Recent zbMATH articles in MSC 42Bhttps://zbmath.org/atom/cc/42B2021-06-15T18:09:00+00:00WerkzeugWeighted and unweighted Solyanik estimates for the multilinear strong maximal function.https://zbmath.org/1460.420292021-06-15T18:09:00+00:00"Qin, Moyan"https://zbmath.org/authors/?q=ai:qin.moyan"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingyingSummary: Let \(\omega\) be a weight in \(A^\ast_\infty\) and let \(\mathcal{M}^m_n(\vec f)\) be the multilinear strong maximal function of \(\vec f=(f_1,\dots,f_m)\), where \(f_1,\dots,f_m\) are functions on \(\mathbb{R}^n\). In this paper, we consider the asymptotic estimates for the distribution functions of \(\mathcal{M}^m_n\). We show that, for \(\lambda\in (0,1)\), if \(\lambda\rightarrow 1^-\), then the multilinear Tauberian constant \(\mathcal{C}^m_n\) and the weighted Tauberian constant \(\mathcal{C}^m_{n,\omega}\) associated with \(\mathcal{M}^m_n\) enjoy the properties that
\[
\mathcal{C}^m_n(\lambda)-1\simeq m(1-\lambda)^{\frac{1}{n}}\text{ and }\mathcal{C}^m_{n,\omega}(\lambda)-1\lesssim m(1-\lambda)^{\left(4n[\omega ]_{A^\ast_\infty}\right)^{-1}}.
\]
Reviewer: Reviewer (Berlin)Two weight commutators on spaces of homogeneous type and applications.https://zbmath.org/1460.420312021-06-15T18:09:00+00:00"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinh"Gong, Ruming"https://zbmath.org/authors/?q=ai:gong.ruming"Kuffner, Marie-Jose S."https://zbmath.org/authors/?q=ai:kuffner.marie-jose-s"Li, Ji"https://zbmath.org/authors/?q=ai:li.ji.1"Wick, Brett D."https://zbmath.org/authors/?q=ai:wick.brett-d"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyongSummary: In this paper, we establish the two weight commutator theorem of Calderón-Zygmund operators in the sense of Coifman-Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for \(A_2\) weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderón-Zygmund operators: Cauchy integral operator on \(\mathbb{R}\), Cauchy-Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).
Reviewer: Reviewer (Berlin)An abstract Logvinenko-Sereda type theorem for spectral subspaces.https://zbmath.org/1460.420392021-06-15T18:09:00+00:00"Egidi, Michela"https://zbmath.org/authors/?q=ai:egidi.michela"Seelmann, Albrecht"https://zbmath.org/authors/?q=ai:seelmann.albrechtSummary: We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean \(L^2\)-space. Our result then provides a bound for the \(L^2\)-norm of such functions in terms of their \(L^2\)-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by \textit{O. Kovrijkine} [Proc. Am. Math. Soc. 129, No. 10, 3037--3047 (2001; Zbl 0976.42004)] and is based on Bernstein-type inequalities for the respective functions, complemented with a suitable covering of the underlying domain.
Reviewer: Reviewer (Berlin)Lower bounds and fixed points for the centered Hardy-Littlewood maximal operator.https://zbmath.org/1460.420302021-06-15T18:09:00+00:00"Zbarsky, Samuel"https://zbmath.org/authors/?q=ai:zbarsky.samuelSummary: For all \(p>1\) and all centrally symmetric convex bodies, \(K\subset\mathbb{R}^d\) defined \(Mf\) as the centered maximal function associated to \(K\). We show that when \(d=1\) or \(d=2\), we have \(||Mf||_p\geq (1+\epsilon(p,K))||f||_p\). For \(d\geq 3\), let \(q_0(K)\) be the infimum value of \(p\) for which \(M\) has a fixed point. We show that for generic shapes \(K\), we have \(q_0(K)>q_0(B(0,1))\).
Reviewer: Reviewer (Berlin)Approximating mixed Hölder functions using random samples.https://zbmath.org/1460.410162021-06-15T18:09:00+00:00"Marshall, Nicholas F."https://zbmath.org/authors/?q=ai:marshall.nicholas-fSummary: Suppose \(f:[0,1]^2\rightarrow \mathbb{R}\) is a \((c,\alpha)\)-mixed Hölder function that we sample at \(l\) points \(X_1,\ldots ,X_l\) chosen uniformly at random from the unit square. Let the location of these points and the function values \(f(X_1),\ldots ,f(X_l)\) be given. If \(l\ge c_1n\log^2n\), then we can compute an approximation \(\tilde{f}\) such that \[\|f-\tilde{f}\|_{L^2}=\mathcal{O}\big(n^{-\alpha}\log^{3/2}n\big),\] with probability at least \(1-n^{2-c_1} \), where the implicit constant only depends on the constants \(c>0\) and \(c_1>0\).
Reviewer: Reviewer (Berlin)Incidence estimates for well spaced tubes.https://zbmath.org/1460.520172021-06-15T18:09:00+00:00"Guth, Larry"https://zbmath.org/authors/?q=ai:guth.larry"Solomon, Noam"https://zbmath.org/authors/?q=ai:solomon.noam"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.6Let \(\mathfrak{L}\) be a finite set of lines in the plane. The Szemerédi-Trotter theorem gives sharp bounds for the number of points that lie on at least \(r \ge 2\) points of \(\mathfrak{L}\). The authors prove analogues of this theorem using \(\delta\)-tubes instead of straight lines. In this case, the estimates do not hold in general, and so the authors make some assumptions about how the tubes are spaced. Under very strong spacing conditions they obtain nearly sharp incidence estimates.
The authors' methods can be generalized to three dimensions, and they prove a special case of the Kakeya conjecture in \(\mathbb{R}^3\).
Reviewer: Norbert Knarr (Stuttgart)Weak Hardy-type spaces associated with ball quasi-Banach function spaces. II: Littlewood-Paley characterizations and real interpolation.https://zbmath.org/1460.420332021-06-15T18:09:00+00:00"Wang, Songbai"https://zbmath.org/authors/?q=ai:wang.songbai"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen"Zhang, Yangyang"https://zbmath.org/authors/?q=ai:zhang.yangyangSummary: Let \(X\) be a ball quasi-Banach function space on \(\mathbb{R}^n\). In this article, assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on \(X\) as well as it is bounded on both the weak ball quasi-Banach function space \(WX\) and the associated space, the authors establish various Littlewood-Paley function characterizations of \(WH_X(\mathbb{R}^n)\) under some weak assumptions on the Littlewood-Paley functions. The authors also prove that the real interpolation intermediate space \((H_X(\mathbb{R}^n),L^\infty(\mathbb{R}^n))_{\theta,\infty}\), between the Hardy space associated with \(X, H_X(\mathbb{R}^n)\), and the Lebesgue space \(L^\infty(\mathbb{R}^n)\), is \(WH_{X^{1/(1-\theta)}}(\mathbb{R}^n)\), where \(\theta\in (0,1)\). All these results are of wide applications. Particularly, when \(X:=M_q^p(\mathbb{R}^n)\) (the Morrey space), \(X:=L^{\vec{p}}(\mathbb{R}^n)\) (the mixed-norm Lebesgue space) and \(X:=(E_\Phi^q)_t(\mathbb{R}^n)\) (the Orlicz-slice space), all these results are even new; when \(X:=L_\omega^\Phi(\mathbb{R}^n)\) (the weighted Orlicz space), the result on the real interpolation is new and, when \(X:=L^{p(\cdot)}(\mathbb{R}^n)\) (the variable Lebesgue space) and \(X:=L_\omega^\Phi (\mathbb{R}^n)\), the Littlewood-Paley function characterizations of \(WH_X(\mathbb{R}^n)\) obtained in this article improves the existing results via weakening the assumptions on the Littlewood-Paley functions.
Reviewer: Reviewer (Berlin)Convergence of ergodic-martingale paraproducts.https://zbmath.org/1460.600302021-06-15T18:09:00+00:00"Kovač, Vjekoslav"https://zbmath.org/authors/?q=ai:kovac.vjekoslav"Stipčić, Mario"https://zbmath.org/authors/?q=ai:stipcic.marioSummary: In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of \(\text{L}^p\)-norms and leave its a.s.convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.
Reviewer: Reviewer (Berlin)\(L^2\) boundedness for commutators of fractional differential type Marcinkiewicz integral with rough variable kernel and BMO Sobolev spaces.https://zbmath.org/1460.420152021-06-15T18:09:00+00:00"Chen, Yanping"https://zbmath.org/authors/?q=ai:chen.yanping.1"Ding, Yong"https://zbmath.org/authors/?q=ai:ding.yong"Zhu, Kai"https://zbmath.org/authors/?q=ai:zhu.kaiSummary: In this paper, for \(0< \gamma< 1\) and \(b\in I_{\gamma}(\text{BMO})\), the authors give the \(L^2(\mathbb{R}^n)\) boundedness of \(\mu_{\gamma;b} \), the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.
Reviewer: Reviewer (Berlin)Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework.https://zbmath.org/1460.420192021-06-15T18:09:00+00:00"Langowski, Bartosz"https://zbmath.org/authors/?q=ai:langowski.bartosz"Nowak, Adam"https://zbmath.org/authors/?q=ai:nowak.adamSummary: We prove sharp power-weighted \(L^p\), weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator \(B_\nu\) in the exotic range of the parameter \(- \infty < \nu < 1\). Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical \(g\)-function and fractional integrals (Riesz potential operators).
Reviewer: Reviewer (Berlin)A \(q\)-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application.https://zbmath.org/1460.420382021-06-15T18:09:00+00:00"Song, Liang"https://zbmath.org/authors/?q=ai:song.liang"Wu, Liangchuan"https://zbmath.org/authors/?q=ai:wu.liangchuanSummary: The theory of tent spaces on \(\mathbb{R}^n\) was introduced by \textit{R. R. Coifman} et al. [Lect. Notes Math. 992, 1--15 (1983; Zbl 0523.42016), ``Some new functions and their applications to harmonic analysis'', J. Funct. Anal. 62, 304--335 (1985)], including atomic decomposition, duality theory and so on. \textit{E. Russ} [``The atomic decomposition for tent spaces on spaces of homogeneous type'' (2006), \url{https://maths.anu.edu.au/files/CMAProc42-r.pdf}] generalized the atomic decomposition for tent spaces to the case of spaces of homogeneous type \((X,d,\mu)\). The main purpose of this paper is to extend the results of Coifman et al. [loc. cit.] and Russ [loc. cit.] to weighted version. More precisely, we obtain a \(q\)-atomic decomposition for the weighted tent spaces \(T^p_{2,w}(X)\), where \(0<p\leq 1\), \(1<q<\infty\), and \(w\in A_\infty\). As an application, we give an atomic decomposition for weighted Hardy spaces associated to non-negative self-adjoint operators on \(X\).
Reviewer: Reviewer (Berlin)Monge-Ampère singular integral operators acting on Triebel-Lizorkin spaces.https://zbmath.org/1460.420142021-06-15T18:09:00+00:00"Cheng, Meifang"https://zbmath.org/authors/?q=ai:cheng.meifang"Lee, Ming-Yi"https://zbmath.org/authors/?q=ai:lee.ming-yi"Lin, Chin-Cheng"https://zbmath.org/authors/?q=ai:lin.chincheng"Qu, Meng"https://zbmath.org/authors/?q=ai:qu.mengSummary: We study the Triebel-Lizorkin spaces associated with sections which are closely related to the Monge-Ampère equations and show that Monge-Ampère singular integral operators are bounded on these Triebel-Lizorkin spaces.
Reviewer: Reviewer (Berlin)Spectrality of self-affine Sierpinski-type measures on \(\mathbb{R}^2\).https://zbmath.org/1460.420082021-06-15T18:09:00+00:00"Dai, Xin-Rong"https://zbmath.org/authors/?q=ai:dai.xinrong"Fu, Xiao-Ye"https://zbmath.org/authors/?q=ai:fu.xiaoye"Yan, Zhi-Hui"https://zbmath.org/authors/?q=ai:yan.zhi-huiSummary: In this paper, we study the spectral property of a class of self-affine measures \(\mu_{R,\mathcal{D}}\) on \(\mathbb{R}^2\) generated by the iterated function system \(\{\phi_d(\cdot)=R^{-1}(\cdot+d)\}_{d\in\mathcal{D}}\) associated with the real expanding matrix \(R=\begin{pmatrix} b_1 & 0 \\ 0 & b_2\end{pmatrix}\) and the digit set \(\mathcal{D}=\{\binom{0}{0},\binom{1}{0},\binom{0}{1}\}\). We show that \(\mu_{R,\mathcal{D}}\) is a spectral measure if and only if \(3|b_i\), \(i=1,2\). This extends the result of \textit{Q.-R. Deng} and \textit{K.-S. Lau} [J. Funct. Anal. 269, No. 5, 1310--1326 (2015; Zbl 1323.28011)], where they considered the case \(b_1=b_2\). And we also give a tree structure for any spectrum of \(\mu_{R,\mathcal{D}}\) by providing a decomposition property on it.
Reviewer: Reviewer (Berlin)Restriction inequalities for the hyperbolic hyperboloid.https://zbmath.org/1460.420102021-06-15T18:09:00+00:00"Bruce, Benjamin Baker"https://zbmath.org/authors/?q=ai:bruce.benjamin-baker"Oliveira e. Silva, Diogo"https://zbmath.org/authors/?q=ai:oliveira-e-silva.diogo"Stovall, Betsy"https://zbmath.org/authors/?q=ai:stovall.betsySummary: In this article we establish new inequalities, both conditional and unconditional, for the restriction problem associated to the hyperbolic, or one-sheeted, hyperboloid in three dimensions, endowed with a Lorentz-invariant measure. These inequalities are unconditional (and optimal) in the bilinear range \(q>\frac{10}{3}\).
Reviewer: Reviewer (Berlin)Commutators of bilinear pseudo-differential operators on local Hardy spaces with variable exponents.https://zbmath.org/1460.420212021-06-15T18:09:00+00:00"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghuiSummary: The aim of this paper is to establish the boundedness of the commutator \([b_1, b_2,T_{\sigma}]\) generated by the bilinear pseudo-differential operator \(T_{\sigma}\) with smooth symbols and \(b_1,b_2\in \mathrm{BMO}(\mathbb{R}^n)\) on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator \(T_{\sigma}\) is bounded from the Lebesgue space \(L^p(\mathbb{R}^n)\) into \(h^{p_1(\cdot)}(\mathbb{R}^n)\times h^{p_2(\cdot)}(\mathbb{R}^n)\). Moreover, the boundedness of the commutator \([b_1, b_2, T_{\sigma}]\) on product of local Hardy spaces with variable exponents is also obtained.
Reviewer: Reviewer (Berlin)Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces.https://zbmath.org/1460.353272021-06-15T18:09:00+00:00"Hirayama, Hiroyuki"https://zbmath.org/authors/?q=ai:hirayama.hiroyuki"Kinoshita, Shinya"https://zbmath.org/authors/?q=ai:kinoshita.shinya"Okamoto, Mamoru"https://zbmath.org/authors/?q=ai:okamoto.mamoruSummary: In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations introduced by \textit{M. Colin} and \textit{T. Colin} [Differ. Integral Equ. 17, No. 3--4, 297--330 (2004; Zbl 1174.35528)]. We determine an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, except for the scaling critical case. This result covers a gap left open in [the first author, Commun. Pure Appl. Anal. 13, No. 4, 1563--1591 (2014; Zbl 1294.35139); with the second author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 178, 205--226 (2019; Zbl 1406.35357)].
Reviewer: Reviewer (Berlin)Areas spanned by point configurations in the plane.https://zbmath.org/1460.520182021-06-15T18:09:00+00:00"Mcdonald, Alex"https://zbmath.org/authors/?q=ai:mcdonald.alexSummary: We consider an over-determined Falconer type problem on \((k+1)\)-point configurations in the plane using the group action framework introduced in [\textit{A. Greenleaf} et al., Rev. Mat. Iberoam. 31, No. 3, 799--810 (2015; Zbl 1329.52015)]. We define the area type of a \((k+1)\)-point configuration in the plane to be the vector in \(\mathbb{R}^{\binom{k+1}{2}}\) with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is \(2k-1\) dimensional, and prove that a compact set \(E\subset\mathbb{R}^d\) of sufficiently large Hausdorff dimension determines a positve measure set of area types.
Reviewer: Reviewer (Berlin)Boundedness of differential transforms for one-sided fractional Poisson-type operator sequence.https://zbmath.org/1460.420132021-06-15T18:09:00+00:00"Chao, Zhang"https://zbmath.org/authors/?q=ai:chao.zhang"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.tao"Torrea, José L."https://zbmath.org/authors/?q=ai:torrea.jose-luisSummary: Let \(\mathcal{P}_{\tau}^\alpha f\) be given by
\[
\mathcal{P}_{\tau}^\alpha f(t)=\frac{1}{4^\alpha\Gamma(\alpha)}\int_0^{+\infty}\frac{\tau^{2\alpha}e^{-{\tau^2}/(4s)}}{s^{1+\alpha}}f(t-s)\text{d}s,\,\tau >0,\, t\in\mathbb{R},\, 0<\alpha<1.
\]
It is known that the function \(U^\alpha(t,\tau)=\mathcal{P}^\alpha_\tau f(t)\) is a classical solution to the extension problem
\[
-D_{\text{left}}U^\alpha+\frac{1-2\alpha}{\tau}\,U^\alpha_\tau+U^\alpha_{\tau\tau}=0,\text{ in }\mathbb{R}\times (0,\infty)
\]
and
\[
\lim\limits_{\tau\rightarrow 0^+}\mathcal{P}_\tau^\alpha f(t)=f(t),\quad a.e. \text{ and in }L^p(\mathbb{R},w)\text{-norm},\,w\in A_p^-.
\]
In this paper, we analyze the convergence speed of a series related with \(\mathcal{P}_\tau^\alpha f\) by discussing the behavior of the family of operators
\[
T_N^\alpha f(t)=\sum\limits_{j=N_1}^{N_2} v_j(\mathcal{P}_{a_{j+1}}^\alpha f(t)-\mathcal{P}_{a_j}^\alpha f(t)),\quad N=(N_1,N_2)\in\mathbb{Z}^2\text{ with }N_1<N_2,
\]
where \(\{v_j\}_{j\in\mathbb{Z}}\) is a bounded number sequence, and \(\{a_j\}_{j\in\mathbb{Z}}\) is a \(\rho\)-lacunary sequence of positive numbers, that is, \(1<\rho\leq a_{j+1}/a_j\), for all \(j\in\mathbb{Z}\). We shall show the boundedness of the maximal operator
\[
T^*f(t)=\sup\limits_N\left|T_N^\alpha f(t)\right|,\quad t\in\mathbb{R},
\]
in the one-sided weighted Lebesgue spaces \(L^p(\mathbb{R},\omega)(\omega\in A_p^-)\), \(1<p<\infty\). As a consequence we infer the existence of the limit, in norm and almost everywhere, of the family \(T_N^\alpha f\) for functions in \(L^p(\mathbb{R},\omega)\). Results for \(L^1(\mathbb{R},\omega)(\omega\in A_1^-)\), \(L^\infty (\mathbb{R})\) and BMO\((\mathbb{R})\) are also obtained. It is also shown that the local size of \(T^*f\), for functions \(f\) having local support, is the same with the order of a singular integral. Moreover, if \(\{v_j\}_{j\in\mathbb{Z}}\in \ell^p(\mathbb{Z})\), we get an intermediate size between the local size of singular integrals and Hardy-Littlewood maximal operator.
Reviewer: Reviewer (Berlin)Erdélyi-Kober fractional integrals on Hardy space and BMO.https://zbmath.org/1460.420322021-06-15T18:09:00+00:00"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punThe paper under review establishes the boundedness of a wide class of fractional integral operators on the Hardy space \(H^{1}(\mathbb{R})\) and its dual \(BMO\). The operators under consideration are called multi Erdélyi-Kober fractional integral operators, and are defined by
\[
Tf(x) = \int _{A} G(\sigma)f(x\sigma ^{\frac{1}{\beta}})d\sigma \quad \forall x \in \mathbb{R},
\]
for an interval \(A\) equal to either \([0,1]\) or \([1,\infty)\), a parameter \(\beta>0\), and a kernel \(G\) which is a Meijer \(G\)-function. These \(G\) functions are defined through a contour integral involving the gamma function and a finite family of points in the complex plane. They capture, as special cases, most usual special functions.
The author proves the boundedness of \(T\) on \(H^{1}\) using the basic definition of the \(H^{1}\) norm: \(\|f\|_{H^{1}}:=\| M_{P}f\|_{L^{1}}\), where \(M_{P}f(x):=\sup_{t>0} |P_{t}\star f(x)|\), and \(P_{t}\) denotes the Poisson kernel. The key step in the proof is to show that
\[
\|M_{P}(Tf)\|_{L^{1}} \leq \int_{A} |G(\sigma)| \sigma^{-\frac{1}{\beta}} \|M_{P}f\|_{L^{1}} d\sigma.
\]
This is done using Fubini's theorem and the fact that \(M_P\) commutes with dilations. The \(H^1\) result then follows by fairly direct computations with the \(G\) function, under appropriate assumptions on its parameters. The \(BMO\) result follows by duality, using the explicit form of the kernel of the adjoint of \(T\).
Reviewer: Pierre Portal (Canberra)Harmonic extension of \(Q_{\mathcal{K}}\)-type spaces via regular wavelets.https://zbmath.org/1460.420362021-06-15T18:09:00+00:00"Han, Fang"https://zbmath.org/authors/?q=ai:han.fang"Li, Pengtao"https://zbmath.org/authors/?q=ai:li.pengtaoSummary: In this paper, we use regular wavelets to investigate the harmonic extension of a class of \(Q_{\mathcal{K}}\)-type spaces \(Q_{\mathcal{K},\gamma}(\mathbb{R})\). For a locally integrable function \(f\), we apply regular wavelets to decompose and estimate the Poisson integral of \(f\). Then, by the aid of a reproducing formula, we characterize the harmonic extension of \(Q_{\mathcal{K},\gamma}(\mathbb{R})\).
Reviewer: Reviewer (Berlin)Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces \(M_{p,q}^s(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\).https://zbmath.org/1460.353182021-06-15T18:09:00+00:00"Chaichenets, Leonid"https://zbmath.org/authors/?q=ai:chaichenets.leonid"Hundertmark, Dirk"https://zbmath.org/authors/?q=ai:hundertmark.dirk"Kunstmann, Peer Christian"https://zbmath.org/authors/?q=ai:kunstmann.peer-christian"Pattakos, Nikolaos"https://zbmath.org/authors/?q=ai:pattakos.nikolaosSummary: We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in [\textit{M. Sugimoto} et al., Integral Transforms Spec. Funct. 22, No. 4--5, 351--358 (2011; Zbl 1221.44007)], of the intersection \(M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\) for \(d\in\mathbb{N}\), \(p,q\in [1,\infty]\) and \(s\geq 0\). We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves a theorem by \textit{Á. Bényi} and \textit{K. A. Okoudjou} [Bull. Lond. Math. Soc. 41, No. 3, 549--558 (2009; Zbl 1173.35115)], where only the case \(q=1\) is considered, and closes a gap in the literature. If \(q>1\) and \(s>d\left(1-\frac{1}{q}\right)\) or if \(q=1\) and \(s\geq 0\) then \(M^s_{p,q}(\mathbb{R}^d)\hookrightarrow M_{\infty,1}(\mathbb{R}^d)\) and the above intersection is superfluous. For this case we also reobtain a Hölder-type inequality for modulation spaces.
For the entire collection see [Zbl 1457.35005].
Reviewer: Reviewer (Berlin)The \(A\)-integral and restricted complex Riesz transform.https://zbmath.org/1460.440052021-06-15T18:09:00+00:00"Aliev, R. A."https://zbmath.org/authors/?q=ai:aliev.rashid-a"Nebiyeva, Kh. I."https://zbmath.org/authors/?q=ai:nebiyeva.khanim-iSummary: In this paper, we prove that the restricted complex Riesz transform of a Lebesgue integrable function is \(A\)-integrable and we obtain an analogue of Riesz's equality.
Reviewer: Reviewer (Berlin)Improved blow up criterion for the three dimensional incompressible magnetohydrodynamics system.https://zbmath.org/1460.352852021-06-15T18:09:00+00:00"Han, Bin"https://zbmath.org/authors/?q=ai:han.bin|han.bin.1"Zhao, Na"https://zbmath.org/authors/?q=ai:zhao.naSummary: In this work, we study the regularity criterion for the 3D incompressible MHD equations. By making use of the structure of the system, we obtain a criterion that is imposed on the magnetic vector field and only one component of the velocity vector field, both in scaling invariant spaces. Moreover, the norms imposed on the magnetic vector field are the Lebesgue and anisotropic Lebesgue norms. This improved the result of our previous blow up criterion in [the authors, Nonlinear Anal., Real World Appl. 51, Article ID 103000, 26 p. (2020; Zbl 1430.35035)], in which the magnetic vector field is bounded in critical Sobolev spaces.
Reviewer: Reviewer (Berlin)On the local regularity theory for the magnetohydrodynamic equations.https://zbmath.org/1460.352782021-06-15T18:09:00+00:00"Chamorro, Diego"https://zbmath.org/authors/?q=ai:chamorro.diego"Cortez, Fernando"https://zbmath.org/authors/?q=ai:cortez.fernando"He, Jiao"https://zbmath.org/authors/?q=ai:he.jiao"Jarrín, Oscar"https://zbmath.org/authors/?q=ai:jarrin.oscarSummary: Local regularity results are obtained for the MHD equations using as global framework the setting of parabolic Morrey spaces. Indeed, by assuming some local boundedness assumptions (in the sense of parabolic Morrey spaces) for weak solutions of the MHD equations it is possible to obtain a gain of regularity for such solutions in the general setting of the Serrin regularity theory. This is the first step of a wider program that aims to study both local and partial regularity theories for the MHD equations.
Reviewer: Reviewer (Berlin)Weighted estimates for bilinear Fourier multiplier operators with multiple weights.https://zbmath.org/1460.420112021-06-15T18:09:00+00:00"Hu, Guoen"https://zbmath.org/authors/?q=ai:hu.guoen"Wang, Zhidan"https://zbmath.org/authors/?q=ai:wang.zhidan"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingying"Yabuta, Kôzô"https://zbmath.org/authors/?q=ai:yabuta.kozoSummary: In the weighted theory of multilinear operators, the weights class which usually has been considered is the product of \(A_p\) weights. However, it is known that \(\prod_{k=1}^2A_{p_k}(\mathbb{R}^n)\varsubsetneq A_{\vec{p}}(\mathbb{R}^{2n})\), and \(\vec{w}=(w_1,\,w_2)\in A_{\vec{p}}(\mathbb{R}^{2n})\) does not imply that \(w_k\in L^1_{\text{loc}}(\mathbb{R}^n)\) for \(k=1,\,2\). Therefore, it is very interesting to study the weighted theory of multilinear operators with the weights in \(A_{\vec{p}}(\mathbb{R}^{2n})\). In this paper, we consider the weights class \(A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})\), which is more general than \(A_{\vec{p}}(\mathbb{R}^{2n})\). If \(\vec{w}=(w_1,\,w_2)\in A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})\), we show that the bilinear Fourier multiplier operator \(T_{\sigma}\) is bounded from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\) to \(L^p(\nu_{\vec{w}})\) when the symbol \(\sigma\) satisfies the Sobolev regularity that \(\sup_{\kappa\in\mathbb{Z}}\Vert \sigma_k\Vert_{W^{s_1,s_2}(\mathbb{R}^{2n})}<\infty\) with \(s_1,s_2\in (\frac{n}{2},\,n]\).
Reviewer: Reviewer (Berlin)Sparse bounds for the discrete spherical maximal functions.https://zbmath.org/1460.420272021-06-15T18:09:00+00:00"Kesler, Robert"https://zbmath.org/authors/?q=ai:kesler.robert-m"Lacey, Michael T."https://zbmath.org/authors/?q=ai:lacey.michael-t"Mena, Darío"https://zbmath.org/authors/?q=ai:mena.darioThe paper under review considers discrete spherical maximal functions defined by
\[
A_{\ast}f(x):= \sup \{\lambda ^{2-d} \sum _{n \in \mathbb{Z}^{d};\ |n|=\lambda} f(x-n);\ \lambda^{2} \in \mathbb{N}\} \quad \forall x \in \mathbb{Z}^{d}
\]
for all \(f \in \ell_{2}(\mathbb{Z}^{d})\) (where \(d \geq 5\)). These are discrete analogues of the spherical maximal function defined by \( \mathcal{A}_{\ast}f:= \sup_{\lambda>0} d\sigma_{\lambda} \star f \) for the normalised surface measure \(d\sigma_{\lambda}\) of a sphere of radius \(\lambda\). In the continuous case, Lacey has obtained in [\textit{M. T. Lacey}, J. Anal. Math. 139, No. 2, 613--635 (2019; Zbl 1433.42016)] sparse bounds in the optimal range of values \((p,q)\). These are estimates of the form
\[
\langle \mathcal{A}_{\ast}f,g \rangle \lesssim \sup_{\mathcal{S}\text{ sparse}} \sum _{Q \in S} |Q| (\frac{1}{|Q|}\int _{Q} |f|^{p})^{\frac{1}{p}} (\frac{1}{|Q|}\int _{Q} |g|^{q})^{\frac{1}{q}},
\]
where the supremum runs over sparse families of cubes. Establishing sparse bounds has recently become a preferred approach to various problems in harmonic analysis, thanks, in particular, to the pioneering work of \textit{A. K. Lerner} [Int. Math. Res. Not. 2013, No. 14, 3159--3170 (2013; Zbl 1318.42018)]. This is because, on the one hand, proving sparse bounds is often quite natural (a combination of Littlewood-Paley and stopping times arguments reminiscent of \(T(b)\) theorems), and, on the other hand, because sparse bound directly imply various important \(L^{p}\) estimates, including sharp weighted estimates, and vector-valued estimates.
The authors obtain sparse bounds for \(A_{\ast}\) for parameters \(p\), \(q\) such that \((\frac{1}{p},\frac{1}{q})\) lies in the interior of the polygon with vertices \(Z_{0} = (\frac{d-2}{d},\frac{2}{d})\), \(Z_{1} = (\frac{d-2}{d},\frac{d-2}{d})\), \(Z_{2} = (\frac{d^{3}-4d^{2}+4d+1}{d^{3}-2d^{2}+d-2},\frac{d^{3}-4d^{2}+6d-7}{d^{3}-2d^{2}+d-2})\), \(Z_{3}=(0,1)\) (and an endpoint estimate). Interestingly, this region is slightly smaller than its continuous counterpart for \(\mathcal{A}_{\ast}\), and conjectured to be optimal. A counterexample supporting this conjecture is given in Section 5.
The proof given by the authors is striking. Like all sparse domination arguments, it uses stopping times \(\tau\) (selecting an appropriate radius \(\lambda\) given \(f\) and \(g\)) and induction. The terms corresponding to small values of \(\tau\) can be estimated using the continuous result (applied to appropriate piecewise constant functions). For large values of \(\tau\), the authors then fundamentally rely on a version of the Hardy-Littlewood circle method decomposition developed in [\textit{Á. Magyar} et al., Ann. Math. (2) 155, No. 1, 189--208 (2002; Zbl 1036.42018)]. It decomposes \(A_{\tau}f\) (using Fourier multipliers) in pieces localised in regions of the sphere around points of the form \(\frac{\ell}{q}\) where \(q \leq \tau\) and \(\ell \in \mathbb{Z}_{q}^{d}\). Terms that correspond to large values of \(q\) can be directly estimated using \(L^2\) bounds from [Magyar et al., loc. cit.]. Some of the terms that correspond to small values of \(q\) can be estimated using a generic result about maximal functions based on Fourier multipliers, proved by \textit{N. Lohoué} [C. R. Acad. Sci., Paris, Sér. I 312, No. 8, 561--566 (1991; Zbl 0746.58073)]. This leaves the most challenging terms, for which one needs to exploit particularly strong cancellation properties. Fortunately, these terms involve Ramanujan sums of the form
\[
m\mapsto \sum _{a \in \mathbb{Z}_{q} ^{\times}} \exp(2i\pi \frac{am}{q}),
\]
for which subtle estimates are known; see Lemma 2.25 and the core of the proof just after it (tertiary decomposition).
Reviewer: Pierre Portal (Canberra)On the zeros of the spectrogram of white noise.https://zbmath.org/1460.621552021-06-15T18:09:00+00:00"Bardenet, Rémi"https://zbmath.org/authors/?q=ai:bardenet.remi"Flamant, Julien"https://zbmath.org/authors/?q=ai:flamant.julien"Chainais, Pierre"https://zbmath.org/authors/?q=ai:chainais.pierreSummary: In a recent paper, \textit{P. Flandrin} [``Time-frequency filtering based on spectrogram zeros'', IEEE Signal Process. Lett., 22, No. 11, 2137--2141 (2015; \url{doi:10.1109/LSP.2015.2463093})] proposed filtering based on the zeros of a spectrogram with Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time-frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of the zeros of the spectrogram. The crux of our analysis is that the zeros of the spectrogram of white Gaussian noise correspond to the zeros of a Gaussian analytic function, a topic of recent independent mathematical interest [\textit{J. B. Hough} et al., Zeros of Gaussian analytic functions and determinantal point processes. Providence, RI: American Mathematical Society (AMS) (2009; Zbl 1190.60038)].
Reviewer: Reviewer (Berlin)Sublinear operators on mixed-norm Hardy spaces with variable exponents.https://zbmath.org/1460.420372021-06-15T18:09:00+00:00"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punSummary: In this paper, we define and study the mixed-norm Hardy spaces with variable exponents. We establish some general principles for the mapping properties of sublinear operators on the mixed-norm Hardy spaces with variable exponents. By using these principles, we obtain the mapping properties of the Calderón-Zygmund operators, the oscillatory singular integral operators, the multiplier operators, the Littlewood-Paley functions, the intrinsic square functions, the parametric Marcinkiewicz integrals and the maximal Bochner-Riesz means on the mixed-norm Hardy spaces with variable exponents.
Reviewer: Reviewer (Berlin)\(L^p\)-estimates for the heat semigroup on differential forms, and related problems.https://zbmath.org/1460.580152021-06-15T18:09:00+00:00"Magniez, Jocelyn"https://zbmath.org/authors/?q=ai:magniez.jocelyn"Ouhabaz, El Maati"https://zbmath.org/authors/?q=ai:ouhabaz.el-maatiFor a \(M^n\) complete Riemannian manifold and \(\Delta\) the (non-negative) Laplace-Beltrami operator, consider \((e^{-t\Delta})_{t\geq 0}\) the associated heat semigroup acting as a contraction semigroup on \(L^p(M)\) for all \(1\leq p \leq \infty\), and the semigroup is strongly continuous on \(L^p(M)\) for \(1\leq p < \infty\). Now instead of \(\Delta\), one may consider the Hodge-de Rham Laplacian \(\overrightarrow{\Delta}_k=d_k^*d_k+d_{k-1}d_{k-1}^*\) and the associated contraction semigroup \((e^{-t\Delta_k})_{t\geq 0}\) on \(L^2(\Lambda^kT^*M)\). Note that \(\overrightarrow{\Delta}_k\) is non-negative.
The article under review studies \(L^p\)-estimates of the semigroup \((e^{-t\Delta_k})_{t\geq 0}\). Apparently the precise estimate of the \(L^p\)-norm \(\|(e^{-t\Delta_k})_{t\geq 0}\|_{p-p}\) is not easy, and the article proves in Theorem 1.2 (i) such an estimate. To be more specific, consider the Bochner's formula \(\overrightarrow{\Delta}_k=\Delta^*\Delta+R_k\) where \(\Delta\) the Levi-Civita connection and \(R_k\) a symmetric section of \(\text{End}(\Lambda^k T^*M)\). Denote by \(R^\pm_k\) the positive and negative part of \(R_k\). The article, under an assumption of a volume-doubling property (Compare Equation (D) in p.3003), the Gaussian upper bound condition (Compare Equation (G) in p.3004), \(R_k^-\) in the enlarged Kato class \(\hat{K}\) (compare Definition 1.1), shows that \(\|(e^{-t\Delta_k})\|_{p-p}\leq C(t\log t)^{\frac{D}{4}(1-\frac{2}{p})}\) for large \(t\) where \(D\) is a homogeneous dimension appearing in the volume doubling property.
In addition, using an \(\epsilon\)-subcritical condition for \(R_k^-\) for \(\epsilon\in[0,1)\), the authors prove in Theorem 1.4 better estimates than that of Theorem 1.2. The article also provides in Proposition 1.5 a side result that under an additional assumption of uniform boundedness of \((e^{-t\Delta})_{t\geq 0}\) on \(L^p(\Lambda^1 T^*M)\) for some fixed \(p\), the Riesz transformation \(d\Delta^{-1/2}\) is bounded on \(L^r(M)\) for all \(r\in (1,\text{max}(p,p'))\).
Reviewer: Byungdo Park (Cheongju)A short glimpse of the giant footprint of Fourier analysis and recent multilinear advances.https://zbmath.org/1460.420012021-06-15T18:09:00+00:00"Grafakos, Loukas"https://zbmath.org/authors/?q=ai:grafakos.loukasIndeed, the paper provides a concise but to-the-point overview of the genesis and influence of Fourier series and Fourier analysis, not only in itself in mathematics but also in physics, like finding \textit{temperatures}, \textit{radio telecommunications}, \textit{acoustics}, \textit{oceanography},\textit{optics}, \textit{spectroscopy}, \textit{crystallography}. [The words in Italics are the fields of investigations mentioned by the author.] Also, he says that Fourier series nowadays are very important for \textit{signal} and \textit{image processing}. Developments are, in short, mentioned for \textit{wavelets}. As to mathematics, he mentions: summation of Fourier series in higher dimensions, products of Fourier series. To close with, the author indicates some advances in multilinear Fourier analysis.
In the bibliography, one finds for instance, papers of the author himself, Carleson, Fefferman, Leong, Farkas, Kahane, Konyagin, Lacey, Thiele, Riesz etc. As to the fundamental work about wavelets due to Ingrid Daubechies [not at all mentioned by the author] one can best consult [\textit{D. Huylebrouck}, ``Ingrid Daubechies, JPEG inventrix and first female professor in Princeton'', in: België + Wiskunde. Gent: Academia Press. 35--70 (2013)].
For the entire collection see [Zbl 1433.00042].
Reviewer: Robert W. van der Waall (Amsterdam)Differentiating Orlicz spaces with rectangles having fixed shapes in a set of directions.https://zbmath.org/1460.420242021-06-15T18:09:00+00:00"D'Aniello, Emma"https://zbmath.org/authors/?q=ai:daniello.emma"Moonens, Laurent"https://zbmath.org/authors/?q=ai:moonens.laurentSummary: In the present note, we examine the behavior of some homothecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its shape) has to be a fixed real number depending on the angle between its longest side and the horizontal line (yielding a shape-function). Depending on the allowed angles and the corresponding shape-function, a basis may differentiate various Orlicz spaces. We here give some examples of shape-functions so that the corresponding basis differentiates \(L \log (L(\mathbb{R}^2)\), and show that in some ``model'' situations, a fast-growing shape function (whose speed of growth depends on \(\alpha > 0\)) does not allow the differentiation of \(L \log^{\alpha}L(\mathbb R^2)\).
Reviewer: Reviewer (Berlin)Vector valued maximal Carleson type operators on the weighted Lorentz spaces.https://zbmath.org/1460.420162021-06-15T18:09:00+00:00"Duong, Dao Van"https://zbmath.org/authors/?q=ai:duong.dao-van"Dung, Kieu Huu"https://zbmath.org/authors/?q=ai:dung.kieu-huu"Nguyen Minh Chuong"https://zbmath.org/authors/?q=ai:nguyen-minh-chuong.Summary: In this paper, by using the idea of linearizing maximal operators originated by \textit{C. Fefferman} [Ann. Math. (2) 98, 551--571 (1973; Zbl 0268.42009)] and the \(TT^\ast\) method of Stein-Wainger [\textit{E. M. Stein} and \textit{S. Wainger}, Math. Res. Lett. 8, No. 5--6, 789--800 (2001; Zbl 0998.42007)], we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by \textit{K. F. Andersen} and \textit{R. T. John} [Stud. Math. 69, 19--31 (1980; Zbl 0448.42016)] on the weighted Lorentz spaces with vector-valued functions.
Reviewer: Reviewer (Berlin)Corrigendum to ``The maximal operator on generalized Orlicz spaces''.https://zbmath.org/1460.470142021-06-15T18:09:00+00:00"Hästö, Peter A."https://zbmath.org/authors/?q=ai:hasto.peter-aSummary: In this note some flaws in the proofs of the paper [the author, ibid. 269, No. 12, 4038--4048 (2015; Zbl 1338.47032)] are corrected.
Reviewer: Reviewer (Berlin)On the composition of rough singular integral operators.https://zbmath.org/1460.420172021-06-15T18:09:00+00:00"Hu, Guoen"https://zbmath.org/authors/?q=ai:hu.guoen"Lai, Xudong"https://zbmath.org/authors/?q=ai:lai.xudong"Xue, Qingying"https://zbmath.org/authors/?q=ai:xue.qingyingSummary: In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on \(L^p(\mathbb{R}^d,w)\), \(p\in(1,\infty)\), which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint \(p=1\), the \(L\log L\) weighted weak-type bound is also obtained, which has interests of its own in the theory of rough singular integral even in the unweighted case.
Reviewer: Reviewer (Berlin)Sparse bounds for maximally truncated oscillatory singular integrals.https://zbmath.org/1460.420182021-06-15T18:09:00+00:00"Krause, Ben"https://zbmath.org/authors/?q=ai:krause.ben"Lacey, Michael T."https://zbmath.org/authors/?q=ai:lacey.michael-tSummary: For a polynomial \(P(x,y)\), and any Calderón-Zygmund kernel \(K\), the operator below satisfies a \((1,{r})\) sparse bound, for \(1<r \leq 2:\)
\[
\sup_{\epsilon > 0} \left | \int_{|y|>\epsilon} f(x-y)e^{2\pi i P(x,y)}K(y)dy \right | .
\]
The implied bound depends upon \(P(x,y)\) only through the degree of \(P\). We derive from this a range of weighted inequalities, including weak type inequalities on \(L^1(w)\), which are new, even in the unweighted case. The unweighted weak type estimate, without maximal truncations, is due to \textit{S. Chanillo} and \textit{M. Christ} [Duke Math. J. 55, 141--155 (1987; Zbl 0667.42007)].
Reviewer: Reviewer (Berlin)Modelled distributions of Triebel-Lizorkin type.https://zbmath.org/1460.460242021-06-15T18:09:00+00:00"Hensel, Sebastian"https://zbmath.org/authors/?q=ai:hensel.sebastian-c"Rosati, Tommaso"https://zbmath.org/authors/?q=ai:rosati.tommaso-cornelisSpaces of type \(F^\alpha_{p,q} (\mathbb R^d)\), \(1\le p <\infty\), \(1\le q \le \infty\), \(\alpha \in \mathbb R\), are introduced using suitable ball means. It is shown that they can be characterized by Daubechies wavelets. This is taken as a starting point to introduce corresponding spaces on so-called regularity structures instead of \(\mathbb R^d\) (spaces of modelled distributions). There are similar spaces \(B^\alpha_{p,q} (\mathbb R^d)\) and their counterparts on regular structures. The authors study embeddings between these abstract spaces. It is one of the main aims of the paper to show how these abstract spaces on regularity structures can be related to their concrete counterparts on~\(\mathbb R^d\).
Reviewer: Hans Triebel (Jena)BMO solvability and absolute continuity of caloric measure.https://zbmath.org/1460.420252021-06-15T18:09:00+00:00"Genschaw, Alyssa"https://zbmath.org/authors/?q=ai:genschaw.alyssa"Hofmann, Steve"https://zbmath.org/authors/?q=ai:hofmann.steveSummary: We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-\(A_{\infty }\) property) of caloric measure with respect to surface measure, for an open set \(\Omega\subset\mathbb{R}^{n+1}\), assuming as a background hypothesis only that the essential boundary of \(\Omega\) satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak-\(A_{\infty}\) property of the caloric measure is equivalent to \(L^p\) solvability of the initial-Dirichlet problem, we may then deduce that BMO-solvability implies \(L^p\) solvability for some finite \(p\).
Reviewer: Reviewer (Berlin)Local well-posedness for the Klein-Gordon-Zakharov system in 3D.https://zbmath.org/1460.353322021-06-15T18:09:00+00:00"Pecher, Hartmut"https://zbmath.org/authors/?q=ai:pecher.hartmutSummary: We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by \textit{I. Bejenaru} and \textit{S. Herr} [J. Funct. Anal. 261, No. 2, 478--506 (2011; Zbl 1228.42027)] for the Zakharov system and already applied by \textit{S. Kinoshita} [Discrete Contin. Dyn. Syst. 38, No. 3, 1479--1504 (2018; Zbl 1397.35281)] to the Klein-Gordon-Zakharov system in 2D.
Reviewer: Reviewer (Berlin)Complex interpolation of vanishing Morrey spaces.https://zbmath.org/1460.460132021-06-15T18:09:00+00:00"Hakim, Denny Ivanal"https://zbmath.org/authors/?q=ai:hakim.denny-ivanal"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiroSummary: We describe the first and second complex interpolations of vanishing Morrey spaces, introduced in [\textit{A. Almeida} and \textit{S. Samko}, J. Funct. Anal. 272, No. 6, 2392--2411 (2017; Zbl 1368.46027)] and [\textit{F. Chiarenza} and \textit{M. Franciosi}, Ann. Mat. Pura Appl. (4) 161, 285--297 (1992; Zbl 0796.35032)]. In addition, we show that the diamond subspace in [\textit{D. I. Hakim} et al., Constr. Approx. 46, No. 3, 489--563 (2017; Zbl 1385.42023)] and one of the function spaces in [\textit{A. Almeida} and \textit{S. Samko}, loc. cit.] are the same. We also give several examples showing that each of the complex interpolations of these spaces is different.
Reviewer: Reviewer (Berlin)Parabolic equations involving Laguerre operators and weighted mixed-norm estimates.https://zbmath.org/1460.350662021-06-15T18:09:00+00:00"Fan, Huiying"https://zbmath.org/authors/?q=ai:fan.huiying"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.taoSummary: In this paper, we study evolution equation \(\partial_tu=-L_\alpha u+f\) and the corresponding Cauchy problem, where \(L_\alpha\) represents the Laguerre operator \(L_\alpha=\frac{1}{2}(-\frac{d^2}{dx^2}+x^2+\frac{1}{x^2}(\alpha^2-\frac{1}{4}))\), for every \(\alpha\geq-\frac{1}{2}\). We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup \(\{e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0}\). In addition, we define the Poisson operator related to the fractional power \((\partial_t+L_\alpha)^s\) and reveal weighted mixed-norm estimates for revelent maximal operators.
Reviewer: Reviewer (Berlin)Product Besov and Triebel-Lizorkin spaces with application to nonlinear approximation.https://zbmath.org/1460.420262021-06-15T18:09:00+00:00"Georgiadis, Athanasios G."https://zbmath.org/authors/?q=ai:georgiadis.athanasios-g"Kyriazis, George"https://zbmath.org/authors/?q=ai:kyriazis.george-c"Petrushev, Pencho"https://zbmath.org/authors/?q=ai:petrushev.pencho-pSummary: The Littlewood-Paley theory of homogeneous product Besov and Triebel-Lizorkin spaces is developed in the spirit of the \(\varphi\)-transform of Frazier and Jawerth. This includes the frame characterization of the product Besov and Triebel-Lizorkin spaces and the development of almost diagonal operators on these spaces. The almost diagonal operators are used to obtain product wavelet decomposition of the product Besov and Triebel-Lizorkin spaces. The main application of this theory is to nonlinear \(m\)-term approximation from product wavelets in \(L^p\) and Hardy spaces. Sharp Jackson and Bernstein estimates are obtained in terms of product Besov spaces.
Reviewer: Reviewer (Berlin)Higher-order Riesz transforms of Hermite operators on new Besov and Triebel-Lizorkin spaces.https://zbmath.org/1460.420342021-06-15T18:09:00+00:00"Bui, The Anh"https://zbmath.org/authors/?q=ai:bui.the-anh"Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinhSummary: Consider the Hermite operator \(H=-\Delta +|x|^2\) on the Euclidean space \(\mathbb{R}^n\). The aim of this article is to prove the boundedness of higher-order Riesz transforms on appropriate Besov and Triebel-Lizorkin spaces. As an application, we prove certain regularity estimates of second-order elliptic equations in divergence form with the oscillator perturbations.
Reviewer: Reviewer (Berlin)Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space.https://zbmath.org/1460.352912021-06-15T18:09:00+00:00"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Wu, Xing"https://zbmath.org/authors/?q=ai:wu.xing"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Zhu, Weipeng"https://zbmath.org/authors/?q=ai:zhu.weipengSummary: Whether or not the data-to-solution map of the Cauchy problem for the Camassa-Holm equation and Novikov equation in the critical Besov space \(B_{2,1}^{3/2}(\mathbb{R})\) is uniformly continuous remains open. In the paper, we aim at solving the open question left in the previous works [the first author et al., J. Differ. Equations 269, No. 10, 8686--8700 (2020; Zbl 1442.35344); J. Math. Fluid Mech. 22, No. 4, Paper No. 50, 10 p. (2020; Zbl 1448.35402)] and giving a negative answer to this problem.
Reviewer: Reviewer (Berlin)Uncertainty principles for the continuous wavelet transform in the Hankel setting.https://zbmath.org/1460.420532021-06-15T18:09:00+00:00"B. Hamadi, N."https://zbmath.org/authors/?q=ai:b-hamadi.n"Omri, S."https://zbmath.org/authors/?q=ai:omri.slim|omri.salemSummary: Shapiro's dispersion and Umbrella theorems are proved for the continuous Hankel wavelet transform. As a side results, we extend local uncertainty principles for set of finite measure to the latter transform.
Reviewer: Reviewer (Berlin)Iterated weak and weak mixed-norm spaces with applications to geometric inequalities.https://zbmath.org/1460.420352021-06-15T18:09:00+00:00"Chen, Ting"https://zbmath.org/authors/?q=ai:chen.ting"Sun, Wenchang"https://zbmath.org/authors/?q=ai:sun.wenchangSummary: In this paper, we consider two types of weak norms, the weak mixed-norm and the iterated weak norm, in Lebesgue spaces with mixed norms. We study properties of two weak norms and present their relationship. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for the two weak norms, respectively. We study the convergence in the truncated norm, which is a substitution of the convergence in measure for mixed-norm Lebesgue spaces. And we give a characterization of the convergence in the truncated norm. We show that Hölder's inequality is not always true on weak mixed-norm Lebesgue spaces and we give a complete characterization of indices which admit Hölder's inequality. As applications, we establish some geometric inequalities related to fractional integrals in weak mixed-norm spaces and in iterated weak spaces, respectively, which essentially generalize the Hardy-Littlewood-Sobolev inequality.
Reviewer: Reviewer (Berlin)On a \(k\)-fold beta integral formula.https://zbmath.org/1460.420222021-06-15T18:09:00+00:00"Wu, Di"https://zbmath.org/authors/?q=ai:wu.di"Shi, Zuoshunhua"https://zbmath.org/authors/?q=ai:shi.zuoshunhua"Nie, Xudong"https://zbmath.org/authors/?q=ai:nie.xudong"Yan, Dunyan"https://zbmath.org/authors/?q=ai:yan.dunyanSummary: In this paper, we investigate some necessary and sufficient conditions which ensure validity of a \(k\)-fold Beta integral formula
\[
\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt=C_{d_1,\dots,d_k}\prod\limits_{1\leq i<j\leq k}\biggl|x^i-x^j\biggr|^{-\alpha_{ij}}\tag{0.1}
\]
for any \(x^i\in\mathbb{R}^n\) and some nonzero real numbers \(d_i\) with \(i=1,2,\dots,k\). We establish that Eq. (0.1) holds if and only if
\[
\max\{d_1,d_2\}<n<d_1+d_2\text{ when }k=2\tag{0.2}
\]
and
\[
\max\{d_1,d_2,d_3\}<n,d_1+d_2+d_3=2n\text{ when }k=3.\tag{0.3}
\]
This yields a complete answer to the question raised by \textit{L. Grafakos} and \textit{C. Morpurgo} [Pac. J. Math. 191, No. 1, 85--94 (1999; Zbl )1006.42017)]. In addition, it turns out that formula (0.1) does not hold if \(k\geq 4\). For those \(k\), \(d_1, d_2, \ldots , d_k\) not satisfying (0.2) or (0.3), we prove that the real-valued integral \(\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt\) can be represented as a function of distances of consecutive differences of the sequence \(x^1, x^2,\dots, x^k\).
Reviewer: Reviewer (Berlin)Calderón-Zygmund operators on homogeneous product Lipschitz spaces.https://zbmath.org/1460.420232021-06-15T18:09:00+00:00"Zheng, Taotao"https://zbmath.org/authors/?q=ai:zheng.taotao"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiecheng"Dai, Jiawei"https://zbmath.org/authors/?q=ai:dai.jiawei"He, Shaoyong"https://zbmath.org/authors/?q=ai:he.shaoyong"Tao, Xiangxing"https://zbmath.org/authors/?q=ai:tao.xiangxingSummary: The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of product Calderón-Zygmund singular integral operators introduced by Journé on the product Lipschitz spaces. The key idea used in this paper is to develop the Littlewood-Paley theory for the product spaces which includes the characterization of a special product Besov space and a density argument for the product Lipschitz spaces in the weak sense.
Reviewer: Reviewer (Berlin)Time-frequency transforms of white noises and Gaussian analytic functions.https://zbmath.org/1460.420092021-06-15T18:09:00+00:00"Bardenet, Rémi"https://zbmath.org/authors/?q=ai:bardenet.remi"Hardy, Adrien"https://zbmath.org/authors/?q=ai:hardy.adrienSummary: A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of Gaussian white noises [\textit{R. Bardenet} et al., Appl. Comput. Harmon. Anal. 48, No. 2, 682--705 (2020; Zbl 1460.62155)]. This answered pioneering work by \textit{P. Flandrin} [``Time-frequency filtering based on spectrogram zeros'', IEEE Signal Process. Lett. 22, No. 11, 2137--2141 (2015)], who observed that the zeros of the Gabor transform of white noise had a regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises. Our main observation is a correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. Moreover, we discuss subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces and its finite dimensional approximations.
Reviewer: Reviewer (Berlin)Bilinear Hilbert transforms and (sub)bilinear maximal functions along convex curves.https://zbmath.org/1460.420202021-06-15T18:09:00+00:00"Li, Junfeng"https://zbmath.org/authors/?q=ai:li.junfeng"Yu, Haixia"https://zbmath.org/authors/?q=ai:yu.haixiaSummary: We determine the \(L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})\) boundedness of the bilinear Hilbert transform \(H_{\gamma}(f,g)\) along a convex curve \(\gamma\):
\[ H_{\gamma}(f,g)(x):=\text{p.v.}\int_{-\infty}^{\infty}f(x-t)g(x-\gamma(t)) \frac{\operatorname{d}t}{t},\]
where \(p\), \(q\), and \(r\) satisfy \(1/p+1/q =1/r\), and \(r>\frac{1}{2}\), \(p>1\), and \(q>1\). Moreover, the same \(L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})\) boundedness property holds for the corresponding (sub)bilinear maximal function \(M_{\gamma}(f,g)\) along a convex curve
\[\gamma M_{\gamma}(f,g)(x):=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon} |f(x-t)g(x-\gamma(t))|\operatorname{d}t.\]
Reviewer: Reviewer (Berlin)On a precise scaling to Caffarelli-Kohn-Nirenberg inequality.https://zbmath.org/1460.420122021-06-15T18:09:00+00:00"Bazan, Aldo"https://zbmath.org/authors/?q=ai:bazan.aldo"Neves, Wladimir"https://zbmath.org/authors/?q=ai:neves.wladimirSummary: We analyze the general form of Caffarelli-Kohn-Nirenberg inequality. Due to a new introduced parameter, this inequality presents two distinguishable ranges. One of them, the inequality is shown to be the interpolation between Hardy and weighted Sobolev inequalities. The other range, which is no more an interpolation, the positive constant in the inequality is not necessarily bounded for all value of the parameters. In both cases, the precise value of the constants were given.
Reviewer: Reviewer (Berlin)Weighted estimates for maximal functions associated with finite type curves in \(\mathbb{R}^2\).https://zbmath.org/1460.420282021-06-15T18:09:00+00:00"Manna, Ramesh"https://zbmath.org/authors/?q=ai:manna.ramesh"Shrivastava, Saurabh"https://zbmath.org/authors/?q=ai:shrivastava.saurabh"Shuin, Kalachand"https://zbmath.org/authors/?q=ai:shuin.kalachandThe authors obtain weighted boundedness on Lebesgue spaces \(L^p (w)\) for the maximal functions \(\mathcal{M}_{lac}\) and \(\mathcal{M}\) associated with a finite type curve \(\mathcal{C}\) with type \(m\) on the plane (Theorem 2.1). The exponent \(p\) is within the expected range, and \(w\) belongs to a specific class of Muckenhoupt weights and satisfies a precise reverse Hölder condition. From this they characterize the power weights for which the corresponding weighted \(L^p\) boundedness of \(\mathcal{M}_{lac}\) and \(\mathcal{M}\) hold (Theorem 2.2).
They also obtain a suitable sparse dominations for both \(\mathcal{M}_{lac}\) and \(\mathcal{M}\) (Theorem 2.4). This result is key in the proof of Theorem 2.1.
Additionally, the article contains several consequences of Theorem 2.1 (Section 5) and some further generalisation and applications of the main results (Section 7).
Reviewer: Guillermo Flores (Córdoba)Accretivity and form boundedness of second order differential operators.https://zbmath.org/1460.350992021-06-15T18:09:00+00:00"Maz'ya, Vladimir G."https://zbmath.org/authors/?q=ai:mazya.vladimir-gilelevich"Verbitsky, Igor E."https://zbmath.org/authors/?q=ai:verbitsky.igor-eSummary: Le \(\mathcal{L}\) be the general second order differential operator with complex-valued distributional coefficients \(A=(a_{jk})_{j,k=1}^n\), \(\vec{b}=(b_j)_{j=1}^n\), and \(c\) in an open set \(\Omega\subseteq \mathbb{R}^n\) (\(n\ge 1\)), with principal part either in the divergence form, \(\mathcal{L}u=\operatorname{div}(A\nabla u)+\vec{b}\cdot\nabla u+cu\) or non-divergence form, \(\mathcal{L}u=\sum_{j,k=1}^n a_{jk}\partial_j\partial_ku+\vec{b}\cdot\nabla u+cu\).
We give a survey of the results by the authors which characterize the following two properties of \(\mathcal{L}\):
\begin{itemize}
\item[(1)] \(-\mathcal{L}\) is accretive, i.e., Re\(\langle-\mathcal{L}u,u\rangle\ge 0\);
\item[(2)] \(\mathcal{L}\) is form bounded, i.e, \(|\langle\mathcal{L}u,u\rangle|\le C\|\nabla u\|_{L^2(\Omega)}^2\), for all complex-valued \(u\in C_0^\infty(\Omega)\).
\end{itemize}
Reviewer: Reviewer (Berlin)