dmitrishin, Dmitry" https://zbmath.org/authors/?q=ai:dmitrishin.dmitry "Hagelstein, Paul" https://zbmath.org/authors/?q=ai:hagelstein.paul-alton "Stokolos, Alex" https://zbmath.org/authors/?q=ai:stokolos.alex Summary: Let $${\mathcal{B}}$$ be a collection of rectangular parallelepipeds in $${\mathbb{R}}^3$$ whose sides are parallel to the coordinate axes and such that $${\mathcal{B}}$$ contains parallelepipeds with side lengths of the form $$s$$, $$\frac{2^N}{s}$$, $$t$$, where $$s, t > 0$$ and $$N$$ lies in a nonempty subset $$S$$ of the natural numbers. We show that if $$S$$ is an infinite set, then the associated geometric maximal operator $$M_{\mathcal{B}}$$ satisfies the weak type estimate $\left| \left\{ x \in{\mathbb{R}}^3 : M_{{\mathcal{B}}}f(x) > \alpha \right\} \right| \le C \int \nolimits_{{\mathbb{R}}^3} \frac{|f|}{\alpha} \left( 1 + \log^+ \frac{|f|}{\alpha}\right)^2,$ but does not satisfy an estimate of the form $\left| \left\{ x \in{\mathbb{R}}^3 : M_{{\mathcal{B}}}f(x) > \alpha \right\} \right| \le C \int \nolimits_{{\mathbb{R}}^3} \phi \left( \frac{|f|}{\alpha}\right)$ for any convex increasing function $$\phi : \mathbb [0, \infty) \rightarrow [0, \infty)$$ satisfying the condition $\lim_{x \rightarrow \infty}\frac{\phi (x)}{x (\log (1 + x))^2} = 0\;.$ Correction to: A weak reverse Hölder inequality for caloric measure'' https://zbmath.org/1485.42035 2022-06-24T15:10:38.853281Z "Genschaw, Alyssa" https://zbmath.org/authors/?q=ai:genschaw.alyssa "Hofmann, Steve" https://zbmath.org/authors/?q=ai:hofmann.steve From the text: In our paper [ibid. 30, No. 2, 1530--1564 (2020; Zbl 1436.42028)], in which we presented a parabolic version of results of [\textit{B. Bennewitz} and \textit{J. L. Lewis}, Complex Variables, Theory Appl. 49, No. 7--9, 571--582 (2004; Zbl 1068.31001)], the proof of Lemma 2.7 (the main lemma) was divided into several cases, one of which we had inadvertently failed to note and treat. We now rectify this omission. Fortunately, this will be a simple matter. Extrapolation and the boundedness in grand variable exponent Lebesgue spaces without assuming the log-Hölder continuity condition, and applications https://zbmath.org/1485.42036 2022-06-24T15:10:38.853281Z "Kokilashvili, Vakhtang" https://zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m "Meskhi, Alexander" https://zbmath.org/authors/?q=ai:meskhi.alexander Summary: The boundedness of the Hardy-Littlewood maximal operator, and the weighted extrapolation in grand variable exponent Lebesgue spaces are established provided that Hardy-Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue space. Moreover, we give some bounds of the norm of the Hardy-Littlewood maximal operator in these spaces. As corollaries, we have appropriate norm inequalities and the boundedness of operators of Harmonic Analysis such as maximal and sharp maximal functions; Calderón-Zygmund singular integrals, commutators of singular integrals in grand variable exponent Lebesgue spaces. Finally, applying the boundedness results of integral operators of Harmonic Analysis, we have the direct and inverse theorems on the approximation of $$2\pi$$-periodic functions by trigonometric polynomials in the framework of grand variable exponent Lebesgue spaces. Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data https://zbmath.org/1485.42037 2022-06-24T15:10:38.853281Z "Yan, Xiangqian" https://zbmath.org/authors/?q=ai:yan.xiangqian "Zhao, Yajuan" https://zbmath.org/authors/?q=ai:zhao.yajuan "Yan, Wei" https://zbmath.org/authors/?q=ai:yan.wei Summary: In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces $$\hat{H}^{\frac{1}{p},\frac{p}{2}}(\mathbb{R})$$ $$(4\leq p<\infty)$$, $$\hat{H}^{\frac{3s_1}{p},\frac{2p}{3}}(\mathbb{R}^2)$$ $$(s_1>\frac{1}{3},\ 3\leq p<\infty)$$, $$\hat{H}^{\frac{2s_2}{p},p}(\mathbb{R}^n)$$ $$(s_2>\frac{n}{2(n+1)},\ 2\leq p<\infty,\ n\geq 3)$$ with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrödinger equation can fail with data in $$\hat{H}^{s,\frac{p}{2}}(\mathbb{R})$$ $$(s<\frac{1}{p})$$. Finally, we show the stochastic continuity of Schrödinger equation with random data in $$\hat{L}^r(\mathbb{R}^n)$$ $$(2\leq r<\infty)$$ almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates. Endpoint regularity of discrete multilinear fractional nontangential maximal functions https://zbmath.org/1485.42038 2022-06-24T15:10:38.853281Z "Zhang, Daiqing" https://zbmath.org/authors/?q=ai:zhang.daiqing Summary: Given $$m\geq 1$$, $$0\leq \lambda \leq 1$$, and a discrete vector-valued function $$\vec{f}=(f_1,\dots,f_m)$$ with each $$f_j:\mathbb{Z}^d\rightarrow \mathbb{R}$$, we consider the discrete multilinear fractional nontangential maximal operator $\mathrm{M}_{\alpha,\mathcal{B}}^{\lambda}(\vec{f}) (\vec{n})=\mathop{\sup_{r>0, \vec{x}\in \mathbb{R}^d}}_{\vert \vec{n}-\vec{x} \vert \leq \lambda r}\frac{1}{N(B_r(\vec{x}))^{m-\frac{\alpha}{d}}} \prod_{j=1}^m\sum_{\vec{k}\in B_r(\vec{x})\cap \mathbb{Z}^d} \bigl\vert f_j(\vec{k}) \bigr\vert,$ where $$\mathcal{B}$$ is the collection of all open balls $$B\subset \mathbb{R}^d$$, $$B_r(\vec{x})$$ is the open ball in $$\mathbb{R}^d$$ centered at $$\vec{x}\in \mathbb{R}^d$$ with radius $$r$$, and $$N(B_r(\vec{x}))$$ is the number of lattice points in the set $$B_r(\vec{x})$$. We show that the operator $$\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}}^{\lambda}(\vec{f})|$$ is bounded and continuous from $$\ell^1(\mathbb{Z}^d)\times \ell^1(\mathbb{Z}^d)\times \cdots \times \ell^1(\mathbb{Z}^d)$$ to $$\ell^q(\mathbb{Z}^d)$$ if $$0\leq \alpha < md$$ and $$q\geq 1$$ such that $$q>\frac{d}{md- \alpha +1}$$. We also prove that the same result also holds for the discrete multilinear fractional nontangential maximal operators associated with cubes. These results we obtained represent significant and natural extensions of what was known previously. Riesz transform characterizations for multidimensional Hardy spaces https://zbmath.org/1485.42039 2022-06-24T15:10:38.853281Z "Kania-Strojec, Edyta" https://zbmath.org/authors/?q=ai:kania-strojec.edyta "Preisner, Marcin" https://zbmath.org/authors/?q=ai:preisner.marcin Summary: We study Hardy space $$H^1_L(X)$$ related to a self-adjoint operator $$L$$ defined on an Euclidean subspace $$X$$ of $${{\mathbb{R}}^d}$$. We continue study from [\textit{E. Kania-Strojec} et al., Rev. Mat. Complut. 34, No. 2, 409--434 (2021; Zbl 1481.42029)], where, under certain assumptions on the heat semigroup $$\exp (-tL)$$, the atomic characterization of local type for $$H^1_L(X)$$ was proved. In this paper we provide additional assumptions that lead to another characterization of $$H^1_L(X)$$ by the Riesz transforms related to $$L$$. As an application, we prove the Riesz transform characterization of $$H^1_L(X)$$ for multidimensional Bessel and Laguerre operators, and the Dirichlet Laplacian on $${\mathbb{R}}^d_+$$. BMO type space associated with Neumann operator and application to a class of parabolic equations https://zbmath.org/1485.42040 2022-06-24T15:10:38.853281Z "Chao, Zhang" https://zbmath.org/authors/?q=ai:chao.zhang "Yang, Minghua" https://zbmath.org/authors/?q=ai:yang.minghua Let $$\mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)$$ denote a BMO space on $$\mathbb{R}^n$$ associated to a Neumann operator. In this paper the authors show that a function $$f \in \mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)$$ is the trace of $$\mathcal{L}u= u_{t}-\Delta_{N}u=0$$, $$u(x,0)=f(x)$$ where $$u$$ satisfies a Carleson-type condition $\sup_{x_{B}, r_{B}}r_{B}^{-n}\int_{0}^{r_{B}^2}\int_{B(x_{B}, r_{B})}\{ t|\partial_{t} u(x,t)|^2+|\nabla_{x}u(x,t)|^2\}dx dt < \infty.$ Conversely, this Carleson condition characterizes all the $$\mathcal{L}$$-carolic functions whose traces belong to the space $$\mathrm{BMO}_{\Delta_{N}}(\mathbb{R}^n)$$. Furthermore, by this characterization the authors prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the initial data in $$\mathrm{BMO}^{-1}_{\Delta_{N}}(\mathbb{R}^n)$$. Reviewer: Koichi Saka (Akita) Correction to: A sharp Bernstein-type inequality and application to the Carleson embedding theorem with matrix weights'' https://zbmath.org/1485.42041 2022-06-24T15:10:38.853281Z "Kraus, Daniela" https://zbmath.org/authors/?q=ai:kraus.daniela "Moucha, Annika" https://zbmath.org/authors/?q=ai:moucha.annika "Roth, Oliver" https://zbmath.org/authors/?q=ai:roth.oliver From the text: The original published version of this article [the authors, ibid. 12, No. 1, Paper No. 40, 6 p. (2022; Zbl 1482.42063)] contained a number of typos that were introduced in the typesetting process by the publisher. This update corrects those errors. Median-type John-Nirenberg space in metric measure spaces https://zbmath.org/1485.42042 2022-06-24T15:10:38.853281Z "Myyryläinen, Kim" https://zbmath.org/authors/?q=ai:myyrylainen.kim Summary: We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak-type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón-Zygmund decomposition and a good-$$\lambda$$ inequality for medians. A John-Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John-Nirenberg spaces coincide under a Boman-type chaining assumption. Invertibility of frame operators on Besov-type decomposition spaces https://zbmath.org/1485.42043 2022-06-24T15:10:38.853281Z "Romero, José Luis" https://zbmath.org/authors/?q=ai:romero.jose-luis "van Velthoven, Jordy Timo" https://zbmath.org/authors/?q=ai:van-velthoven.jordy-timo "Voigtlaender, Felix" https://zbmath.org/authors/?q=ai:voigtlaender.felix Summary: We derive an extension of the Walnut-Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that $$L^2$$ frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the $$L^2$$ canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces. The CMO-Dirichlet problem for the Schrödinger equation in the upper half-space and characterizations of CMO https://zbmath.org/1485.42044 2022-06-24T15:10:38.853281Z "Song, Liang" https://zbmath.org/authors/?q=ai:song.liang "Wu, Liangchuan" https://zbmath.org/authors/?q=ai:wu.liangchuan Summary: Let $$\mathcal{L}$$ be a Schrödinger operator of the form $$\mathcal{L}=-\Delta +V$$ acting on $$L^2(\mathbb{R}^n)$$ where the non-negative potential $$V$$ belongs to the reverse Hölder class $$\mathrm{RH}_q$$ for some $$q\ge (n+1)/2$$. Let $$\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n)$$ denote the function space of vanishing mean oscillation associated to $$\mathcal{L}$$. In this article, we will show that a function $$f$$ of $$\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n)$$ is the trace of the solution to $$\mathbb{L}u=-u_{tt}+\mathcal{L}u=0, u(x,0)=f(x)$$, if and only if, $$u$$ satisfies a Carleson condition $\sup_{B: \text{balls}}\mathcal{C}_{u,B} :=\sup_{B(x_B,r_B): \text{balls}} r_B^{-n}\int_0^{r_B}\int_{B(x_B, r_B)} \big |t \nabla u(x,t)\big |^2\, \frac{\mathrm{dx}\, \mathrm{dt}}{t} <\infty,$ and $\lim_{a \rightarrow 0}\sup_{B: r_B \le a} \,\mathcal{C}_{u,B} = \lim_{a \rightarrow \infty}\sup_{B: r_B \ge a} \,\mathcal{C}_{u,B} = \lim_{a \rightarrow \infty}\sup_{B: B \subseteq \left( B(0, a)\right)^c} \,\mathcal{C}_{u,B}=0.$ This continues the lines of the previous characterizations by \textit{X. T. Duong} et al. [J. Funct. Anal. 266, No. 4, 2053--2085 (2014; Zbl 1292.35099)] and \textit{R. Jiang} and \textit{B. Li} [On the Dirichlet problem for the Schrödinger equation with boundary value in BMO space'', Preprint, \url{arXiv:2006.05248}] for the $$\mathrm{BMO}_{\mathcal{L}}$$ spaces, which were founded by \textit{E. B. Fabes} et al. [Indiana Univ. Math. J. 25, 159--170 (1976; Zbl 0306.46032)] for the classical BMO space. For this purpose, we will prove two new characterizations of the $$\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n)$$ space, in terms of mean oscillation and the theory of tent spaces, respectively. A characterization of spaces of homogeneous type induced by continuous ellipsoid covers of $$\mathbb{R}^n$$ https://zbmath.org/1485.46032 2022-06-24T15:10:38.853281Z "Bownik, Marcin" https://zbmath.org/authors/?q=ai:bownik.marcin "Li, Baode" https://zbmath.org/authors/?q=ai:li.baode "Weissblat, Tal" https://zbmath.org/authors/?q=ai:weissblat.tal Summary: We study the relationship between the concept of a continuous ellipsoid $$\Theta$$ cover of $$\mathbb{R}^n$$, which was introduced by \textit{W. Dahmen} et al. [Numer. Math. 107, No. 3, 503--532 (2007; Zbl 1129.65092), Constr. Approx. 31, No. 2, 149--194 (2010; Zbl 1195.46030)], (see also \textit{S. Dekel} and \textit{P. Petrushev} [in: Multiscale, nonlinear and adaptive approximation. Dedicated to Wolfgang Dahmen on the occasion of his 60th birthday. Berlin: Springer. 137--167 (2009; Zbl 1196.46019)]), and the space of homogeneous type induced by $$\Theta$$. We characterize the class of quasi-distances on $$\mathbb{R}^n$$ (up to equivalence) which correspond to continuous ellipsoid covers. This places firmly continuous ellipsoid covers as a subclass of spaces of homogeneous type on $$\mathbb{R}^n$$ satisfying quasi-convexity and 1-Ahlfors-regularity. Real interpolation of martingale Orlicz Hardy spaces and BMO spaces https://zbmath.org/1485.46034 2022-06-24T15:10:38.853281Z "Long, Long" https://zbmath.org/authors/?q=ai:long.long "Weisz, Ferenc" https://zbmath.org/authors/?q=ai:weisz.ferenc "Xie, Guangheng" https://zbmath.org/authors/?q=ai:xie.guangheng Summary: In this article, the authors prove that the real interpolation spaces between martingale Orlicz Hardy spaces and martingale BMO spaces are martingale Orlicz-Lorentz Hardy spaces. Using sharp maximal functions, the authors also establish the characterizations of martingale Orlicz Hardy spaces. Weighted Besov spaces with variable exponents https://zbmath.org/1485.46043 2022-06-24T15:10:38.853281Z "Wang, Shengrong" https://zbmath.org/authors/?q=ai:wang.shengrong "Xu, Jingshi" https://zbmath.org/authors/?q=ai:xu.jingshi Summary: In this paper, we introduce Besov spaces with variable exponents and variable Muckenhoupt weights. Then we give a approximation characterization, the lifting property, embeddings, the duality and interpolation of these spaces. Singular integrals, rank one perturbations and Clark model in general situation https://zbmath.org/1485.47016 2022-06-24T15:10:38.853281Z "Liaw, Constanze" https://zbmath.org/authors/?q=ai:liaw.constanze "Treil, Sergei" https://zbmath.org/authors/?q=ai:treil.sergei Summary: We start with considering rank one self-adjoint perturbations $$A_{\alpha}=A+ \alpha (\cdot, \varphi) \varphi$$ with cyclic vector $$\varphi \in \mathcal{H}$$ on a separable Hilbert space $$\mathcal{H}$$. The spectral representation of the perturbed operator $$A_{\alpha}$$ is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators $$A$$ and $$A_{\alpha}$$. Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle. This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings. Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role. Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators. These lecture notes give an account of the mini-course delivered by the authors, which was centered around [\textit{C. Liaw} and \textit{S. Treil}, J. Funct. Anal. 257, No. 6, 1947--1975 (2009; Zbl 1206.42012); Rev. Mat. Iberoam. 29, No. 1, 53--74 (2013; Zbl 1272.42011); J. Anal. Math. 130, 287--328 (2016; Zbl 06697868)]. Unpublished results are restricted to the last part of this manuscript. For the entire collection see [Zbl 1381.00044]. Uncertainty principles associated to sets satisfying the geometric control condition https://zbmath.org/1485.93121 2022-06-24T15:10:38.853281Z "Green, Walton" https://zbmath.org/authors/?q=ai:green.walton "Jaye, Benjamin" https://zbmath.org/authors/?q=ai:jaye.benjamin-j "Mitkovski, Mishko" https://zbmath.org/authors/?q=ai:mitkovski.mishko Summary: In this paper, we study forms of the uncertainty principle suggested by problems in control theory. We obtain a version of the classical Paneah-Logvinenko-Sereda theorem for the annulus. More precisely, we show that a function with spectrum in an annulus of a given thickness can be bounded, in $$L^2$$-norm, from above by its restriction to a neighborhood of a GCC set, with constant independent of the radius of the annulus. We apply this result to obtain energy decay rates for damped fractional wave equations, extending the work of Malhi and Stanislavova to both the higher-dimensional and non-periodic setting. Wigner analysis of operators. I: Pseudodifferential operators and wave fronts https://zbmath.org/1485.94017 2022-06-24T15:10:38.853281Z "Cordero, Elena" https://zbmath.org/authors/?q=ai:cordero.elena "Rodino, Luigi" https://zbmath.org/authors/?q=ai:rodino.luigi Summary: We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the $$\mathcal{A}$$-Wigner distribution defined by $$W_{\mathcal{A}}(f) = \mu(\mathcal{A})(f \otimes \bar{f})$$, where $$\mathcal{A}$$ is a $$4 d \times 4 d$$ symplectic matrix and $$\mu(\mathcal{A})$$ is an associate metaplectic operator. Basic examples are given by the so-called $$\tau$$-Wigner distributions. Such representations provide a new characterization for modulation spaces when $$\tau \in(0, 1)$$. Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjöstrand class (in particular, in the Hörmander class $$S_{0 , 0}^0)$$. The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front. In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given. Uncertainty principles for the two-sided quaternion linear canonical transform https://zbmath.org/1485.94032 2022-06-24T15:10:38.853281Z "Zhu, Xiaoyu" https://zbmath.org/authors/?q=ai:zhu.xiaoyu "Zheng, Shenzhou" https://zbmath.org/authors/?q=ai:zheng.shenzhou Summary: The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy's uncertainty principle, Beurling's uncertainty principle and Donoho-Stark's uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb's uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff-Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.