Recent zbMATH articles in MSC 46Ehttps://zbmath.org/atom/cc/46E2024-04-15T15:10:58.286558ZWerkzeugNon-radial weights and polynomial approximation in spaces of analytic functionshttps://zbmath.org/1530.300512024-04-15T15:10:58.286558Z"Abkar, Ali"https://zbmath.org/authors/?q=ai:abkar.aliSummary: We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights.A characterization of the degenerate complex Hessian equations for functions with bounded \((p,m)\)-energyhttps://zbmath.org/1530.320162024-04-15T15:10:58.286558Z"Åhag, Per"https://zbmath.org/authors/?q=ai:ahag.per"Czyż, Rafal"https://zbmath.org/authors/?q=ai:czyz.rafalLet \((X,\omega)\) be a connected compact Kähler of dimension \(n\), where \(\omega\) is a Kähler form with \(\int_X\omega^n=1\). Put
\[\mathcal E^p_m(X,\omega):=\big\{u\in\mathcal E_m(X,\omega): u\leq0,\;e_{p,m}(u)<+\infty\big\},\] where
\[\mathcal E_m(X,\omega):=\left\{u\in\mathcal{SH}_m(X,\omega): \int_XH_m(u)=1\right\},\] \(e_{p,m}(u):=\int_X(-u)^pH_m(u)\), and \(H_m\) denotes the complex Hessian operator. Assume that \(n\geq2\), \(p>0\), \(1\leq m\leq n\). The main results of the paper are the following two theorems:
\begin{itemize}
\item[\(\bullet\)] Assume that \(\mu\) is a Borel measure and on \(X\), \(q>0\), \(1>\beta>\max\{\frac{pn-n}{pn-n+m}, \frac{p}{p+1}\}\) for \(p>1\), and \(\beta=\frac{p}{p+1}\) for \(p\leq 1\). Then following conditions are equivalent:
(1) \(\mathcal E^p_m(X,\omega)\subset L^q(X,\mu)\);
(2) there exists a \(C>0\) such that for all \(u\in\mathcal E_m(X,\omega)\cap L^\infty(X)\) with \(\sup_Xu=-1\) we have \(\int_X(-u)^qd\mu\leq Ce_{p,m}(u)^{q\beta/p}\);
(3) there exists a \(C>0\) such that for all \(u\in\mathcal E^p_m(X,\omega)\) with \(\sup_Xu=-1\) we have \(\int_X(-u)^qd\mu\leq Ce_{p,m}(u)^{q\beta/p}\).
\item[\(\bullet\)] Assume that \(\mu\) is a probability measure and on \(X\). Then \(\mathcal E^p_m(X,\omega)\subset L^p(X,\mu)\) if and only if there exists a unique \((\omega,m)\)-subharmonic function \(u\in\mathcal E^p_m(X\omega)\) such that \(\sup_X u=-1\) and \(H_m(u)=\mu\).
\end{itemize}
Reviewer: Marek Jarnicki (Kraków)Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flowshttps://zbmath.org/1530.351182024-04-15T15:10:58.286558Z"Cheng, Li-Juan"https://zbmath.org/authors/?q=ai:cheng.lijuan"Thalmaier, Anton"https://zbmath.org/authors/?q=ai:thalmaier.antonSummary: Let \(M\) be a differentiable manifold endowed with a family of complete Riemannian metrics \(g(t)\) evolving under a geometric flow over the time interval \([0,T[\). We give a probabilistic representation for the derivative of the corresponding conjugate semigroup on \(M\) which is generated by a Schrödinger-type operator. With the help of this derivative formula, we derive fundamental Harnack-type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.On the global well-posedness for a multi-dimensional compressible Navier-Stokes-Poisson systemhttps://zbmath.org/1530.352152024-04-15T15:10:58.286558Z"Dong, Junting"https://zbmath.org/authors/?q=ai:dong.junting"Wang, Zheng"https://zbmath.org/authors/?q=ai:wang.zheng"Xu, Fuyi"https://zbmath.org/authors/?q=ai:xu.fuyiSummary: This article is dedicated to the study of the Cauchy problem for compressible Navier-Stokes-Poisson system in spatial dimensions two and higher. We prove the global well-posedness when the initial data are close to a stable equilibrium state in critical \(L^p\) framework.Time-frequency analysis on flat tori and Gabor frames in finite dimensionshttps://zbmath.org/1530.420062024-04-15T15:10:58.286558Z"Abreu, L. D."https://zbmath.org/authors/?q=ai:abreu.luis-daniel"Balazs, P."https://zbmath.org/authors/?q=ai:balazs.peter.2"Holighaus, N."https://zbmath.org/authors/?q=ai:holighaus.nicki"Luef, F."https://zbmath.org/authors/?q=ai:luef.franz"Speckbacher, M."https://zbmath.org/authors/?q=ai:speckbacher.michaelSummary: We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori \(\mathbb{T}_N^2 = \mathbb{R}^2 /(\mathbb{Z} \times N \mathbb{Z}) = [0, 1] \times [0, N]\) act as phase spaces. We work on an \(N\)-dimensional subspace \(S_N\) of distributions periodic in time and frequency in the dual \(S_0^\prime(\mathbb{R})\) of the Feichtinger algebra \(S_0(\mathbb{R})\) and equip it with an inner product. To construct the Hilbert space \(S_N\) we apply a suitable double periodization operator to \(S_0(\mathbb{R})\). On \(S_N\), the STFT is applied as the usual STFT defined on \(S_0^\prime(\mathbb{R})\). This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of \textit{Yu. I Lyubarskij} [Adv. Sov. Math. 11, 167--180 (1992; Zbl 0770.30025)] and \textit{K. Seip} and \textit{R. Wallstén} [J. Reine Angew. Math. 429, 107--113 (1992; Zbl 0745.46033)] for Gabor frames with Gaussian windows and which, for \(N\) odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.Modulus of smoothness and approximation theorems in Clifford analysishttps://zbmath.org/1530.420192024-04-15T15:10:58.286558Z"Tyr, Othman"https://zbmath.org/authors/?q=ai:tyr.othmanSummary: This paper uses some basic results on Clifford analysis introduced by \textit{E. Hitzer} [Clifford Anal. Clifford Algebr. Appl. 2, No. 3, 223--235 (2013; Zbl 1297.43006)], to study some problems in the theory of approximation of functions in the space of square integral functions in the Clifford algebra. The equivalence between the moduli of smoothness of all orders constructed by the Steklov function and the K-functionals constructed from the Sobolev-type space is proved. A consequence of this equivalence theorem is given at the end of this work.Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaceshttps://zbmath.org/1530.420262024-04-15T15:10:58.286558Z"Guliyev, V. S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabir"Omarova, Meriban N."https://zbmath.org/authors/?q=ai:omarova.meriban-n"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraLet \(T\) be a Calderón-Zygmund singular integral operator and \(b \in \mathrm{BMO}(\mathbb R^n)\). Then, it is well known that the commutator operator \([b,T]f=T(bf)-b T f\) is bounded on \(L^p({\mathbb R}^n)\) for \(1<p<\infty\), and also on Orlicz-Morrey spaces. The authors treat a generalization of \(M^{\Phi,\varphi}(\mathbb R^n)\), where \(\Phi\) is a Young function and \(\varphi(x,t)\) is a measurable function on \(\mathbb R^n\times(0,\infty)\) which is decreasing on \(t\) for each \(x\in\mathbb R^n\), and \(\Phi^{-1}(t^{-n})/\varphi(t)\) is almost decreasing. The generalized Orlicz-Morrey space \(M^{\Phi,\varphi}(\mathbb R^n)\) of the third kind is defined as the set of measurable functions \(f\) for which the norm \(\|f\|_{M^{\Phi,\varphi}}=\sup_{x\in\mathbb R^n, r>0}\varphi(x,r)^{-1} \Phi^{-1}(|B(x,r)|)\|f\|_{L^{\Phi}(B(x,r)}\) is finite. \par Under appropriate assumptions on \(\varphi_1\), \(\varphi_2\) and \(\Phi\), they give the boundedness of \([b,T]\) from \(M^{\Phi,\varphi_1}(\mathbb R^n)\) to \(M^{\Phi,\varphi_2}(\mathbb R^n)\). They define weak Orlicz-Morrey space \(WM^{\Phi,\varphi}(\mathbb R^n)\) and discuss the boundedness of \([b,T]\) from \(M^{\Phi,\varphi_1}(\mathbb R^n)\) to \(WM^{\Phi,\varphi_2}(\mathbb R^n)\). Several other related results are discussed.
Reviewer: Kôzô Yabuta (Nishinomiya)Estimates for bilinear generalized fractional integral operator and its commutator on generalized Morrey spaces over RD-spaceshttps://zbmath.org/1530.420272024-04-15T15:10:58.286558Z"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghui"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Wang, Miaomiao"https://zbmath.org/authors/?q=ai:wang.miaomiaoSummary: Let \((X,d,\mu)\) be an RD-space. In this paper, we prove that a bilinear generalized fractional integral \(\widetilde{T}_{\alpha}\) is bounded from the product of generalized Morrey spaces \(\mathcal{L}^{\varphi_1,p_1}(X)\times \mathcal{L}^{\varphi_2,p_2}(X)\) into spaces \(\mathcal{L}^{\varphi ,q}(X)\), and it is also bounded from the product of spaces \(\mathcal{L}^{\varphi_1,p_1}(X)\times \mathcal{L}^{\varphi_2,p_2}(X)\) into generalized weak Morrey spaces \(W\mathcal{L}^{\varphi,q}(X)\), where the Lebesgue measurable functions \(\varphi_1\), \(\varphi_2\) and \(\varphi\) satisfy certain conditions and \(\varphi_1\varphi_2=\varphi, \alpha \in (0,1)\) and \(\frac{1}{q}=\frac{1}{p_1}+\frac{1}{p_2}-2\alpha\) for \(1<p_1, p_2<\frac{1}{\alpha}\). Furthermore, we establish the boundedness of the commutator \(\widetilde{T}_{\alpha ,b_1,b_2}\) formed by \(b_1,b_2\in \mathrm{BMO}(X)\)(or \(\mathrm{Lip}_{\beta}(X)\)) and \(\widetilde{T}_{\alpha}\) on spaces \(\mathcal{L}^{\varphi,q}(X)\) and on spaces \(W\mathcal{L}^{\varphi,q}(X)\). As applications, we show that the \(\widetilde{T}_{\alpha}\) and its commutator \(\widetilde{T}_{\alpha ,b_1,b_2}\) are bounded on grand generalized Morrey spaces \(\mathcal{L}^{\theta,\varphi,p)}(X)\) over \((X,d,\mu)\).Calderón-Zygmund theory in Lorentz mixed-norm spaces and its application to compressible fluidshttps://zbmath.org/1530.420282024-04-15T15:10:58.286558Z"Wei, Wei"https://zbmath.org/authors/?q=ai:wei.wei.11"Wang, Yanqing"https://zbmath.org/authors/?q=ai:wang.yanqing"Ye, Yulin"https://zbmath.org/authors/?q=ai:ye.yulinSummary: In this paper, it is shown that the general Calderón-Zygmund singular integral operators including Riesz transforms are bounded on anisotropic Lorentz spaces \(L^{\vec{p}, \vec{q}}(\mathbb{R}^n)\) with \(1 < \vec{p} < \infty\) and \(1 < \vec{q} \leq \infty\). As an application, we establish some new blowup criteria involving temperature in its scaling-invariant anisotropic Lorentz space for the 3D full compressible Navier-Stokes equations allowing vacuum and without additional conditions on viscosity coefficients except physical restrictions, which improve the previous corresponding results.
{\copyright} 2023 Wiley-VCH GmbH.Endpoint Sobolev regularity of multilinear maximal operatorshttps://zbmath.org/1530.420322024-04-15T15:10:58.286558Z"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng"Zhang, Xiao"https://zbmath.org/authors/?q=ai:zhang.xiao.3"Zhang, Huiyun"https://zbmath.org/authors/?q=ai:zhang.huiyunSummary: In this paper, we establish some new endpoint regularity properties of multilinear maximal operators and their fractional variants, both in the centered and uncentered cases. The main results we obtain not only answer a question of \textit{F. Liu} et al. [Bull. Sci. Math. 179, Article ID 103155, 39 p. (2022; Zbl 1496.42024)], but also provide some new endpoint Sobolev boundedness and continuity results for multilinear fractional maximal operators. More importantly, we present a new approach to deal with the endpoint Sobolev continuity for the above operators.Greedy approximation algorithms for sparse collectionshttps://zbmath.org/1530.420332024-04-15T15:10:58.286558Z"Rey, Guillermo"https://zbmath.org/authors/?q=ai:rey.guillermoSummary: We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The constructive nature of the algorithm gives additional information about the almost disjoint structure of sparse collections.
As applications, we give three results for collections of axis-parallel rectangles in every dimension. The first is a constructive proof of the equivalence between Carleson and sparse collections, first shown by \textit{T. S. Hänninen} [Ark. Mat. 56, No. 2, 333--339 (2018; Zbl 1406.42028)]. The second is a structure theorem proving that every finite collection \(\mathcal{E}\) can be partitioned into \(\mathcal{O}(N)\) sparse subfamilies, where \(N\) is the Carleson constant of \(\mathcal{E}\). We also give examples showing that such a decomposition is impossible when the geometric assumptions are dropped. The third application is a characterization of the Carleson constant involving only \(L^{1,\infty}\) estimates.Littlewood-Paley characterization of BMO and Triebel-Lizorkin spaceshttps://zbmath.org/1530.420342024-04-15T15:10:58.286558Z"Tselishchev, Anton"https://zbmath.org/authors/?q=ai:tselishchev.anton"Vasilyev, Ioann"https://zbmath.org/authors/?q=ai:vasilyev.ioannThe authors prove a generalization of the Littlewood-Paley characterisation of the BMO space by weakening the smoothness of the Schwartz functions significantly. They also prove similar results for a certain family of Triebel-Lizorkin spaces.
Reviewer: Ferenc Weisz (Budapest)A weighted weak-type inequality for one-sided maximal operatorshttps://zbmath.org/1530.420352024-04-15T15:10:58.286558Z"Wang, J."https://zbmath.org/authors/?q=ai:wang.j.25"Ren, Y."https://zbmath.org/authors/?q=ai:ren.yanbo"Zhang, E."https://zbmath.org/authors/?q=ai:zhang.erxinSummary: We obtain necessary and sufficient conditions for a weighted weak-type inequality of the form
\[\underset{\left\{{M}_g^+\left(f\right)>\lambda \right\}}{\int }\widetilde{\varphi }\left(\frac{\lambda }{{\omega }_3\left(x\right){\omega }_4\left(x\right)}\right){\omega }_4\left(x\right)dx\le{C}_1\underset{-\infty }{\overset{+\infty }{\int }}\widetilde{\varphi }\left(\frac{\left|f\left(x\right)\right|}{{\omega }_1\left(x\right){\omega }_2\left(x\right)}\right){\omega }_2\left(x\right)dx\]
to be true, which generalize some known results.Spherical maximal function on local Morrey spaces with variable exponentshttps://zbmath.org/1530.420362024-04-15T15:10:58.286558Z"Yee, Tat-Leung"https://zbmath.org/authors/?q=ai:yee.tat-leung"Cheung, Ka Luen"https://zbmath.org/authors/?q=ai:cheung.ka-luen"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-pun"Suen, Chun Kit"https://zbmath.org/authors/?q=ai:suen.chun-kitSummary: We establish the boundedness of the spherical maximal function on local Morrey spaces with variable exponents.Inequalities for weighted spaces with variable exponentshttps://zbmath.org/1530.420382024-04-15T15:10:58.286558Z"Rocha, Pablo"https://zbmath.org/authors/?q=ai:rocha.pablo-alejandroSummary: In this article we obtain an ``off-diagonal'' version of the Fefferman-Stein vectorvalued maximal inequality on weighted Lebesgue spaces with variable exponents. As an application of this result and the atomic decomposition developed in [\textit{K.-P. Ho}, Tôhoku Math. J. (2) 69, No. 3, 383--413 (2017; Zbl 1378.42012)] we prove, for certain exponents \(q(\cdot)\) in \(\mathcal{P}^{\log} (\mathbb{R}^n)\) and certain weights \(\omega\), that the Riesz potential \(I_\alpha\), with \(0<\alpha <n\), can be extended to a bounded operator from \(H^{p(\cdot)}_\omega (\mathbb{R}^n)\) into \(L^{q(\cdot)}_\omega (\mathbb{R}^n)\), for \(\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}\).Operators on Herz-Morrey spaces with variable exponentshttps://zbmath.org/1530.420412024-04-15T15:10:58.286558Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punSummary: This paper studies the nonlinear operators on Herz-Morrey spaces with variable exponents. We obtain our results by extending the extrapolation theory of Rubio de Francia to Herz-Morrey spaces with variable exponents. As applications of our main results, we obtain the boundedness of the spherical maximal functions, the nonlinear commutators of Rochberg and Weiss and the geometrical maximal operators on the Herz-Morrey spaces with variable exponents.Weighted local Hardy spaces with variable exponentshttps://zbmath.org/1530.420422024-04-15T15:10:58.286558Z"Izuki, Mitsuo"https://zbmath.org/authors/?q=ai:izuki.mitsuo"Nogayama, Toru"https://zbmath.org/authors/?q=ai:nogayama.toru"Noi, Takahiro"https://zbmath.org/authors/?q=ai:noi.takahiro"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiroSummary: This paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood-Paley characterization, and wavelet decomposition.
{\copyright} 2023 Wiley-VCH GmbH.Local grand variable exponent Lebesgue spaceshttps://zbmath.org/1530.420442024-04-15T15:10:58.286558Z"Rafeiro, Humberto"https://zbmath.org/authors/?q=ai:rafeiro.humberto"Samko, Stefan"https://zbmath.org/authors/?q=ai:samko.stefan-gLet \(\Omega \subset \mathbb{R}^n\) be an open set, \(F \subset \bar{\Omega}\) be a closed non-empty set with measure zero, \(p\) a measurable exponent with \(1 \leqslant p_{-}<p_{+}<\infty\) and \(\theta>0\). Assume that \(a\) is a function in \(G\left(\mathbb{R}_{+}\right)\), the set of non-negative functions \(a\) in \(L^{\infty}\left(\mathbb{R}_{+}\right)\) such that
\begin{itemize}
\item [(i)] \(a\) is continuous in a neighborhood of the origin with \(a(0+)=0\);
\item [(ii)] \(\inf _{t \in(\kappa, \infty)} a(t)>0\) for every \(\kappa \in \mathbb{R}_{+}\).
\end{itemize}
Define the local grand variable exponent Lebesgue space \(L_{F, a}^{p(\cdot)), \theta}(\Omega)\) by
\[
\begin{aligned}
L_{F, a, \ell}^{p(\cdot)), \theta}(\Omega) & =L_{F, a}^{p(\cdot)), \theta}(\Omega) \\
& =\left\{f \in L^0(\Omega) \mid \sup _{0<\varepsilon<\ell} \varepsilon^\theta\left\|f(\cdot)\left[a\left(\delta_F(\cdot)\right)\right]^{\varepsilon}\right\|_{L^{p(\cdot)}(\Omega)}<\infty\right\},
\end{aligned}
\]
where \(\delta_F(x):=\inf _{y \in F}|x-y|\). In the paper under review, the authors first show that for any quasi-monotone functions \(a\), \(b\) on \((0, \kappa)\) for some \(\kappa \in(0, \infty)\), if their Matuszewska-Orlicz indices satisfy that \(0<m(a) \leqslant M(a)<\infty\) and \(0<m(b) \leqslant M(b)<\infty\), then
\[
L_{F, a}^{p(\cdot)), \theta}(\Omega)=L_{F, b}^{p(\cdot)), \theta}(\Omega),
\]
with equivalence of norms. Moreover, the authors also show that the centered Hardy-Littlewood maximal operator, singular operators, and maximal singular operators are all bounded on such spaces.
Reviewer: Dongyong Yang (Xiamen)Equivalence of subcritical and critical Adams inequalities in \(W^{m,2}(\mathbb{R}^{2m})\) and existence and non-existence of extremals for Adams inequalities under inhomogeneous constraintshttps://zbmath.org/1530.420462024-04-15T15:10:58.286558Z"Zhang, Caifeng"https://zbmath.org/authors/?q=ai:zhang.caifeng"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.luSummary: Though there have been extensive works on the existence of maximizers for sharp Trudinger-Moser inequalities under homogeneous and inhomogeneous constraints, and sharp Adams inequalities under homogeneous constraints, much less is known for that of the maximizers for Adams inequalities under inhomogeneous constraints. Furthermore, whether exists equivalence between subcritical and critical Adams inequalities in \(W^{m,2}(\mathbb{R}^{2m})\) also remains open. In this paper, we shall give partial answers to these problems. We first establish the equivalence of subcritical Adams inequalities and critical Adams inequalities under inhomogeneous constraints through exploiting the scaling invariance of Adams inequalities in \(W^{m,2}(\mathbb{R}^{2m})\) (see Theorem 1.1). Then we consider the existence and non-existence of extremals for sharp Adams inequalities under inhomogeneous constraints in Theorem 1.2, 1.3 and 1.4. Our methods are based on Fourier rearrangement inequality and careful analysis for vanishing phenomenon of radially maximizing sequence for Adams inequalities in \(W^{m,2}(\mathbb{R}^{2m})\).On the democracy inequality for Haar systems and some geometric and analytical properties in Herz spaces on dyadic settingshttps://zbmath.org/1530.420532024-04-15T15:10:58.286558Z"Fernández, Daniela"https://zbmath.org/authors/?q=ai:fernandez.daniela"Nowak, Luis"https://zbmath.org/authors/?q=ai:nowak.luis"Perini, Alejandra"https://zbmath.org/authors/?q=ai:perini.alejandraSummary: In this paper we explore geometric condition to obtain that the Democracy inequality
of Haar systems on Herz spaces implies that these spaces are Lebesgue spaces in the setting of
spaces of homogeneous type. For this purpose, we give previously a construction of dyadic Herz
spaces and prove some analytic properties.Dilational symmetries of decomposition and coorbit spaceshttps://zbmath.org/1530.420562024-04-15T15:10:58.286558Z"Führ, Hartmut"https://zbmath.org/authors/?q=ai:fuhr.hartmut"Raisi-Tousi, Reihaneh"https://zbmath.org/authors/?q=ai:tousi.reihaneh-raisiSummary: We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings. We then apply the general results to a particular class of dilation groups, the so-called shearlet dilation groups. We present a general, algebraic characterization of matrices that are coorbit compatible with a given shearlet dilation group. We explicitly determine the groups of compatible dilations, for a variety of concrete examples.Wiener amalgam spaces with respect to Orlicz spaces on the affine grouphttps://zbmath.org/1530.430012024-04-15T15:10:58.286558Z"Arıs, Büşra"https://zbmath.org/authors/?q=ai:aris.busra"Öztop, Serap"https://zbmath.org/authors/?q=ai:oztop.serapSummary: Let \(\mathbb{A}\) be the affine group, \(\Phi\) be a Young function and \(L^{\Phi}(\mathbb{A})\) be the corresponding Orlicz space. We study the Orlicz amalgam spaces \(W(L^{\Phi} (\mathbb{A}), L^1 (\mathbb{A}))\) and \(W(L^{\infty} (\mathbb{A}), L^{\Phi} (\mathbb{A}))\), where the local components are \(L^{\Phi}(\mathbb{A}), L^{\infty} (\mathbb{A})\) and the global components are \(L^1(\mathbb{A})\), \(L^{\Phi} (\mathbb{A})\), respectively. We obtain an equivalent discrete-type norm on the spaces \(W(L^{\Phi} (\mathbb{A}), L^1 (\mathbb{A}))\) and \(W(L^{\infty} (\mathbb{A}), L^{\Phi} (\mathbb{A}))\). This allows us to prove new convolution relations.Correction to: ``On harmonic Hilbert spaces on compact abelian groups''https://zbmath.org/1530.430022024-04-15T15:10:58.286558Z"Das, Suddhasattwa"https://zbmath.org/authors/?q=ai:das.suddhasattwa"Giannakis, Dimitrios"https://zbmath.org/authors/?q=ai:giannakis.dimitrios"Montgomery, Michael R."https://zbmath.org/authors/?q=ai:montgomery.michael-rFrom the text: We correct an error in Theorem 6 of the first and second author's paper [ibid. 29, No. 1, Paper No. 12, 26 p. (2023; Zbl 1515.43002)]. That theorem claimed that given a suitable (e.g., absolutely summable, symmetric, and subconvolutive) function \(\lambda : \hat{G}\to\mathbb{C}\) on the dual group \(\hat{G}\), there is an associated harmonic Hilbert space \(\mathcal{H}_\lambda\) of complex-valued functions on \(G\), which is a Banach \(^\ast\)-algebra
with respect to pointwise function multiplication and complex conjugation, and the Gelfand spectrum \(\sigma (\mathcal{H}_\lambda)\) is homeomorphic to \(G\). However, the stated assumptions on \(\lambda\) in that theorem are actually not sufficient to deduce that \(G\cong \sigma (\mathcal{H}_\lambda)\). Here, we show that [loc. cit., Theorem 6] remains valid if and only if \(\lambda\) satisfies the Gelfand-Raikov-Shilov condition. Aside from this modification on the assumptions of Theorem 6, the results of [loc. cit.] remain unchanged.Conductive homogeneity of compact metric spaces and construction of \(p\)-energyhttps://zbmath.org/1530.460012024-04-15T15:10:58.286558Z"Kigami, Jun"https://zbmath.org/authors/?q=ai:kigami.junStarting in the second half of the 1990s, the concept of Sobolev spaces on metric spaces has emerged. The book [\textit{J. Heinonen} et al., Sobolev spaces on metric measure spaces. An approach based on upper gradients. Cambridge: Cambridge University Press (2015; Zbl 1332.46001)] provides a panoramic view on the topic. The Sobolev spaces described therein mainly consist of a function and an associated function being in some sense a generalization of the weak derivative. Having the concept of a derivative asks for the introduction of \(p\)-energies akin the situation in the Euclidean setting.
Barlow and Bass (e.g., [\textit{M.~T. Barlow} and \textit{R.~F. Bass}, Ann. Inst. Henri Poincaré, Probab. Stat. 25, No.~3, 225--257 (1989; Zbl 0691.60070)]) have studied Brownian motions in the metric setting. The inconvenience is that there are some metric spaces that do not satisfy the Gaussian heat kernel estimate.
The book under review offers a different approach to Sobolev spaces on metric spaces as a remedy to this deficiency by defining Sobolev spaces in such a way that the estimate becomes available in a larger class of metric spaces. The starting point is the following fact mentioned at the very beginning of the book. Namely, if \(I=[0,1]\) and
\[
\mathcal{E}_p^n(f)=\sum_{i=1}^{2^n}\left|f\left(\frac{i-1}{2^n}\right)-f\left(\frac{i}{2^n}\right)\right|^p \tag{1}
\]
for \(n\geq 1\) and \(f\in W^{1,p}(I)\), then, denoting by \(\nabla f\) the weak derivative of \(f\), it follows that
\[
\lim_{n \to \infty}(2^{p-1})^n\mathcal{E}_p^n(f)\to \int_{0}^{1}|\nabla f|^p\, dx. \tag{2}
\]
By looking only at the left-hand side, we obtain a way to talk about an energy without having to refer to the derivative of \(f\). The idea is to understand the points \(\frac{i}{2^n}\) as nodes. More precisely, the author is approximating metric spaces by graphs and then evaluating the function (or rather an approximation) at the nodes. The process to obtain the graphs is by using finer and finer partitions of the metric space into compact sets. Given one such approximation, two of the nodes are connected by an edge if and only if the corresponding sets meet. This leads to a version of (1) for metric spaces. Denoting by \(P_n f\) an approximation of \(f\) urges us to find a proper scaling constant \(\sigma\) such that \(\sigma^n \mathcal{E}_p^n(P_n f)\) converges. This then gives rise to a Sobolev space.
After the introduction, the author carefully introduces the partitions and their descriptions via trees. Furthermore, we get acquainted with the standard assumptions. One of them is more restrictive but contains all others.
A large part of the book deals with generalizing the energy and finding a proper constant \(\sigma\) to obtain an analogue of (2).
The author gives a definition for the energy and its corresponding Sobolev space. The obtained results depend on the parameter \(p\). The situation is much better understood for large enough \(p\). However, the author also shares what is known for small \(p\). In particular, the author proves an existence theorem concerning the heat kernel.
Now that the foundations are laid and the theory is developed, in the later part of the book, the author details many examples for the reader to get a feeling for the introduced concepts.
The book ends with a nice touch: a section concerning open problems and an appendix containing many useful facts and results.
Reviewer: Thomas Zürcher (Katowice)Phase-isometries on the unit sphere of CL-spaceshttps://zbmath.org/1530.460102024-04-15T15:10:58.286558Z"Tan, Dongni"https://zbmath.org/authors/?q=ai:tan.dongni"Zhang, Fan"https://zbmath.org/authors/?q=ai:zhang.fan.12|zhang.fan.1|zhang.fan"Huang, Xujian"https://zbmath.org/authors/?q=ai:huang.xujianSummary: A mapping \(f:S_X\to S_Y\) between the unit spheres of two real Banach spaces \(X\) and \(Y\) is said to be a phase-isometry if it satisfies \(\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}\) for all \(x,y\in S_X\). When \(f\) is surjective, \(X\) is a real CL-space and \(Y\) is an arbitrary real Banach space, we establish in this paper that there exists a phase-function \(\varepsilon:S_X\to\{-1,1\}\) such that \(\varepsilon\cdot f\) is an isometry which is the restriction of a linear isometry from \(X\) to \(Y\).How the distance between subspaces in the metric of a spherical opening affects the geometric structure of a symmetric spacehttps://zbmath.org/1530.460142024-04-15T15:10:58.286558Z"Strakhov, Stepan Igorevich"https://zbmath.org/authors/?q=ai:strakhov.stepan-igorevichSummary: A relationship is found between the metric of a spherical opening on the space of all subspaces of a symmetric space and some numerical characteristic of the subspace. It is known that, for example, in \(L_1\) this characteristic takes only two values (i.e. this is a binary space), while in \(L_2\) there are infinitely many values. Using the connection found, the necessary conditions for the binarity of a symmetric space were generalized.Spaceablity in weak Morrey spaceshttps://zbmath.org/1530.460202024-04-15T15:10:58.286558Z"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiro"Tabatabaie, Seyyed Mohammad"https://zbmath.org/authors/?q=ai:tabatabaie.seyyed-mohammadSummary: In this paper we prove that \(\mathrm{w}\mathcal{M}^p_q(\mathbb{R}^n)\setminus\mathcal{M}^p_q(\mathbb{R}^n)\) is spaceable in the weak Morrey space \(\mathrm{w}\mathcal{M}^p_q(\mathbb{R}^n)\), where \(1 \leq q \leq p< \infty\).Multipliers on spaces of holomorphic functionshttps://zbmath.org/1530.460212024-04-15T15:10:58.286558Z"Trybuła, Maria"https://zbmath.org/authors/?q=ai:trybula.mariaSummary: We consider multipliers on the space of holomorphic functions of one variable \(H(\Omega)\), \(\Omega\subset\mathbb{C}\) open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to \(\Omega\) and \(\Omega\) is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on \(H(\Omega)\). We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when \(\Omega\) is convex.Unconditional basis constructed from parameterised Szegö kernels in analytic \(\mathbb{H}^p(D)\)https://zbmath.org/1530.460222024-04-15T15:10:58.286558Z"Hon, Chitin"https://zbmath.org/authors/?q=ai:hon.chitin"Leong, Ieng Tak"https://zbmath.org/authors/?q=ai:leong.iengtak|leong.ieng-tak"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.tao"Yang, Haibo"https://zbmath.org/authors/?q=ai:yang.haibo"Zou, Bin"https://zbmath.org/authors/?q=ai:zou.binSummary: Rational orthogonal systems in approximating analytic functions have attracted considerable interest. Among which adaptive Fourier decomposition, abbreviated as AFD, was recently established. An AFD is a sparse representation using a Takenaka-Malmquist (TM) system whose parameters are optimally selected according to the given signal. TM systems have been proved to be Schauder systems in the corresponding Banach spaces \(\mathbb{H}^p\), \(1 < p < \infty\). In the present paper, from the methodology point of view we give an alternative definition of the Hardy spaces by using the periodic Lusin area function. We extend the Botchkariev-Meyer-Wojtaszcyk Theorem to rational function systems. By using Meyer's bimodal wavelet and the Fefferman-Stein vector valued maximum operator we prove that under certain conditions the rational systems become unconditional bases in the Banach space \(\mathbb{H}^p(D)\), \(1 < p < \infty\).Nonlinear disjointness/supplement preservers of nonnegative continuous functionshttps://zbmath.org/1530.460232024-04-15T15:10:58.286558Z"Li, Lei"https://zbmath.org/authors/?q=ai:li.lei.5"Liao, Ching-Jou"https://zbmath.org/authors/?q=ai:liao.chingjou"Shi, Luoyi"https://zbmath.org/authors/?q=ai:shi.luoyi"Wang, Liguang"https://zbmath.org/authors/?q=ai:wang.liguang"Wong, Ngai-Ching"https://zbmath.org/authors/?q=ai:wong.ngai-chingSummary: Let \(F(X), F(Y)\) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces \(X, Y\), respectively. Let \(\Phi : F(X) \to F(Y)\) be a surjective map satisfying that
\[
f \vee g > \mathbf{0} \quad \Longleftrightarrow \quad \Phi (f) \vee \Phi (g) > \mathbf{0}, \quad \forall f, g \in F (X).
\]
In many cases, we show that there is a homeomorphism \(\tau : Y \to X\) such that
\[
\Phi(f)(y) \neq 0 \quad \Longleftrightarrow \quad f(\tau(y)) \neq 0, \quad \forall f \in F (X), \forall y \in Y.
\]
Assume \(X, Y\) are locally compact Hausdorff (resp. separable and metrizable) and \(\Phi : C_0 (X)_+ \to C_0 (Y)_+\) (resp. \(\Phi : C^b (X)_+ \to C^b (Y)_+\)) is a surjective map. We show that \(\Phi\) preserves the norms of infima, i.e.,
\[
\| \Phi (f) \wedge \Phi (g) \| = \| f \wedge g \|, \quad \forall f, g \in C_0 (X)_+ \text{ (resp. } C^b (X)_+),
\]
if and only if there is a homeomorphism \(\tau : Y \to X\) such that
\[
\Phi(f)(y) = f(\tau(y)), \quad \forall f \in C_0 (X)_+ \text{ (resp. } C^b (X)_+), \forall y \in Y.
\]\(\theta\)-type Calderón-Zygmund operator and its commutator on (grand) generalized weighted variable exponent Morrey space over RD-spaceshttps://zbmath.org/1530.460242024-04-15T15:10:58.286558Z"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghui"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangping"Liu, Ronghui"https://zbmath.org/authors/?q=ai:liu.ronghuiSummary: Let \((\mathcal{X}, d, \mu)\) be a RD-space satisfying both the doubling and reverse doubling conditions. In this setting, the authors first obtain the definitions of a generalized weighted variable exponent Morrey space \(\mathcal{L}^{p (\cdot), \varphi}_{\omega}(\mathcal{X})\) and a grand generalized weighted variable exponent Morrey space \(\mathcal{L}^{p(\cdot), \varphi, \alpha}_{\omega} (\mathcal{X})\) on RD-spaces. Second, under assumption that \(\varphi\) satisfies some certain condition, the authors prove that the \(T_{\theta}\) and its commutator generated by \(b \in \mathrm{BMO}(\mathcal{X})\) and the \(T_{\theta}\) are bounded on spaces \(\mathcal{L}^{p(\cdot), \varphi}_{\omega}(\mathcal{X})\). Finally, via some known results, the boundedness of operators \(T_{\theta}\) and \([b, T_{\theta}]\) on spaces \(\mathcal{L}^{p(\cdot), \varphi, \alpha}_{\omega} (\mathcal{X})\) is also established.Fractional variable exponents Sobolev trace spaces and Dirichlet problem for the regional fractional \(p(.)\)-Laplacianhttps://zbmath.org/1530.460252024-04-15T15:10:58.286558Z"Berghout, Mohamed"https://zbmath.org/authors/?q=ai:berghout.mohamedThe author considers fractional variable exponent Sobolev spaces. Variable exponent spaces were introduced by \textit{W.~Orlicz} [Stud. Math. 3, 200--211 (1931; JFM 57.0251.02)] and their properties were further developed by Nakano as special cases of the theory of modular spaces. The main results of the paper are Theorems~1.1, 1.3 and 1.4, where the author applies the relative capacity to characterize completely the zero trace fractional variable exponents Sobolev spaces and presents a relative capacity criterion for removable sets. In fact, the author proves relations between traces in various senses and capacities, i.e., he uses the fractional relative \((s, q(\cdot), p(\cdot, \cdot))\)-capacity to characterize completely the space \(H^{s,q(\cdot),p(\cdot,\cdot)}_0 (\Omega)\). He presents a necessary and sufficient condition for the equality \(H^{s,q(\cdot),p(\cdot,\cdot)}_0 (\Omega) =\tilde{W}^{s,q(\cdot),p(\cdot,\cdot)}_0 (\Omega)\).
By the above argument one can study the existence of weak solutions for the Dirichlet problem for the regional fractional \(p(\cdot)\)-Laplacian.
Reviewer: Abdolrahman Razani (Qazvin)Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaceshttps://zbmath.org/1530.460262024-04-15T15:10:58.286558Z"Dao, Nguyen Anh"https://zbmath.org/authors/?q=ai:dao.nguyen-anhSummary: Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.Bi-Lipschitz invariance of planar \(BV\)- and \(W^{1,1}\)-extension domainshttps://zbmath.org/1530.460272024-04-15T15:10:58.286558Z"García-Bravo, Miguel"https://zbmath.org/authors/?q=ai:garcia-bravo.miguel"Rajala, Tapio"https://zbmath.org/authors/?q=ai:rajala.tapio"Zhu, Zheng"https://zbmath.org/authors/?q=ai:zhu.zhengThe authors consider the problem of extending Sobolev functions defined on a particular domain \(\Omega \subset \mathbb{R}^{d}\) to Sobolev functions (of the same regularity) defined on the entire space \(\mathbb{R}^{d}\). A domain \(\Omega \subset \mathbb{R}^{d}\) for which such an extension is possible is called an \textit{extension domain}. More precisely, given a parameter \(1\leq p\leq \infty \), \(\Omega \subset \mathbb{ R}^{d}\) is a \(W^{1,p}\)-extension domain if, for any \(u\in W^{1,p}\left( \Omega \right) \), there exists \(\tilde{u}\in W^{1,p}\left( \mathbb{R} ^{d}\right) \) such that \(\tilde{u}=u\) on \(\Omega \) and
\[
\left\Vert \tilde{u}\right\Vert _{W^{1,p}(\mathbb{R}^{d})}\leq C\left\Vert u\right\Vert _{W^{1,p}(\Omega )},
\]
where \(C>0\) is a constant depending only on \(\Omega \). Some classical examples of \(W^{1,p}\)-extension domains (for any \(1\leq p\leq \infty \)) are the Lipschitz domains (as it was proved by \textit{A. P. Calderón} [Proc. Sympos. Pure Math. 4, 33--49 (1961; Zbl 0195.41103)] and \textit{E. M. Stein} [Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press (1970; Zbl 0207.13501)]) or more generally the \((\varepsilon ,\delta )\)-domains introduced by \textit{P. W. Jones} [Acta Math. 147, 71--88 (1981; Zbl 0489.30017)].
The authors are motivated by the following result of \textit{P.~Hajłasz} et al. [Rev. Mat. Iberoam. 24, No.~2, 645--669 (2008; Zbl 1226.46029)]:
{Theorem 1.1.} If \(\Omega\) and \(\Omega ^{\prime }\) are bi-Lipschitz equivalent, for \(1<p\leq \infty \), then \(\Omega\) is a \(W^{1,p}\)-extension domain if and only if \(\Omega'\) is a \(W^{1,p}\)-extension domain.
It is natural to ask if this statement remains true in the case \(p=1\). Under some additional hypotheses on the domain \(\Omega\), \textit{P.~Koskela} et al. [in: Around the research of Vladimir Maz'ya. I. Function spaces. Dordrecht: Springer; Novosibirsk: Tamara Rozhkovskaya Publisher. 255--272 (2010; Zbl 1196.46025)] proved that Theorem~1.1 can be extended to cover the situation when \(p=1\). The authors of the present paper are able to remove those additional hypotheses on \(\Omega\). They prove the following:
{Theorem 1.3 (or Corollary 1.4).} Let \(\Omega\subset \mathbb{R}^{2}\) be a bounded \(BV\) (or \(W^{1,1}\))-extension domain and \(f:\Omega\rightarrow \Omega^{\prime }\) a bi-Lipschitz map. Then \(\Omega^{\prime }\) is also a \(BV\) (or \(W^{1,1}\))-extension domain.
The proof relies on the results of \textit{J.~Väisälä} [Conform. Geom. Dyn. 12, 58--66 (2008; Zbl 1192.30005)] and on quasiconvexity arguments.
Reviewer: Eduard Curca (Lyon)Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operatorhttps://zbmath.org/1530.460282024-04-15T15:10:58.286558Z"Gonçalves, Helena F."https://zbmath.org/authors/?q=ai:goncalves.helena-f"Haroske, Dorothee D."https://zbmath.org/authors/?q=ai:haroske.dorothee-d"Skrzypczak, Leszek"https://zbmath.org/authors/?q=ai:skrzypczak.leszekThe authors first construct Rychkov's linear, bounded universal extension operator for Besov-type and Triebel-Lizorkin-type spaces on bounded Lipschitz domains. Then by the extension operator and the wavelet decomposition in Besov-type and Triebel-Lizorkin-type spaces, necessary and sufficient conditions for the continuity of limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on bounded Lipschitz domains are given.
Reviewer: Jingshi Xu (Guilin)Embedding theorems for functional spaces associated with a class of Hermitian formshttps://zbmath.org/1530.460292024-04-15T15:10:58.286558Z"Peicheva, Anastasiya S."https://zbmath.org/authors/?q=ai:peicheva.anastasiya-sSummary: We prove embedding theorems into the scale of Sobolev-Slobodetskii spaces for functional spaces associated with a class of Hermitian forms. More precisely we consider the Hermitian forms constructed with the use of the first order differential matrix operators with injective principal symbol. The results are valid for both coercive and non-coercive forms.Correction to: ``Commutator estimates for vector fields on variable Triebel-Lizorkin spaces''https://zbmath.org/1530.460302024-04-15T15:10:58.286558Z"Salah, Ben Mahmoud"https://zbmath.org/authors/?q=ai:salah.ben-mahmoud"Douadi, Drihem"https://zbmath.org/authors/?q=ai:drihem.douadiFrom the text: In the original publication [ibid. 72, No. 1, 21--36 (2023; Zbl 1517.46026)] second author first name and last name has been interchanged.
This has been updated in the original publication.The superposition operator and Strauss lemma in logarithmic Besov spaceshttps://zbmath.org/1530.460312024-04-15T15:10:58.286558Z"Wu, Suqing"https://zbmath.org/authors/?q=ai:wu.suqingSummary: In this article, the author studies the superposition operator and Strauss lemma in Besov spaces with logarithmic smoothness. As consequences, the author generalizes the corresponding classical results, especially in critical case.Hölder continuity of the traces of Sobolev functions to hypersurfaces in Carnot groups and the \(\mathcal{P} \)-differentiability of Sobolev mappingshttps://zbmath.org/1530.460322024-04-15T15:10:58.286558Z"Basalaev, S. G."https://zbmath.org/authors/?q=ai:basalaev.sergey-g"Vodopyanov, S. K."https://zbmath.org/authors/?q=ai:vodopyanov.serguei-kSummary: We study the behavior of Sobolev functions and mappings on the Carnot groups with the left invariant sub-Riemannian metric. We obtain some sufficient conditions for a Sobolev function to be locally Hölder continuous (in the Carnot-Carathéodory metric) on almost every hypersurface of a given foliation. As an application of these results we show that a quasimonotone contact mapping of class \(W^{1,\nu}\) of Carnot groups is continuous, \( \mathcal{P} \)-differentiable almost everywhere, and has the \(\mathcal{N} \)-Luzin property.New atomic decomposition for Besov type space \(\dot{B}^0_{1, 1}\) associated with Schrödinger type operatorshttps://zbmath.org/1530.460332024-04-15T15:10:58.286558Z"Bui, The Anh"https://zbmath.org/authors/?q=ai:the-anh-bui."Duong, Xuan Thinh"https://zbmath.org/authors/?q=ai:duong.xuan-thinhSummary: Let \((X, d, \mu )\) be a space of homogeneous type. Let \(L\) be a nonnegative self-adjoint operator on \(L^2(X)\) satisfying certain conditions on the heat kernel estimates which are motivated from the heat kernel of the Schrödinger operator on \(\mathbb{R}^n\). The main aim of this paper is to prove a new atomic decomposition for the Besov space \(\dot{B}^{0, L}_{1, 1}(X)\) associated with the operator \(L\). As a consequence, we prove the boundedness of the Riesz transform associated with \(L\) on the Besov space \(\dot{B}^{0, L}_{1, 1}(X)\).Metric space mappings connected with Sobolev-type function classeshttps://zbmath.org/1530.460342024-04-15T15:10:58.286558Z"Romanov, A. S."https://zbmath.org/authors/?q=ai:romanov.alexandr-sergeevich|romanov.aleksandr-sergeevichSummary: We study some properties of the metric space mappings connected with the Sobolev-type function classes \(M^1_p(X,d,\mu) \).The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson's inequalitieshttps://zbmath.org/1530.460352024-04-15T15:10:58.286558Z"Sodini, Giacomo Enrico"https://zbmath.org/authors/?q=ai:sodini.giacomo-enricoSummary: We show that the algebra of cylinder functions in the Wasserstein Sobolev space \(H^{1, q}(\mathcal{P}_p(X,\mathsf{d}), W_{p, \mathsf{d}}, \mathfrak{m})\) generated by a finite and positive Borel measure \(\mathfrak{m}\) on the \((p, \mathsf{d})\)-Wasserstein space \((\mathcal{P}_p(X, \mathsf{d}), W_{p, \mathsf{d}})\) on a complete and separable metric space \((X, \mathsf{d})\) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space \(\mathbb{B}\), then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if \(\mathbb{B}\) is reflexive (resp. if the dual of \(\mathbb{B}\) is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space.On generalized definitions of ultradifferentiable classeshttps://zbmath.org/1530.460362024-04-15T15:10:58.286558Z"Jiménez-Garrido, Javier"https://zbmath.org/authors/?q=ai:jimenez-garrido.javier"Nenning, David Nicolas"https://zbmath.org/authors/?q=ai:nenning.david-nicolas"Schindl, Gerhard"https://zbmath.org/authors/?q=ai:schindl.gerhardAn ultradifferentiable class is a class of functions that are more regular than \(C^\infty\). They come up in the investigation of differential or convolution operators.
As the references of the present paper show, there is a long history of definitions of ultradifferentiable classes.
One such generalization has been given by \textit{S.~Pilipović} et al. [Novi Sad J. Math. 45, No.~1, 125--142 (2015; Zbl 1460.46028)]. Their approach leads to ultradifferentiable classes that are larger than any Gevrey class.
In the present paper it is shown that these classes of ultradifferentiable functions are a subset of the classes introduced in [\textit{A.~Rainer} and \textit{G.~Schindl}, Stud. Math. 224, No.~2, 97--131 (2014; Zbl 1318.26053)].
Reviewer: Rüdiger W. Braun (Düsseldorf)Holomorphic Hörmander-type functional calculus on sectors and stripshttps://zbmath.org/1530.470192024-04-15T15:10:58.286558Z"Haase, Markus"https://zbmath.org/authors/?q=ai:haase.markus"Pannasch, Florian"https://zbmath.org/authors/?q=ai:pannasch.florianSummary: In this paper, the abstract multiplier theorems for 0-sectorial and 0-strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405-444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and ``holomorphic Schwartz'' functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937-974] to obtain an improvement -- with respect to the smoothness condition -- of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.Supercyclicity and resolvent condition for weighted composition operatorshttps://zbmath.org/1530.470322024-04-15T15:10:58.286558Z"Mengestie, Tesfa"https://zbmath.org/authors/?q=ai:mengestie.tesfa-y"Seyoum, Werkaferahu"https://zbmath.org/authors/?q=ai:seyoum.werkaferahuLet \(u\) and \(\psi\) be entire functions. This article presents a few results about weighted composition operators \(W_{u,\psi} f := u (f \circ \psi)\) acting on the Fock space \(\mathcal{F}_p, \ 1 \leq p < \infty\). It is proved that no weighted composition operator on Fock spaces is supercyclic. Conditions under which the operators satisfy Ritt's resolvent growth condition are also identified. In particular, it is shown that a non-trivial composition operator \(C_{\psi}\) on a Fock space satisfies such a growth condition if and only if it is compact.
Reviewer: José Bonet (València)Toeplitz operators on the polyharmonic Bergman spacehttps://zbmath.org/1530.470382024-04-15T15:10:58.286558Z"Zhang, Bo"https://zbmath.org/authors/?q=ai:zhang.bo.7"Yang, Yixin"https://zbmath.org/authors/?q=ai:yang.yixin"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufengThe purpose of the article is to get the reproducing kernel for the polyharmonic Bergman space and the study properties of the Toeplitz operator on this latter space.
Reviewer: Mohammed El Aïdi (Bogotá)On Poisson semigroup hypercontractivity for higher-dimensional sphereshttps://zbmath.org/1530.470572024-04-15T15:10:58.286558Z"Huang, Yi. C."https://zbmath.org/authors/?q=ai:huang.yi-cThe author considers the Poisson semigroup \(R(t)=e^{-t\sqrt{-\Delta-(n-1)P}}\) on the \(n\)-sphere. Here, \(\Delta\) is the Laplace-Beltrami operator on the \(n\)-sphere \(S_n\) and \(P\) is the projection operator onto spherical harmonics of degree \(\geq 1\). It is proven that
\[
\|R(t)f\|_{L_p(S_n)}\leq \|f\|_{L_q(S_n)} \text{ for all }f \ \Leftrightarrow \ \ e^{-t}< \sqrt{\frac{p-1}{q-1}}.
\]
The last property is called hypercontractivity. Here, \(1<p\leq q<\infty\) and \(n\geq 1\).
Reviewer: Sergey G. Pyatkov (Khanty-Mansiysk)The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functionshttps://zbmath.org/1530.470602024-04-15T15:10:58.286558Z"Carlone, Raffaele"https://zbmath.org/authors/?q=ai:carlone.raffaele"Fiorenza, Alberto"https://zbmath.org/authors/?q=ai:fiorenza.alberto"Tentarelli, Lorenzo"https://zbmath.org/authors/?q=ai:tentarelli.lorenzoSummary: For kernels \(\nu\) which are positive and integrable we show that the operator \(g \mapsto J_\nu g = \int_0^x \nu(x - s) g(s) ds\) on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a ``contractive'' effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor \(N(x) = \int_0^x \nu(s) d s\). For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator \(J_\nu\) ``shrinks'' the norm of the argument by a factor that, as in the Hölder case, depends on the function \(N\) (whereas no regularization result can be obtained).
These results can be applied, for instance, to Abel kernels and to the Volterra function \(\mathcal{I}(x) = \mu(x, 0, - 1) = \int_0^\infty x^{s - 1} /\Gamma(s) d s\), the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in \(\mathbb{R}^2\).No Lavrentiev gap for some double phase integralshttps://zbmath.org/1530.490232024-04-15T15:10:58.286558Z"De Filippis, Filomena"https://zbmath.org/authors/?q=ai:de-filippis.filomena"Leonetti, Francesco"https://zbmath.org/authors/?q=ai:leonetti.francescoSummary: We prove the absence of the Lavrentiev gap for non-autonomous functionals
\[
\mathcal{F}(u) := \int\limits_{\Omega}f(x,Du(x))\,dx,
\]
where the density \(f(x,z)\) is \(\alpha \)-Hölder continuous with respect to \(x \in \Omega \subset\mathbb{R}^n \), it satisfies the \((p,q)\)-growth conditions
\[
\vert z\vert^p \leqslant f(x,z) \leqslant L(1+\vert z\vert^q),
\]
where \(1<p<q<p(\frac{n+\alpha}{n})\), and it can be approximated from below by suitable densities \(f_k \).Quantitative estimates for fractional Sobolev mappings in rational homotopy groupshttps://zbmath.org/1530.550112024-04-15T15:10:58.286558Z"Park, Woongbae"https://zbmath.org/authors/?q=ai:park.woongbae"Schikorra, Armin"https://zbmath.org/authors/?q=ai:schikorra.arminLet \( \mathcal N\subset \mathbb R ^M\) be a compact simply connected manifold without boundary. For a map \(f: \mathbb S^N \to \mathcal N\) an estimate of its rational homotopy group element \(\deg([f]) \in \mathbb R\) in terms of its fractional Sobolev norm is given:
\[
|\deg[f]|<C(\deg)[f]^{\frac{N+L(\deg)}{\beta}}_{W^{\beta, \frac{N}{\beta}} (\mathbb S^N)}
\]
for all \(\beta\in (\beta_0(\deg)),1]\), where \(C(\deg),\beta_0(\deg)\) are some constants which are computable in terms of the homomorphism \(\deg: \pi_N(\mathcal N)\to \mathbb R \).
This extends an earlier work by \textit{A. Schikorra} and \textit{J. van Schaftingen} [Proc. Am. Math. Soc. 148, No. 7, 2877--2891 (2020; Zbl 1487.55020)].
Reviewer: Zdzisław Dzedzej (Gdańsk)The equivalence principle applicability boundaries, measurability, and UVD in QFThttps://zbmath.org/1530.811172024-04-15T15:10:58.286558Z"Shalyt-Margolin, Alexander"https://zbmath.org/authors/?q=ai:shalyt-margolin.alexander-e(no abstract)Pricing options on flow forwards by neural networks in a Hilbert spacehttps://zbmath.org/1530.915622024-04-15T15:10:58.286558Z"Benth, Fred Espen"https://zbmath.org/authors/?q=ai:benth.fred-espen"Detering, Nils"https://zbmath.org/authors/?q=ai:detering.nils"Galimberti, Luca"https://zbmath.org/authors/?q=ai:galimberti.lucaSummary: We propose a new methodology for pricing options on flow forwards by applying infinite-dimensional neural networks. We recast the pricing problem as an optimisation problem in a Hilbert space of real-valued functions on the positive real line, which is the state space for the term structure dynamics. This optimisation problem is solved by using a feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural network is built upon the basis of the Hilbert space. We provide case studies that show its numerical efficiency, with superior performance over that of a classical neural network trained on sampling the term structure curves.Reproducing kernel Hilbert space method for the numerical solutions of fractional cancer tumor modelshttps://zbmath.org/1530.920362024-04-15T15:10:58.286558Z"Attia, Nourhane"https://zbmath.org/authors/?q=ai:attia.nourhane"Akgül, Ali"https://zbmath.org/authors/?q=ai:akgul.ali"Seba, Djamila"https://zbmath.org/authors/?q=ai:seba.djamila"Nour, Abdelkader"https://zbmath.org/authors/?q=ai:nour.abdelkaderSummary: This research work is concerned with the new numerical solutions of some essential fractional cancer tumor models, which are investigated by using reproducing kernel Hilbert space method (RKHSM). The most valuable advantage of the RKHSM is its ease of use and its quick calculation to obtain the numerical solutions of the considered problem. We make use of the Caputo fractional derivative. Our main tools are reproducing kernel theory, some important Hilbert spaces, and a normal basis. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results clearly show the superior performance of the RKHSM.
{{\copyright} 2020 John Wiley \& Sons, Ltd.}Sampling and reconstruction of concentrated reproducing kernel signals in mixed Lebesgue spaceshttps://zbmath.org/1530.940132024-04-15T15:10:58.286558Z"Jiang, Yingchun"https://zbmath.org/authors/?q=ai:jiang.yingchun"Zhang, Yajing"https://zbmath.org/authors/?q=ai:zhang.yajingSummary: In this paper, we mainly study the deterministic or random sampling and reconstruction of concentrated signals in the reproducing kernel subspaces of mixed Lebesgue spaces \(L^{p,q}(\mathbb{R}\times X)\), where \(X\) is a metric space with non-negative Borel measure. We first revisit and reformulate the iterative reconstruction algorithms in reproducing kernel subspaces of mixed Lebesgue spaces. Then, we establish a weighted sampling stability inequality and propose an algorithm to provide a good approximation to the concentrated signals in the reproducing kernel subspaces. Finally, we prove that the concentrated signals can also be approximately reconstructed from the random samples with high probability.