We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must require an additional regularity assumption on the initial data to ensure the decay.
Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.
{"title":"Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach","authors":"Timothée Crin-Barat , Lorenzo Liverani , Ling-Yun Shou , Enrique Zuazua","doi":"10.1016/j.matpur.2025.103757","DOIUrl":"10.1016/j.matpur.2025.103757","url":null,"abstract":"<div><div>We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of <em>inhomogeneous hypocoercivity</em>. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must require an additional regularity assumption on the initial data to ensure the decay.</div><div>Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103757"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103753
Purba Das , Donghan Kim
We study the concept of (generalized) p-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of -norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to -norm of the Schauder coefficients. This result provides an isomorphism between the space of α-Hölder continuous functions with finite (generalized) p-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.
{"title":"On isomorphism of the space of continuous functions with finite p-th variation along a partition sequence","authors":"Purba Das , Donghan Kim","doi":"10.1016/j.matpur.2025.103753","DOIUrl":"10.1016/j.matpur.2025.103753","url":null,"abstract":"<div><div>We study the concept of (generalized) <em>p</em>-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the <em>p</em>-th variation of a given function is closely related to the finiteness of <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-norm of the Schauder coefficients. This result provides an isomorphism between the space of <em>α</em>-Hölder continuous functions with finite (generalized) <em>p</em>-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103753"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.matpur.2025.103763
Vincenzo Antonelli , Francesco Malaspina , Simone Marchesi , Joan Pons-Llopis
In this work we study the moduli spaces of instanton bundles on the flag twistor space . We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) ‘t Hooft bundle on F. In particular we prove that there exist μ-stable ‘t Hooft bundles for each admissible charge k. We completely describe the geometric structure of the moduli space of (special) ‘t Hooft bundles for arbitrary charge k. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in F as well as the family of del Pezzo surfaces realized as hyperplane sections of F. Finally we investigate the splitting behavior of ‘t Hooft bundles when restricted to conics.
{"title":"‘t Hooft bundles on the complete flag threefold and moduli spaces of instantons","authors":"Vincenzo Antonelli , Francesco Malaspina , Simone Marchesi , Joan Pons-Llopis","doi":"10.1016/j.matpur.2025.103763","DOIUrl":"10.1016/j.matpur.2025.103763","url":null,"abstract":"<div><div>In this work we study the moduli spaces of instanton bundles on the flag twistor space <span><math><mi>F</mi><mo>:</mo><mo>=</mo><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) ‘t Hooft bundle on <em>F</em>. In particular we prove that there exist <em>μ</em>-stable ‘t Hooft bundles for each admissible charge <em>k</em>. We completely describe the geometric structure of the moduli space of (special) ‘t Hooft bundles for arbitrary charge <em>k</em>. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in <em>F</em> as well as the family of del Pezzo surfaces realized as hyperplane sections of <em>F</em>. Finally we investigate the splitting behavior of ‘t Hooft bundles when restricted to conics.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103763"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.matpur.2025.103761
Jean-Michel Coron , Shengquan Xiang
In this paper, we study the controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we can also prove the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding open problem for nonlinear heat equations.
{"title":"Global controllability to harmonic maps of the heat flow from a circle to a sphere","authors":"Jean-Michel Coron , Shengquan Xiang","doi":"10.1016/j.matpur.2025.103761","DOIUrl":"10.1016/j.matpur.2025.103761","url":null,"abstract":"<div><div>In this paper, we study the controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we can also prove the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding open problem for nonlinear heat equations.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103761"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.matpur.2025.103759
Lei Zhang , Xiao-Qiang Zhao
In this paper, we study the propagation dynamics for a large class of time and space heterogeneous reaction-diffusion equations , where represents the shifting distance, and the nonlinearity is asymptotically of KPP type as and is negative as . Let be the spreading speed of the limiting equation . Under the assumption that the shifting speed admits a uniform mean c, we show that the solutions with compactly supported initial data go to zero eventually when , the leftward spreading speed is when , and the rightward spreading speed is c and when and , respectively. We also establish the existence, uniqueness and nonexistence of the forced traveling wave in terms of the sign of .
{"title":"Spreading properties and forced traveling waves of reaction-diffusion equations in a time-heterogeneous shifting environment","authors":"Lei Zhang , Xiao-Qiang Zhao","doi":"10.1016/j.matpur.2025.103759","DOIUrl":"10.1016/j.matpur.2025.103759","url":null,"abstract":"<div><div>In this paper, we study the propagation dynamics for a large class of time and space heterogeneous reaction-diffusion equations <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span>, where <span><math><mi>ω</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> represents the shifting distance, and the nonlinearity <span><math><mi>u</mi><mi>g</mi><mo>(</mo><mi>ξ</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span> is asymptotically of KPP type as <span><math><mi>ξ</mi><mo>→</mo><mo>−</mo><mo>∞</mo></math></span> and is negative as <span><math><mi>ξ</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span>. Let <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> be the spreading speed of the limiting equation <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>g</mi><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span>. Under the assumption that the shifting speed <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo></math></span> admits a uniform mean <em>c</em>, we show that the solutions with compactly supported initial data go to zero eventually when <span><math><mi>c</mi><mo>≤</mo><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the leftward spreading speed is <span><math><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> when <span><math><mi>c</mi><mo>></mo><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, and the rightward spreading speed is <em>c</em> and <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> when <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> and <span><math><mi>c</mi><mo>≥</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, respectively. We also establish the existence, uniqueness and nonexistence of the forced traveling wave in terms of the sign of <span><math><mi>c</mi><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103759"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.matpur.2025.103760
Yaxiong Liu , Zhuo Liu , Hui Yang , Xiangyu Zhou
In this paper, we obtain a Le Potier-type isomorphism theorem twisted with multiplier submodule sheaves, which relates a holomorphic vector bundle endowed with a strongly Nakano semi-positive singular Hermitian metric to the tautological line bundle with the induced metric. As applications, we obtain a Kollár-type injectivity theorem, a Nadel-type vanishing theorem, and a singular holomorphic Morse inequality for holomorphic vector bundles and so on.
{"title":"A Le Potier-type isomorphism twisted with multiplier submodule sheaves","authors":"Yaxiong Liu , Zhuo Liu , Hui Yang , Xiangyu Zhou","doi":"10.1016/j.matpur.2025.103760","DOIUrl":"10.1016/j.matpur.2025.103760","url":null,"abstract":"<div><div>In this paper, we obtain a Le Potier-type isomorphism theorem twisted with multiplier submodule sheaves, which relates a holomorphic vector bundle endowed with a strongly Nakano semi-positive singular Hermitian metric to the tautological line bundle with the induced metric. As applications, we obtain a Kollár-type injectivity theorem, a Nadel-type vanishing theorem, and a singular holomorphic Morse inequality for holomorphic vector bundles and so on.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103760"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.matpur.2025.103762
Yu-Chu Lin , Haitao Wang , Kung-Chien Wu
Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.
{"title":"Stability of background perturbation for Boltzmann equation","authors":"Yu-Chu Lin , Haitao Wang , Kung-Chien Wu","doi":"10.1016/j.matpur.2025.103762","DOIUrl":"10.1016/j.matpur.2025.103762","url":null,"abstract":"<div><div>Consider the Boltzmann equation in the perturbation regime. Since the macroscopic quantities in the background global Maxwellian are obtained through measurements, there are typically some errors involved. This paper investigates the effect of background variations on the solution for a given initial perturbation. Our findings demonstrate that the solution changes continuously with variations in the background and provide a sharp time decay estimate of the associated errors. The proof relies on refined estimates for the linearized solution operator and a proper decomposition of the nonlinear solution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103762"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-13DOI: 10.1016/j.matpur.2025.103764
Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring
We consider local weak solutions to the fractional p-Poisson equation of order s, i.e. . In the range and we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any that The qualitative result is accompanied by a local quantitative estimate.
{"title":"Gradient estimates for the fractional p-Poisson equation","authors":"Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring","doi":"10.1016/j.matpur.2025.103764","DOIUrl":"10.1016/j.matpur.2025.103764","url":null,"abstract":"<div><div>We consider local weak solutions to the fractional <em>p</em>-Poisson equation of order <em>s</em>, i.e. <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi></math></span>. In the range <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span> we prove Calderón & Zygmund type estimates at the gradient level. More precisely, we show for any <span><math><mi>q</mi><mo>></mo><mn>1</mn></math></span> that<span><span><span><math><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mfrac><mrow><mi>q</mi><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup><mspace></mspace><mo>⟹</mo><mspace></mspace><mi>∇</mi><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>loc</mi></mrow><mrow><mi>q</mi><mi>p</mi></mrow></msubsup><mo>.</mo></math></span></span></span> The qualitative result is accompanied by a local quantitative estimate.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103764"},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-28DOI: 10.1016/j.matpur.2025.103725
The Anh Bui
Let L be the Dunkl Laplacian on the Euclidean space associated with a normalized root R and a multiplicity function . In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian L are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type , where . Next, consider the Dunkl transform denoted by . We introduce the multiplier operator , defined as , where m is a bounded function defined on . Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type . Importantly, our findings present novel results, even in the specific case of the Hardy spaces.
{"title":"Harmonic analysis in Dunkl settings","authors":"The Anh Bui","doi":"10.1016/j.matpur.2025.103725","DOIUrl":"10.1016/j.matpur.2025.103725","url":null,"abstract":"<div><div>Let <em>L</em> be the Dunkl Laplacian on the Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> associated with a normalized root <em>R</em> and a multiplicity function <span><math><mi>k</mi><mo>(</mo><mi>ν</mi><mo>)</mo><mo>≥</mo><mn>0</mn><mo>,</mo><mi>ν</mi><mo>∈</mo><mi>R</mi></math></span>. In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian <em>L</em> are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type <span><math><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo><mo>‖</mo><mo>⋅</mo><mo>‖</mo><mo>,</mo><mi>d</mi><mi>w</mi><mo>)</mo></math></span>, where <span><math><mi>d</mi><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>ν</mi><mo>∈</mo><mi>R</mi></mrow></msub><msup><mrow><mo>〈</mo><mi>ν</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow><mrow><mi>k</mi><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></msup><mi>d</mi><mi>x</mi></math></span>. Next, consider the Dunkl transform denoted by <span><math><mi>F</mi></math></span>. We introduce the multiplier operator <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, defined as <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>f</mi><mo>=</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>m</mi><mi>F</mi><mi>f</mi><mo>)</mo></math></span>, where <em>m</em> is a bounded function defined on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type <span><math><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo><mo>‖</mo><mo>⋅</mo><mo>‖</mo><mo>,</mo><mi>d</mi><mi>w</mi><mo>)</mo></math></span>. Importantly, our findings present novel results, even in the specific case of the Hardy spaces.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"199 ","pages":"Article 103725"},"PeriodicalIF":2.1,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-28DOI: 10.1016/j.matpur.2025.103720
Maxence Cassier , Patrick Joly , Luis Alejandro Rosas Martínez
This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in [4], we develop a method based on the spectral analysis of the underlying non selfadjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation.
{"title":"Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (II) A modal approach","authors":"Maxence Cassier , Patrick Joly , Luis Alejandro Rosas Martínez","doi":"10.1016/j.matpur.2025.103720","DOIUrl":"10.1016/j.matpur.2025.103720","url":null,"abstract":"<div><div>This work concerns the analysis of electromagnetic dispersive media modelled by generalized Lorentz models. More precisely, this paper is the second of two articles dedicated to the long time behaviour of solutions of Maxwell's equations in dissipative Lorentz media, via the decay rate of the electromagnetic energy for the corresponding Cauchy problem. In opposition to the frequency dependent Lyapunov functions approach used in <span><span>[4]</span></span>, we develop a method based on the spectral analysis of the underlying non selfadjoint operator of the model. Although more involved, this approach is closer to physics, as it uses the dispersion relation of the model, and has the advantage to provide more precise and more optimal results, leading to distinguish the notion of weak and strong dissipation.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"201 ","pages":"Article 103720"},"PeriodicalIF":2.1,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144117002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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