Recent zbMATH articles in MSC 42Bhttps://zbmath.org/atom/cc/42B2023-09-22T14:21:46.120933ZWerkzeugLocally random groupshttps://zbmath.org/1517.220032023-09-22T14:21:46.120933Z"Mallahi-Karai, Keivan"https://zbmath.org/authors/?q=ai:karai.keivan-mallahi"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir"Salehi Golsefidy, Alireza"https://zbmath.org/authors/?q=ai:salehi-golsefidy.alirezaSummary: In this work, we introduce and study the notion of \textit{local randomness} for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional \textit{dimension condition} on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood-Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain-Gamburd and Tao.Stochastic homogenization: a short proof of the annealed Calderón-Zygmund estimatehttps://zbmath.org/1517.350202023-09-22T14:21:46.120933Z"Josien, Marc"https://zbmath.org/authors/?q=ai:josien.marcSummary: A building block for many field theories in continuum physics are second-order elliptic operators in divergence form, as given through a coefficient field which may be assimilated to a metric tensor field on \(\mathbb{R}^d\). The mapping properties of these linear operators are a crucial ingredient for analysis. In this paper, we focus on Calderón-Zygmund estimates, that is, on the boundedness of the corresponding Helmholtz projection in \(\mathrm{L}^p (\mathbb{R}^d)\)-spaces. Even when the coefficient field is uniformly smooth, this estimate may fail for \(p\) not close to 2. We seek an intrinsic criterion on the validity of the Calderón-Zygmund estimate in the whole range of \(p\in (1,\infty)\); intrinsic in the sense that it is formulated in terms of the scalar and vector potentials of the harmonic coordinates. We seek genericity in form of a statistical statement, and thus consider general ensembles of coefficient fields. Our criterion comes in form of finite stochastic moments for the potentials, or rather their corrections from being affine. In line with this, the Calderón-Zygmund estimates we obtain are \textit{annealed} as opposed to \textit{quenched}, meaning that there is an inner norm in form of a stochastic moment next to the (outer) \(\mathrm{L}^p\)-norm in space. This result grows out of recent progress in quantitative stochastic homogenization; it is ultimately inspired by the classical large-scale regularity theory of \textit{M. Avellaneda} and \textit{F.-H. Lin} [Commun. Pure Appl. Math. 40, No. 6, 803--847 (1987; Zbl 0632.35018)]. More specifically, we provide an easier version of the proof given by us in [with \textit{F. Otto}, J. Funct. Anal. 283, No. 7, Article ID 109594, 74 p. (2022; Zbl 1494.35022)], albeit under stronger assumptions. Annealed Calderon-Zygmund estimates were first established in [\textit{M. Duerinckx} and \textit{F. Otto}, Stoch. Partial Differ. Equ., Anal. Comput. 8, No. 3, 625--692 (2020; Zbl 1456.35017)], and are a very convenient tool for error estimates in stochastic homogenization.Dispersive estimates for kinetic transport equation in Besov spaceshttps://zbmath.org/1517.350752023-09-22T14:21:46.120933Z"He, Cong"https://zbmath.org/authors/?q=ai:he.cong"Chen, Jingchun"https://zbmath.org/authors/?q=ai:chen.jingchun"Fang, Houzhang"https://zbmath.org/authors/?q=ai:fang.houzhang"He, Huan"https://zbmath.org/authors/?q=ai:he.huan(no abstract)On the instantaneous radius of analyticity of \(L^p\) solutions to 3D Navier-Stokes systemhttps://zbmath.org/1517.351642023-09-22T14:21:46.120933Z"Zhang, Ping"https://zbmath.org/authors/?q=ai:zhang.ping.5|zhang.ping.2|zhang.ping.1|zhang.ping|zhang.ping.3Summary: In this paper, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier-Stokes system with initial data in the Besov spaces \(\dot{B}^s_{p,q}({\mathbb{R}}^3)\) for \(p\in ]1,\infty [\), \(q\in [1,\infty ]\) and \(s\in \left [-1+\frac{3}{p},\frac{3}{p}\right [\). Then for initial data \(u_0\in L^p({\mathbb{R}}^3)\) with \(p\) in \(]3, 6[\), we prove that 3D Navier-Stokes system has a unique solution \(u=u_L+v\) with \(u_L \stackrel{\text{def}}= e^{t \Delta} u_0\) and \(v\in{\widetilde{L}^\infty_T\Big (\dot{B}^{1-\frac{3}{p}}_{p,\frac{p}{2}}\Big )}\cap{\widetilde{L}^1_T\Big (\dot{B}^{3-\frac{3}{p}}_{p,\frac{p}{2}}\Big )}\) for some positive time \(T\). Furthermore, we derive an explicit lower bound for the radius of space analyticity of \(v\), which in particular extends the corresponding results in [\textit{R. Hu} and \textit{P. Zhang}, Chin. Ann. Math., Ser. B 43, No. 5, 749--772 (2022; Zbl 1502.35082)] with initial data in \(L^p({\mathbb{R}}^3)\) for \(p\in [3, 18/5[\).Global \(L^p\)-boundedness of rough Fourier integral operatorshttps://zbmath.org/1517.352752023-09-22T14:21:46.120933Z"Sindayigaya, Joachim"https://zbmath.org/authors/?q=ai:sindayigaya.joachimSummary: In this paper, we establish the \(L^p\)-boundedness of Fourier integral operators \(T_{\phi, a}\) with rough amplitude and phase function, which satisfies the new class of rough non-degeneracy condition. In this study, under the conditions \(a\in L^\infty S^m_\rho\), \(\phi\in L^\infty\Phi^2\) and when \(1 - \frac{\delta}{2} \leqslant \rho \leq 1\), we show that \(T_{\phi, a}\) is bounded from \(L^p\) to itself for \(p\in[1, \infty]\) with some measure conditions on \(m\). Our main results extend and improve some known results about \(L^p\)-boundedness of Fourier integral operators.Geometric harmonic analysis I. A sharp divergence theorem with nontangential pointwise traceshttps://zbmath.org/1517.420012023-09-22T14:21:46.120933Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusThe present book is the first in a series of five volumes, at the confluence of Harmonic Analysis,
Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically
branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles:
Volume I: A Sharp Divergence Theorem with Nontangential Pointwise Traces;
Volume II: Function Spaces Measuring Size and Smoothness on Rough Sets;
Volume III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering;
Volume IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis;
Volume V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems.
The present review is concerned with the first volume. In the first chapter, starting from the classical work
of De Giorgi-Federer, the authors develop a new generation of divergence theorems both in the Euclidean space
as well as in the setting of Riemannian manifolds. The most striking feature is that the vector field in question is
strictly defined in the underlying open set and its boundary trace is considered in a pointwise nontangential fashion.
In the second chapter, a wealth of examples and counter-examples are presented, indicating that the main results are
optimal from a variety of perspectives.
The third chapter gathers foundational material from measure theory and topology.
Chapter four contains a variety of selected topics from (or inspired by) distribution theory.
For example, the authors develop a brand of distribution theory on arbitrary subsets of the Euclidean space, taking
Lipschitz functions with bounded support as test functions. Here they also coin the notion of
``bullet product'' which, in essence, is a weak version (modeled upon integration by parts) of the inner product of
the normal vector to a domain with a given vector field satisfying only some very mild integrability properties in that domain.
Among other things, a proof of Leibniz's product rule for weak derivatives is provided, and the chapter ends with what the
authors call the contribution at infinity of a vector field.
In the fifth chapter, the author discusses basic results from Geometric Measure Theory, including thick sets,
the corkscrew condition, the geometric measure theoretic boundary, area and coarea formulas, countable rectifiability,
approximate tangent planes, functions of bounded variation, sets of locally finite perimeter, Ahlfors regularity,
uniformly rectifiable sets, the local John condition, and nontangentially accessible domains.
The sixth chapter is focused on tools from Harmonic Analysis, such as the regularized distance function,
Whitney's Extension Theorem, and the fractional Hardy-Littlewood maximal operator in non-metric settings.
This chapter also contains an informative review of Clifford algebras (which are higher-dimensional versions
of the field of complex numbers, that happen to be highly non-commutative, in which a brand of complex analysis may be developed),
and a discussion of reverse Hölder inequalities and interior estimates. The authors close this chapter by introducing
the solid maximal function and defining maximal Lebesgue spaces.
The seventh chapter, entitled Quasi-Metric Spaces and Spaces of Homogeneous Type, consists of the following sections:
Quasi-Metric Spaces and a Sharp Metrization Result; Estimating Integrals Involving the Quasi-Distance;
Hölder Spaces on Quasi-Metric Spaces; Functions of Bounded Mean Oscillations on Spaces of Homogeneous Type;
Whitney Decompositions on Geometrically Doubling Quasi-Metric Spaces; The Hardy-Littlewood Maximal Operator
on Spaces of Homogeneous Type; Muckenhoupt Weights on Spaces of Homogeneous Type; The Fractional Integration Theorem.
The eighth chapter is entitled Open Sets with Locally Finite Surface Measures and Boundary Behavior.
The first section focuses on nontangential approach regions in arbitrary open sets. The second and third sections
deal with the basic properties of the nontangential maximal operator. The fourth section contains size estimates
for the nontangential maximal operator involving a doubling measure, while the fifth one is reserved for a
comparison between the nontangential and tangential maximal operators. In the sixth section, the authors establish
off-diagonal Carleson measure estimates of reverse Hölder type, which are crucial ingredients in the proofs of the main results.
The seventh section elaborates on estimates for Marcinkiewicz type integrals and applications. The eighth and the ninth sections
are on what the authors call the nontangentially accessible boundary and, respectively, the nontangential boundary trace operator.
The tenth section treats the averaged nontangential maximal operator.
The last chapter contains the proofs of the main results pertaining to the family of divergence theorems stated in chapter one.
Reviewer: Mohammed El Aïdi (Bogotá)Almost periodic type functions of several variables and applicationshttps://zbmath.org/1517.420062023-09-22T14:21:46.120933Z"Chávez, Alan"https://zbmath.org/authors/?q=ai:chavez.alan"Khalil, Kamal"https://zbmath.org/authors/?q=ai:khalil.kamal"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.marko"Pinto, Manuel"https://zbmath.org/authors/?q=ai:pinto.manuelFor functions of several real variables, new types of almost periodic functions are introduced. Their principal structural characterizations, their composition principles, and the invariance under convolution products are studied. The obtained notions and results are applied to partial differential equations and to abstract integral equations in Banach spaces.
Reviewer: Elijah Liflyand (Ramat-Gan)Band-limited maximizers for a Fourier extension inequality on the circle. IIhttps://zbmath.org/1517.420082023-09-22T14:21:46.120933Z"Barker, James"https://zbmath.org/authors/?q=ai:barker.james"Thiele, Christoph"https://zbmath.org/authors/?q=ai:thiele.christoph-martin"Zorin-Kranich, Pavel"https://zbmath.org/authors/?q=ai:zorin-kranich.pavelSummary: Among the class of functions on the circle with Fourier modes up to degree 120, constant functions are the unique real-valued maximizers for the endpoint Tomas-Stein inequality.
For Part I see [\textit{D. Oliveira e Silva} et al., ibid. 31, No. 1, 192--198 (2022; Zbl 1503.42008)].Beurling's theorem in the Clifford algebrashttps://zbmath.org/1517.420092023-09-22T14:21:46.120933Z"Tyr, Othman"https://zbmath.org/authors/?q=ai:tyr.othman"Daher, Radouan"https://zbmath.org/authors/?q=ai:daher.radouanSummary: In this research, the Clifford-Fourier transform introduced by \textit{E. Hitzer} [Clifford Anal. Clifford Algebr. Appl. 2, No. 3, 223--235 (2013; Zbl 1297.43006)], satisfies some uncertainty principles similar to the Euclidean Fourier transform. An analog of the Beurling-Hörmander's theorem for the Clifford-Fourier transform is obtained. As a straightforward consequence of Beurling's theorem, other versions of the uncertainty principle, such as the Hardy, Gelfand-Shilov and Cowling-Price theorems are also deduced.Tighter Heisenberg-Weyl type uncertainty principle associated with quaternion wavelet transformhttps://zbmath.org/1517.420102023-09-22T14:21:46.120933Z"Wang, Xinyu"https://zbmath.org/authors/?q=ai:wang.xinyu.1"Zheng, Shenzhou"https://zbmath.org/authors/?q=ai:zheng.shenzhouIn this paper, the authors present three tighter Heisenberg-Weyl-type uncertainty principles related to the two-dimensional continuous quaternion wavelet transform. These results are based on the relationship between the quaternion Fourier transform and the continuous quaternion wavelet transform for the setting of the polar coordinate form of a quaternion-valued function.
Reviewer: Nikhil Khanna (Masqaṭ)Weak and strong type estimates for square functions associated with operatorshttps://zbmath.org/1517.420112023-09-22T14:21:46.120933Z"Cao, Mingming"https://zbmath.org/authors/?q=ai:cao.mingming"Si, Zengyan"https://zbmath.org/authors/?q=ai:si.zengyan"Zhang, Juan"https://zbmath.org/authors/?q=ai:zhang.juan.7|zhang.juan.1|zhang.juan.2Summary: Let \(L\) be a linear operator on \(L^2(\mathbb{R}^n)\) which generates a semigroup \(e^{-tL}\) whose kernels \(p_t(x,y)\) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function \(S_{\alpha,L}\) associated with an abstract operator \(L\). We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding to Sawyer's conjecture [\textit{E. T. Sawyer}, Stud. Math. 75, 253--263 (1983; Zbl 0528.44002)] by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type \((p, p)\) for \(S_{\alpha,L}\) and the endpoint estimate for commutators of \(S_{\alpha,L}\). Finally, all the conclusions aforementioned can be applied to a number of square functions associated to \(L\).Weighted variational inequalities for singular integrals on spaces of homogeneous typehttps://zbmath.org/1517.420122023-09-22T14:21:46.120933Z"Huang, Hongwei"https://zbmath.org/authors/?q=ai:huang.hongwei"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyong"Zhang, Feng"https://zbmath.org/authors/?q=ai:zhang.feng.2The study of the variational inequalities of various classical operators has been an active topic of current research. This article belongs to this topic. In fact, the authors study the weighted variational inequalities for singular integrals on spaces of homogeneous type. More precisely, let \(V_q K\), \(q\in(2,\infty)\), be the variation operator of a family of truncated operators of a singular integral \(K\) on an RD-space \((X,d,\mu)\) with additional layer decay property. Under the assumption that the kernel of \(K\) satisfies a Dini condition and \(V_q K\) is bounded on \(L^{p_0}(\mu)\) for some \(p_0\in(1,\infty)\), the authors establish the boundedness of \(V_q K\) on \(L^p(w)\) for \(w\in A_p\) and from \(L^1(w)\) to \(L^{1,\infty}(w)\) for \(w\in A_1\). These results also hold for oscillation operators of \(K\). The new contribution of this article is to extend some known singular integrals to the spaces of homogeneous type.
Generally speaking, this paper is interesting and well written. Particularly, the outline of this paper and the proofs are clear. It is worth mentioning that the method used in proving the main results involves some nontrivial techniques of analysis. In my opinion, this article is a nice piece of work.
Reviewer: Feng Liu (Qingdao)Boundedness of Marcinkiewicz integral with rough kernel and their commutator on weighted Herz space with variable exponenthttps://zbmath.org/1517.420132023-09-22T14:21:46.120933Z"Khalil, Omer"https://zbmath.org/authors/?q=ai:khalil.omer"Tao, Shuang Ping"https://zbmath.org/authors/?q=ai:tao.shuangping"Bechir, Acyl Mahamat"https://zbmath.org/authors/?q=ai:bechir.acyl-mahamatSummary: In this paper, we will study the boundedness of Marcinkiewicz integral with rough kernel and their commutators generated by Marcinkiewicz integral and \(\mathrm{Lip}_\gamma(\mathbb{R}^n)\) function on weighted Herz spaces with variable exponent.\( L^{ p(\cdot )} - L^{q(\cdot )}\) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimensionhttps://zbmath.org/1517.420142023-09-22T14:21:46.120933Z"Urciuolo, Marta"https://zbmath.org/authors/?q=ai:urciuolo.marta-susana"Vallejos, Lucas"https://zbmath.org/authors/?q=ai:vallejos.lucas-alejandroSummary: Let \(\alpha_i, \beta_i>0\), \(1\leq i\leq n\), and for \(t>0\) and \(x=(x_1,\dots,x_n)\in\mathbb{R}^n\), let
\[
t\cdot x=(t^{\alpha_1}x_1,\dots,t^{\alpha_n}x_n),\quad t\circ x=(t^{\beta_1}x_1,\dots,t^{\beta_n}x_n),
\]
and let
\[
\alpha=\alpha_1+\dots+\alpha_n,\quad\lVert x\rVert_{\alpha}=\sum_{i=1}^n\vert x_i\vert^{\frac{1}{\alpha_i}}.
\]
Let \(\varphi_1,\dots,\varphi_n\) be real functions in \(C^{\infty}(\mathbb{R}^n\setminus\{0\})\) such that \(\varphi=(\varphi_1,\dots,\varphi_n)\) is a homogeneous function with respect to these groups of dilations, i.e., \(\varphi(t\cdot x)=t\circ\varphi(x)\). Let \(\gamma>0\) and let \(\mu\) be the Borel measure in \(\mathbb{R}^{2n}\) given by
\[
\mu(E)=\int\chi_E(x,\varphi(x))\lVert x\rVert_{\alpha}^{\gamma-\alpha}\,dx.
\]
Let \(T_{\mu}f=\mu\ast f\), \(f\in S(\mathbb{R}^{2n})\). In this paper, we study the boundedness of \(T_{\mu}\) from \( L^{ p(\cdot )}(\mathbb{R}^{2n})\) into \( L^{ q(\cdot )}(\mathbb{R}^{2n})\) for certain variable exponents \(p(\cdot)\) and \(q({\cdot})\).On certain estimates for rough generalized parametric Marcinkiewicz integralshttps://zbmath.org/1517.420152023-09-22T14:21:46.120933Z"Zhang, Daiqing"https://zbmath.org/authors/?q=ai:zhang.daiqingIn this paper, the authors obtain the \(\dot{F}^0_{p,q}(\mathbb{R}^n)\) to \(L^p(\mathbb{R}^n)\) boundednes for the generalized parametric Marcinkiewicz integral operators associated to surfaces generated by polynomial compound mappings with rough kernels. Some related papers can be found in [\textit{F. Liu}, Bull. Korean Math. Soc. 56, No. 5, 1099--1115 (2019; Zbl 1423.42028); \textit{F. Liu} et al., J. Math. Inequal. 14, No. 1, 187--209 (2020; Zbl 1439.42022); Math. Inequal. Appl. 18, No. 2, 453--469 (2015; Zbl 1323.42014)].
Reviewer: Haixia Yu (Beijing)Sharp bounds for multilinear fractional integral operators on Morrey type spaces: the endpoint caseshttps://zbmath.org/1517.420162023-09-22T14:21:46.120933Z"Zheng, Suting"https://zbmath.org/authors/?q=ai:zheng.suting"Wang, Dinghuai"https://zbmath.org/authors/?q=ai:wang.dinghuai"Hu, Xi"https://zbmath.org/authors/?q=ai:hu.xiSummary: In this paper, the Strichartz's result of the exponential integrability of fractional integral operators [\textit{R. S. Strichartz}, Indiana Univ. Math. J. 21, 841--842 (1972; Zbl 0241.46028)] is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that \(p_i \neq 1\) for all \(i=1,\ldots, m\) and extend some known results.Discrete analogues in harmonic analysis: directional maximal functions in \(\mathbb{Z}^2\)https://zbmath.org/1517.420172023-09-22T14:21:46.120933Z"Cladek, Laura"https://zbmath.org/authors/?q=ai:cladek.laura"Krause, Ben"https://zbmath.org/authors/?q=ai:krause.benThis paper studies the directional maximal functions in \({\mathbb Z}^{2}\). The proof relies on a version of the Kakeya-type problem and a combination of geometric and number-theoretic methods. It also gives a result on a discrete directional maximal operator along polynomial orbits.
Reviewer: Kwok Pun Ho (Hong Kong)Almost everywhere convergence for Lebesgue differentiation processes along rectangleshttps://zbmath.org/1517.420182023-09-22T14:21:46.120933Z"D'Aniello, E."https://zbmath.org/authors/?q=ai:daniello.emma"Gauvan, A."https://zbmath.org/authors/?q=ai:gauvan.anthony"Moonens, L."https://zbmath.org/authors/?q=ai:moonens.laurent"Rosenblatt, J."https://zbmath.org/authors/?q=ai:rosenblatt.joseph-maxSummary: In this paper, we study Lebesgue differentiation processes along rectangles \(R_k\) shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in \(L^p\) spaces. In particular, classes of examples of such processes failing to converge a.e., in \(L^\infty\) are provided, for which \(R_k\) is known to be oriented along the slope \(k^{-s}\) for \(s > 0\), yielding an interesting counterpart to the fact that the directional maximal operator associated to the set \(\{k^{-s}: k\in\mathbb{N}^\ast\}\) fails to be bounded in \(L^p\) for any \(1 \leq p < \infty\).Application of Perron trees to geometric maximal operatorshttps://zbmath.org/1517.420192023-09-22T14:21:46.120933Z"Gauvan, Anthony"https://zbmath.org/authors/?q=ai:gauvan.anthonySummary: We characterize the \(L^p(\mathbb{R}^2)\) boundedness of the geometric maximal operator \(M_{a,b}\) associated to the basis \(\mathcal{B}_{a,b}\) \((a,b > 0)\) which is composed of rectangles \(R\) whose eccentricity and orientation are of the form
\[
\left (e_R ,\omega_R \right) = \left (\frac{1}{n^a}, \frac{\pi}{4n^b} \right)
\]
for some \(n \in \mathbb{N}^\ast\). The proof involves generalized Perron trees, as constructed by \textit{K. E. Hare} and \textit{J.-O. Rönning} [J. Fourier Anal. Appl. 4, No. 2, 215--227 (1998; Zbl 0911.42010)].A unified version of weighted weak type inequalities for one-sided maximal function on \({\mathbb{R}}^2\)https://zbmath.org/1517.420202023-09-22T14:21:46.120933Z"Zhang, Erxin"https://zbmath.org/authors/?q=ai:zhang.erxin"Ren, Yanbo"https://zbmath.org/authors/?q=ai:ren.yanboSummary: Let \(\varphi_1\), \(\gamma\) be nondecreasing functions on \([0,\infty)\), \(\varphi_2\) be a quasi-convex function and \(M^+f\) be the one-sided Hardy-Littlewood maximal function on \(\mathbb{R}^2\). In this paper, we give a characterization theorem for a weighted weak type inequality of the form
\[
\varphi_1(\lambda)\omega (\{x \in \mathbb{R}^2: M^+f(x)> \lambda \}) \le C\int_{\mathbb{R}^2}\varphi_2 \left(C\frac{|f(x)|\varrho (x)}{\gamma (\lambda)} \right) \sigma (x)dx,
\]
which generalizes and unifies some known results.Grand Morrey spaces and grand Hardy-Morrey spaces on Euclidean spacehttps://zbmath.org/1517.420212023-09-22T14:21:46.120933Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punIn this paper, the author introduces and investigates grand Morrey spaces and grand Hardy-Morrey spaces on \(\mathbb R^n\). He shows that whenever a grand Morrey space satisfies some mild conditions, the characteristic functions of balls belong to a grand Morrey space. Hence, a grand Morrey space is a ball Banach function space. Moreover, the small block space is defined, which is the block-type space built on the small Lebesgue space. Then the author establishes a duality result between grand Morrey spaces and small block spaces and obtains the boundedness of the Hardy-Littlewood maximal operator on small block spaces. With the help of these properties of these new spaces and the refined Rubio de Francia extrapolation theory, the author establishes the boundedness of the Calderón-Zygmund operator and its linear and nonlinear commutator on grand Morrey spaces. Furthermore, the author obtains the boundedness of Calderón-Zygmund operators and the parametric Marcinkiewicz integrals on grand Hardy-Morrey spaces by extending the extrapolation theory. It is worth mentioning that the results in this paper show that the extrapolation theory is applied not only to linear operators on grand Morrey spaces and grand Hardy-Morrey spaces.
Reviewer: Jian Tan (Nanjing)Nontriviality of John-Nirenberg-Campanato spaceshttps://zbmath.org/1517.420222023-09-22T14:21:46.120933Z"Zeng, Zongze"https://zbmath.org/authors/?q=ai:zeng.zongze"Chang, Der-Chen"https://zbmath.org/authors/?q=ai:chang.der-chen-e"Tao, Jin"https://zbmath.org/authors/?q=ai:tao.jin"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachunSummary: Let \(p, q\in[1, \infty)\), \(\alpha\in\mathbb{R}\), and \(s\) be a non-negative integer. This article addresses the nontriviality of the John-Nirenberg-Campanato space \(JN_{(p, q, s)_\alpha}(\mathcal{X})\), where \(\mathcal{X}\) denotes \(\mathbb{R}^n\) or any given cube of \(\mathbb{R}^n\). First, the authors obtain some (non-)trivial ranges of \(\{p, q, s, \alpha\}\), which sheds some light on the independence of \(JN_{(p, q, s)_\alpha}(\mathcal{X})\) over the second subindex \(q\). Second, for positive \(\alpha\), the authors show that \(JN_{(p, q, s)_\alpha}(\mathcal{X})\) is different from the Campanato space via establishing an embedding from the fractional Sobolev space to \(JN_{(p, q, s)_\alpha}(\mathcal{X})\). Third, for negative \(\alpha\), the authors show that the Riesz-Morrey space \(RM_{p, q, \alpha}(\mathcal{X})\) is a proper subspace of \(JN_{(p, q, s)_\alpha}(\mathcal{X})\), which gives a negative answer to the conjecture on the equivalence between John-Nirenberg-Campanato spaces and Riesz-Morrey spaces. Moreover, the nontriviality of local John-Nirenberg-Campanato spaces is also presented.Rough pseudodifferential operators on Hardy spaces for Fourier integral operatorshttps://zbmath.org/1517.420232023-09-22T14:21:46.120933Z"Rozendaal, Jan"https://zbmath.org/authors/?q=ai:rozendaal.janSummary: We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols \(a(x,\eta)\) are elements of \(C_\ast^r S_{1,\delta}^m\) classes that have limited regularity in the \(x\) variable. We show that the associated pseudodifferential operator \(a(x, D)\) maps between Sobolev spaces \(\mathcal{H}_{\mathrm{FIO}}^{s,p} (\mathbb{R}^n)\) and \(\mathcal{H}_{\mathrm{FIO}}^{t,p} (\mathbb{R}^n)\) over the Hardy space for Fourier integral operator \(\mathcal{H}_{\mathrm{FIO}}^p (\mathbb{R}^n)\). Our main result implies that for \(m = 0\), \(\delta =1/2\) and \(r > n - 1\), \(a(x, D)\) acts boundedly on \(\mathcal{H}_{\mathrm{FIO}}^p (\mathbb{R}^n)\) for all \(p \in (1, \infty)\).Two-dimensional martingale transforms and their applications in summability of Walsh-Fourier serieshttps://zbmath.org/1517.420272023-09-22T14:21:46.120933Z"Goginava, Ushangi"https://zbmath.org/authors/?q=ai:goginava.ushangi"Nagy, Károly"https://zbmath.org/authors/?q=ai:nagy.karolySummary: In the paper, we are going to prove that the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh-Fourier series is bounded from \(L\log L\left((0,1]\times (0,1]\right)\) to \(L_{1,\infty}((0,1]\times (0,1])\). As a consequence, it can be obtained that \(L\log L\left((0,1]\times (0,1]\right)\) is maximal Orlicz space, where the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh-Fourier series for the functions from this space converge in measure.Bilinear wavelet representation of Calderón-Zygmund formshttps://zbmath.org/1517.420342023-09-22T14:21:46.120933Z"Di Plinio, Francesco"https://zbmath.org/authors/?q=ai:di-plinio.francesco"Green, Walton"https://zbmath.org/authors/?q=ai:green.walton"Wick, Brett D."https://zbmath.org/authors/?q=ai:wick.brett-duaneThe authors establish a finite representation of a bilinear Calderón-Zygmund form \(\Lambda(f,g,h)=\int_{\mathbb R^{3d}} K(x_0,x_1,x_2)f(x_1)g(x_2)h(x_0)\,\mathrm{d}x\) (where the supports of \(f\), \(g\) and \(h\) have no point in common) as a sum of wavelet forms and paraproduct forms. A trilinear form \(U\) is called a wavelet form if \(U(\pi(f,g,h))= \int_{Z^d}\langle f\otimes f,\nu_z\rangle \langle h,\phi_z\rangle {d}\mu(z)\) where \(\pi\) is a permutation, \(Z^d= \mathbb R^d\times (0,\infty)\), \(\varphi_z= \frac 1{t^d}\varphi(\frac{\cdot-\omega}{t})\), \(\mathrm{d}\mu(z)= \frac 1t\mathrm{d}\omega \mathrm{d}t\) when \(z= (\omega,t)\in Z^d\) and where \(\phi\) is a mother wavelet and \(\nu\) is a function that is connected to the smoothness of the wavelet form. The form \(\Lambda\) has \(0\)-th order paraproducts if there exist BMO-functions \(b_0^i\) such that \(\Lambda(1,1,\psi)=\langle b_0^0,\psi\rangle\), \(\Lambda(\psi,1,1)=\langle b_0^1,\psi\rangle\) and \(\Lambda(1,\psi,1)=\langle b_0^2,\psi\rangle\) for all Schwartz functions \(\psi\) with \(\int_{\mathbb R^d}\psi(x)\,\mathrm{d}x=0\), that is the standard bilinear \(T(1,1)\) condition holds. It is assumed that form \(\Lambda\) has paraproducts up to order \((k_1,k_2)\), and this concept is defined iteratively.
Using this representation the authors derive a sparse \(T(1)\)-bound and then sharp weighted bilinear estimates in Lebesgue and Sobolev spaces as well as weighted fractional Sobolev space estimates.
Finally, the authors consider how these results can be extended to \(m\)-linear operators and discuss some further questions.
Reviewer: Gustaf Gripenberg (Aalto)Heisenberg-type uncertainty inequalities for the Dunkl wavelet transformhttps://zbmath.org/1517.420352023-09-22T14:21:46.120933Z"Ghobber, Saifallah"https://zbmath.org/authors/?q=ai:ghobber.saifallahSummary: The aim of this paper is to prove Heisenberg-type uncertainty inequalities for the Dunkl wavelet transform, involving the time and scale dispersions, showing that, the Dunkl wavelet transform of a nonzero function cannot be concentrated both in time and in scale in the time-scale domain. These inequalities are the consequences of other stronger uncertainty inequalities, which are the local and logarithmic uncertainty inequalities for the Dunkl wavelet transform.Interpolation of abstract Hardy-type spaceshttps://zbmath.org/1517.460192023-09-22T14:21:46.120933Z"Borovitsky, V. A."https://zbmath.org/authors/?q=ai:borovitskii.vyacheslav-a"Kislyakov, S. V."https://zbmath.org/authors/?q=ai:kislyakov.sergei-vitalevichSummary: Interpolation theorems are proved for Hardy-type spaces arising form certain uniform algebras more general than weak\(^\ast\)-Dirichlet algebras. It is shown that, in a sense, the entire setting is not sensitive to the introduction of a weight. Some generalizations that model the case of two variables are also discussed.Spaces of Besov-Sobolev type and a problem on nonlinear approximationhttps://zbmath.org/1517.460242023-09-22T14:21:46.120933Z"Domínguez, Óscar"https://zbmath.org/authors/?q=ai:dominguez.oscar"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreas"Street, Brian"https://zbmath.org/authors/?q=ai:street.brian"Van Schaftingen, Jean"https://zbmath.org/authors/?q=ai:van-schaftingen.jean"Yung, Po-Lam"https://zbmath.org/authors/?q=ai:yung.polamLet \(\{\phi_j \}^\infty_{- \infty}\) be the usual dyadic resolution of unity in \(\mathbb R^n\), underlying the study of the well-known homogeneous spaces \(\dot{B}^s_{p,q}\) and \(\dot{F}^s_{p,q}\) in \(\mathbb R^n\). In particular,
\[
\|f \, | \dot{B}^s_{p,q} \| = \Big( \sum^\infty_{j=-\infty} 2^{jsq} \big\| (\phi_j \hat{f} )^\vee \,| L_p (\mathbb R^n) \|^q \Big)^{1/q}
\]
is the standard quasi-norm in \(\dot{B}^s_{p,q}\). There are numerous modifications of these spaces. In the paper under review, the authors introduce spaces \(\mathcal{B}^s_p (\gamma, r)\), replacing \(L_p\) in \(B^s_{p,p}\) by \(L_{p,r} (\mu_\gamma)\), where \(L_{p,r}\) are Lorentz spaces and \(\mu_\gamma\) are special measures. In particular, \(\mathcal{B}^s_p (\gamma, p) = \dot{B}^s_{p,p}\), \(\gamma \in \mathbb R\). It is the main aim of this paper to study these spaces in detail, mainly parallel to related classical questions.
Reviewer: Hans Triebel (Jena)Commutator estimates for vector fields on variable Triebel-Lizorkin spaceshttps://zbmath.org/1517.460262023-09-22T14:21:46.120933Z"Salah, Ben Mahmoud"https://zbmath.org/authors/?q=ai:salah.ben-mahmoud"Drihem, Douadi"https://zbmath.org/authors/?q=ai:drihem.douadiSummary: In this paper we present a bilinear estimate for commutators on Triebel-Lizorkin spaces with variable smoothness and integrability, and under no vanishing assumptions on the divergence of vector fields.Extrapolation of compactness on weighted spaceshttps://zbmath.org/1517.470622023-09-22T14:21:46.120933Z"Hytönen, Tuomas"https://zbmath.org/authors/?q=ai:hytonen.tuomas-p"Lappas, Stefanos"https://zbmath.org/authors/?q=ai:lappas.stefanosSummary: Let \(T\) be a linear operator that, for some \(p_1 \in (1, \infty)\), is bounded on \(L^{p_1} (\tilde{w})\) for all \(\tilde{w} \in A_{p_1} (\mathbb{R}^d)\) and, in addition, compact on \(L^{p_1} (w_1)\) for some \(w_1 \in A_{p_1} (\mathbb{R}^d)\). Then \(T\) is bounded and compact on \(L^p (w)\) for all \(p \in (1, \infty)\) and all \(w \in A_p (\mathbb{R}^d)\). This ``compact version'' of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one (``off-diagonal estimates'') or only in a limited range of the \(L^p\) scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner-Riesz multipliers.Maximal operators, Littlewood-Paley functions and variation operators associated with nonsymmetric Ornstein-Uhlenbeck operatorshttps://zbmath.org/1517.470712023-09-22T14:21:46.120933Z"Almeida, Víctor"https://zbmath.org/authors/?q=ai:almeida.victor-m"Betancor, Jorge J."https://zbmath.org/authors/?q=ai:betancor.jorge-j"Quijano, Pablo"https://zbmath.org/authors/?q=ai:quijano.pablo"Rodríguez-Mesa, Lourdes"https://zbmath.org/authors/?q=ai:rodriguez-mesa.lourdesSummary: In this paper, we establish \(L^p\) boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix \(-\lambda(I+R)\), being \(\lambda > 0\) and \(R\) a skew-adjoint matrix. The semigroups associated with these Ornstein-Uhlenbeck operators are the basic building blocks of all the normal Ornstein-Uhlenbeck semigroups.Embedding derivatives and integration operators on Hardy type tent spaceshttps://zbmath.org/1517.470732023-09-22T14:21:46.120933Z"Wang, Mao Fa"https://zbmath.org/authors/?q=ai:wang.maofa"Zhou, Lv"https://zbmath.org/authors/?q=ai:zhou.lvSummary: In this paper, we completely characterize the positive Borel measures \(\mu\) on the unit ball \(\mathbb{B}_n\) such that the differential type operator \(\mathcal{R}^m\) of order \(m \in \mathbb{N}\) is bounded from Hardy type tent space \(\mathcal{HT}_{q,\alpha}^p (\mathbb{B}_n)\) into \(L^s (\mu)\) for full range of \(p, q, s, \alpha\). Subsequently, the corresponding compact description of differential type operator \(\mathcal{R}^m\) is also characterized. As an application, we obtain the boundedness and compactness of integration operator \(J_g\) from \(\mathcal{HT}_{q,\alpha}^p (\mathbb{B}_n)\) to \(\mathcal{HT}_{s,\beta}^t (\mathbb{B}_n)\), and the methods used here are adaptable to the Hardy spaces.Weighted inequalities for commutators of \(p\)-adic Hausdorff operators on Herz spaceshttps://zbmath.org/1517.471232023-09-22T14:21:46.120933Z"Tran Luu Cuong"https://zbmath.org/authors/?q=ai:tran-luu-cuong."Kieu Huu Dung"https://zbmath.org/authors/?q=ai:kieu-huu-dung."Pham Thi Kim Thuy"https://zbmath.org/authors/?q=ai:pham-thi-kim-thuy.Summary: In this paper, we establish the boundedness of commutators of \(p\)-adic matrix Hausdorff operators and \(p\)-adic rough Hausdorff operators on the block Herz spaces.Orlicz-Hardy weak martingale spaces for two-parameterhttps://zbmath.org/1517.600472023-09-22T14:21:46.120933Z"Liu, Kaituo"https://zbmath.org/authors/?q=ai:liu.kaituo"Lu, Jianzhong"https://zbmath.org/authors/?q=ai:lu.jianzhong"Wu, Jun"https://zbmath.org/authors/?q=ai:wu.jun.3|wu.jun.1|wu.jun.2|wu.jun"Yue, Tian"https://zbmath.org/authors/?q=ai:yue.tianSummary: In this paper, we investigate several two-parameter weak Orlicz-Hardy martingale spaces generated by the \(p\)-convex and \(q\)-concave functions, and establish their atomic decomposition theorems. Using the atomic decomposition, we obtain a sufficient condition for the boundedness of a sublinear operator defined on the two-parameter weak Orlicz-Hardy martingale spaces. Furthermore, the dual spaces of the two-parameter weak Orlicz-Hardy martingale spaces are considered.Short communication: Weak sparse superresolution is well-conditionedhttps://zbmath.org/1517.651332023-09-22T14:21:46.120933Z"Hockmann, Mathias"https://zbmath.org/authors/?q=ai:hockmann.mathias"Kunis, Stefan"https://zbmath.org/authors/?q=ai:kunis.stefanSummary: This paper considers superresolution (SR) as the mapping of Fourier coefficients of a discrete measure on \([0,1)^d\) to its support and weights. We focus on weak SR assuming a condition on the separation of the involved measures similar to the Rayleigh criterion and prove that the reconstruction map satisfies a local Lipschitz property giving explicit estimates for the Lipschitz constant depending on the dimension \(d\) and the sampling effort while improving a bound on the assumed separation from [6]. With the Wasserstein distance as the metric on the space of measures we even conclude that weak SR is globally Lipschitz continuous and hence well-conditioned.Coupled fractional Wigner distribution with applications to LFM signalshttps://zbmath.org/1517.940312023-09-22T14:21:46.120933Z"Teali, Aajaz A."https://zbmath.org/authors/?q=ai:teali.aajaz-a"Shah, Firdous A."https://zbmath.org/authors/?q=ai:shah.firdous-ahmad"Tantary, Azhar Y."https://zbmath.org/authors/?q=ai:tantary.azhar-y"Nisar, Kottakkaran S."https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran(no abstract)