Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110761
Pierre Gilles Lemarié-Rieusset
We construct non-trivial steady solutions in for the 2D Navier–Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to introduce a notion of special (non square integrable) solutions.
{"title":"Highly singular (frequentially sparse) steady solutions for the 2D Navier–Stokes equations on the torus","authors":"Pierre Gilles Lemarié-Rieusset","doi":"10.1016/j.jfa.2024.110761","DOIUrl":"10.1016/j.jfa.2024.110761","url":null,"abstract":"<div><div>We construct non-trivial steady solutions in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> for the 2D Navier–Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to introduce a notion of special (non square integrable) solutions.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110761"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110760
Filippo Boni , Simone Dovetta , Enrico Serra
We investigate the existence of normalized ground states for Schrödinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear δ-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every -subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on -periodic graphs and on a prototypical -periodic graph, the two–dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.
{"title":"Normalized ground states for Schrödinger equations on metric graphs with nonlinear point defects","authors":"Filippo Boni , Simone Dovetta , Enrico Serra","doi":"10.1016/j.jfa.2024.110760","DOIUrl":"10.1016/j.jfa.2024.110760","url":null,"abstract":"<div><div>We investigate the existence of normalized ground states for Schrödinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear <em>δ</em>-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on <span><math><mi>Z</mi></math></span>-periodic graphs and on a prototypical <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-periodic graph, the two–dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110760"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110755
Hendrik De Bie , Pan Lian , Frederick Maes
In this paper, we study the pointwise bounds for the kernel of the -generalized Fourier transform with , introduced by Ben Saïd, Kobayashi and Ørsted. We present explicit formulas for the case , which show that the kernels can exhibit polynomial growth. Subsequently, we provide a polynomial bound for the even dimensional kernel for this transform, focusing on the cases with finite order. Furthermore, by utilizing an estimation for the Prabhakar function, it is found that the -generalized Fourier kernel is bounded by a constant when and , except within an angular domain that diminishes as . As a byproduct, we prove that the -generalized Fourier kernel is uniformly bounded, when and .
在本文中，我们研究了Ben Saïd， Kobayashi和Ørsted引入的(κ,a)-广义傅里叶变换（κ≡0）核的点向界。我们给出了a=4情况下的显式公式，表明核可以呈现多项式增长。随后，我们给出了该变换的偶维核的多项式界，重点讨论了有限阶的情况。进一步，通过对Prabhakar函数的估计，我们发现(0,a)-广义傅里叶核在a>；1和m≥2时被一个常数限定，除了在角域内随着a→∞而减小。作为副产物，我们证明了(0,2 r /n)-广义傅里叶核是一致有界的，当m=2且r，n∈n。
{"title":"Bounds for the kernel of the (κ,a)-generalized Fourier transform","authors":"Hendrik De Bie , Pan Lian , Frederick Maes","doi":"10.1016/j.jfa.2024.110755","DOIUrl":"10.1016/j.jfa.2024.110755","url":null,"abstract":"<div><div>In this paper, we study the pointwise bounds for the kernel of the <span><math><mo>(</mo><mi>κ</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></span>-generalized Fourier transform with <span><math><mi>κ</mi><mo>≡</mo><mn>0</mn></math></span>, introduced by Ben Saïd, Kobayashi and Ørsted. We present explicit formulas for the case <span><math><mi>a</mi><mo>=</mo><mn>4</mn></math></span>, which show that the kernels can exhibit polynomial growth. Subsequently, we provide a polynomial bound for the even dimensional kernel for this transform, focusing on the cases with finite order. Furthermore, by utilizing an estimation for the Prabhakar function, it is found that the <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mi>a</mi><mo>)</mo></math></span>-generalized Fourier kernel is bounded by a constant when <span><math><mi>a</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>, except within an angular domain that diminishes as <span><math><mi>a</mi><mo>→</mo><mo>∞</mo></math></span>. As a byproduct, we prove that the <span><math><mo>(</mo><mn>0</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>ℓ</mi></mrow></msup><mo>/</mo><mi>n</mi><mo>)</mo></math></span>-generalized Fourier kernel is uniformly bounded, when <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>ℓ</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110755"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.jfa.2024.110758
Panu Lahti
We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for that at -a.e. , the mapping f “behaves non-quasiconformally”.
{"title":"Alberti's rank one theorem and quasiconformal mappings in metric measure spaces","authors":"Panu Lahti","doi":"10.1016/j.jfa.2024.110758","DOIUrl":"10.1016/j.jfa.2024.110758","url":null,"abstract":"<div><div>We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BV</mi></mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi>Y</mi><mo>)</mo></math></span> that at <span><math><msup><mrow><mo>‖</mo><mi>D</mi><mi>f</mi><mo>‖</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>-a.e. <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>, the mapping <em>f</em> “behaves non-quasiconformally”.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110758"},"PeriodicalIF":1.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110763
A. Fovelle
We prove an optimal result of stability under -sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.
{"title":"Asymptotic smoothness, concentration properties in Banach spaces and applications","authors":"A. Fovelle","doi":"10.1016/j.jfa.2024.110763","DOIUrl":"10.1016/j.jfa.2024.110763","url":null,"abstract":"<div><div>We prove an optimal result of stability under <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110763"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110757
Damian Głodkowski , Agnieszka Widz
We define a σ-centered notion of forcing that forces the existence of a Boolean algebra with the Grothendieck property and without the Nikodym property. In particular, the existence of such an algebra is consistent with the negation of the continuum hypothesis. The algebra we construct consists of Borel subsets of the Cantor set and has cardinality . We also show how to apply our method to streamline Talagrand's construction of such an algebra under the continuum hypothesis.
{"title":"Epic math battle of history: Grothendieck vs Nikodym","authors":"Damian Głodkowski , Agnieszka Widz","doi":"10.1016/j.jfa.2024.110757","DOIUrl":"10.1016/j.jfa.2024.110757","url":null,"abstract":"<div><div>We define a <em>σ</em>-centered notion of forcing that forces the existence of a Boolean algebra with the Grothendieck property and without the Nikodym property. In particular, the existence of such an algebra is consistent with the negation of the continuum hypothesis. The algebra we construct consists of Borel subsets of the Cantor set and has cardinality <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. We also show how to apply our method to streamline Talagrand's construction of such an algebra under the continuum hypothesis.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110757"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110759
Kai Toyosawa
We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.
{"title":"Weak exactness and amalgamated free product of von Neumann algebras","authors":"Kai Toyosawa","doi":"10.1016/j.jfa.2024.110759","DOIUrl":"10.1016/j.jfa.2024.110759","url":null,"abstract":"<div><div>We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110759"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.jfa.2024.110756
Ondřej F.K. Kalenda, Jiří Spurný
We investigate simpliciality of function spaces without constants. We prove, in particular, that several properties characterizing simpliciality in the classical case differ in this new setting. We also show that it may happen that a given point is not represented by any measure pseudosupported by the Choquet boundary, illustrating so limits of possible generalizations of the representation theorem. Moreover, we address the abstract Dirichlet problem in the new setting and establish some common points and nontrivial differences with the classical case.
{"title":"On simpliciality of function spaces not containing constants","authors":"Ondřej F.K. Kalenda, Jiří Spurný","doi":"10.1016/j.jfa.2024.110756","DOIUrl":"10.1016/j.jfa.2024.110756","url":null,"abstract":"<div><div>We investigate simpliciality of function spaces without constants. We prove, in particular, that several properties characterizing simpliciality in the classical case differ in this new setting. We also show that it may happen that a given point is not represented by any measure pseudosupported by the Choquet boundary, illustrating so limits of possible generalizations of the representation theorem. Moreover, we address the abstract Dirichlet problem in the new setting and establish some common points and nontrivial differences with the classical case.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110756"},"PeriodicalIF":1.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-08DOI: 10.1016/j.jfa.2024.110746
Luke D. Edholm , Yonatan Shelah
New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in . Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.
{"title":"The Leray transform: Distinguished measures, symmetries and polygamma inequalities","authors":"Luke D. Edholm , Yonatan Shelah","doi":"10.1016/j.jfa.2024.110746","DOIUrl":"10.1016/j.jfa.2024.110746","url":null,"abstract":"<div><div>New symmetries, norm computations and spectral information are obtained for the Leray transform on a class of unbounded hypersurfaces in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Emphasis is placed on certain distinguished measures, with results on operator norm monotonicity established by proving new polygamma inequalities. Classical techniques of Bernstein-Widder and Euler-Maclaurin play crucial roles in our analysis. Underpinning this work is a projective geometric theory of duality, which manifests here in the form of Hölder invariance.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110746"},"PeriodicalIF":1.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-08DOI: 10.1016/j.jfa.2024.110745
Vicente Asensio , Enrique Jordá , Thomas Kalmes
<div><div>We characterize those pairs <span><math><mo>(</mo><mi>ψ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> of smooth mappings <span><math><mi>ψ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>C</mi><mo>,</mo><mi>φ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for which the corresponding weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>ψ</mi><mo>⋅</mo><mo>(</mo><mi>f</mi><mo>∘</mo><mi>φ</mi><mo>)</mo></math></span> acts continuously on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in terms of <span><math><mi>ψ</mi><mo>,</mo><mi>φ</mi></math></span>. Among other things, as an application of our results we show that for a univariate polynomial <em>φ</em> with <span><math><mtext>deg</mtext><mo>(</mo><mi>φ</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>, power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for every <span><math><mi>ψ</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> only depends on <em>φ</em> and that in this case power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> is equivalent to <span><math><msub><mrow><mo>(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> converging to 0 in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> as well as to the uniform mean ergodicity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for which neither the multiplication operator <span><math><mi>f</mi><mo>↦</mo><mi>ψ</mi><mi>f</mi></math></span> nor the composition
{"title":"Power boundedness and related properties for weighted composition operators on S(Rd)","authors":"Vicente Asensio , Enrique Jordá , Thomas Kalmes","doi":"10.1016/j.jfa.2024.110745","DOIUrl":"10.1016/j.jfa.2024.110745","url":null,"abstract":"<div><div>We characterize those pairs <span><math><mo>(</mo><mi>ψ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> of smooth mappings <span><math><mi>ψ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><mi>C</mi><mo>,</mo><mi>φ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for which the corresponding weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>=</mo><mi>ψ</mi><mo>⋅</mo><mo>(</mo><mi>f</mi><mo>∘</mo><mi>φ</mi><mo>)</mo></math></span> acts continuously on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> in terms of <span><math><mi>ψ</mi><mo>,</mo><mi>φ</mi></math></span>. Among other things, as an application of our results we show that for a univariate polynomial <em>φ</em> with <span><math><mtext>deg</mtext><mo>(</mo><mi>φ</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>, power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for every <span><math><mi>ψ</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> only depends on <em>φ</em> and that in this case power boundedness of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> is equivalent to <span><math><msub><mrow><mo>(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> converging to 0 in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> as well as to the uniform mean ergodicity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ψ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> on <span><math><mi>S</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for which neither the multiplication operator <span><math><mi>f</mi><mo>↦</mo><mi>ψ</mi><mi>f</mi></math></span> nor the composition","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 3","pages":"Article 110745"},"PeriodicalIF":1.7,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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