Pub Date : 2025-02-27DOI: 10.1016/j.jmaa.2025.129428
Lili Chen, Yunyi Jiang, Yanfeng Zhao
In this work, a series of fixed point results of the Ishikawa iterative algorithm and the SP iterative algorithm are presented in graphical convex metric spaces. First, we introduce the concepts of mean nonexpansive set-valued mappings in the above space. Furthermore, we study the existence and uniqueness of fixed points for mean nonexpansive set-valued mappings in graphical convex metric spaces. It is shown that the proposed two iterative sequences can both converge to a fixed point of the mean nonexpansive set-valued mapping. Meanwhile, we demonstrate the hypotheses of the existence theorem of fixed points for mean nonexpansive set-valued mappings by providing an example in G-complete graphical convex metric spaces are sufficient but not necessary.
{"title":"Iterative algorithms and fixed point theorems for mean nonexpansive set-valued mappings in graphical convex metric spaces","authors":"Lili Chen, Yunyi Jiang, Yanfeng Zhao","doi":"10.1016/j.jmaa.2025.129428","DOIUrl":"10.1016/j.jmaa.2025.129428","url":null,"abstract":"<div><div>In this work, a series of fixed point results of the Ishikawa iterative algorithm and the SP iterative algorithm are presented in graphical convex metric spaces. First, we introduce the concepts of mean nonexpansive set-valued mappings in the above space. Furthermore, we study the existence and uniqueness of fixed points for mean nonexpansive set-valued mappings in graphical convex metric spaces. It is shown that the proposed two iterative sequences can both converge to a fixed point of the mean nonexpansive set-valued mapping. Meanwhile, we demonstrate the hypotheses of the existence theorem of fixed points for mean nonexpansive set-valued mappings by providing an example in <em>G</em>-complete graphical convex metric spaces are sufficient but not necessary.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 1","pages":"Article 129428"},"PeriodicalIF":1.2,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143519596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-26DOI: 10.1016/j.jmaa.2025.129419
Alina Shalukhina
We prove the self-improvement property of the Hardy–Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion obtained in 2010 by Lerner and Ombrosi for maximal operators on quasi-Banach function spaces over Euclidean spaces. The specialty of the proof for spaces of homogeneous type lies in using adjacent grids of Hytönen–Kairema dyadic cubes and studying the maximal operator alongside its “dyadic” version. Then we apply the obtained result to variable Lebesgue spaces over spaces of homogeneous type.
{"title":"Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type","authors":"Alina Shalukhina","doi":"10.1016/j.jmaa.2025.129419","DOIUrl":"10.1016/j.jmaa.2025.129419","url":null,"abstract":"<div><div>We prove the self-improvement property of the Hardy–Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion obtained in 2010 by Lerner and Ombrosi for maximal operators on quasi-Banach function spaces over Euclidean spaces. The specialty of the proof for spaces of homogeneous type lies in using adjacent grids of Hytönen–Kairema dyadic cubes and studying the maximal operator alongside its “dyadic” version. Then we apply the obtained result to variable Lebesgue spaces over spaces of homogeneous type.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129419"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143521296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-26DOI: 10.1016/j.jmaa.2025.129416
Xiong Lin , Jianfei Wang , Mingxin Chen , Qihua Ruan
In this paper, we prove a new boundary Schwarz lemma for holomorphic mappings between nonequidimensional unit balls. As an application, a new rigidity theorem for holomorphic mappings between the unit ball to is established, where . In particular, when , our result reduces to that of Liu and Tang.
{"title":"Rigidity of boundary Schwarz lemma between nonequidimensional unit balls","authors":"Xiong Lin , Jianfei Wang , Mingxin Chen , Qihua Ruan","doi":"10.1016/j.jmaa.2025.129416","DOIUrl":"10.1016/j.jmaa.2025.129416","url":null,"abstract":"<div><div>In this paper, we prove a new boundary Schwarz lemma for holomorphic mappings between nonequidimensional unit balls. As an application, a new rigidity theorem for holomorphic mappings between the unit ball <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is established, where <span><math><mi>N</mi><mo>≥</mo><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>. In particular, when <span><math><mi>N</mi><mo>=</mo><mi>n</mi></math></span>, our result reduces to that of Liu and Tang.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129416"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143521295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-26DOI: 10.1016/j.jmaa.2025.129422
Tianlan Chen
We construct the bifurcation of interval of positive radial solutions from the trivial solution to the following Minkowski-curvature problems on unbounded domains where f is not necessarily linearizable at zero. The proof of main results are based on the topological degree and global bifurcation techniques.
{"title":"Bifurcation from interval and positive solutions of Minkowski-curvature on unbounded domain","authors":"Tianlan Chen","doi":"10.1016/j.jmaa.2025.129422","DOIUrl":"10.1016/j.jmaa.2025.129422","url":null,"abstract":"<div><div>We construct the bifurcation of interval of positive radial solutions from the trivial solution to the following Minkowski-curvature problems on unbounded domains<span><span><span><math><mo>−</mo><mtext>div</mtext><mo>(</mo><mfrac><mrow><mi>∇</mi><mi>u</mi></mrow><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>)</mo><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mspace></mspace><mi>u</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></math></span></span></span><span><span><span><math><mi>u</mi><mo>→</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mtext>as</mtext><mspace></mspace><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>+</mo><mo>∞</mo><mo>,</mo></math></span></span></span> where <em>f</em> is not necessarily linearizable at zero. The proof of main results are based on the topological degree and global bifurcation techniques.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129422"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-25DOI: 10.1016/j.jmaa.2025.129409
Zalina A. Kusraeva
The necessary and sufficient conditions under which the cone of positive homogeneous polynomials between vector lattices coincides with the closed convex hull of the set of sums of monomials in lattice homomorphisms are found. Incidentally the answer for the following question is established: when the cone of positive multilinear operators between vector lattices serves as a point-wise uniformly closed convex hull of the set of lattice multimorphisms.
{"title":"Extremal structure of cones of positive homogeneous polynomials. Part II","authors":"Zalina A. Kusraeva","doi":"10.1016/j.jmaa.2025.129409","DOIUrl":"10.1016/j.jmaa.2025.129409","url":null,"abstract":"<div><div>The necessary and sufficient conditions under which the cone of positive homogeneous polynomials between vector lattices coincides with the closed convex hull of the set of sums of monomials in lattice homomorphisms are found. Incidentally the answer for the following question is established: when the cone of positive multilinear operators between vector lattices serves as a point-wise uniformly closed convex hull of the set of lattice multimorphisms.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129409"},"PeriodicalIF":1.2,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143521297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.jmaa.2025.129413
Zhigang Wu , Juanzi Cai , Mengqian Liu
This paper is dedicated to the study of the inviscid liquid-gas two-phase flow model in . We investigate the global existence of strong solutions to the Cauchy problem in the critical regularity framework by constructing local solution and the a priori estimate. Additionally, by establishing Lyapunov energy functional, we obtain the decay estimates of solutions in -norms without additional smallness assumptions on the initial data.
{"title":"Global strong solution to the inviscid liquid-gas two-phase flow model in Lp framework","authors":"Zhigang Wu , Juanzi Cai , Mengqian Liu","doi":"10.1016/j.jmaa.2025.129413","DOIUrl":"10.1016/j.jmaa.2025.129413","url":null,"abstract":"<div><div>This paper is dedicated to the study of the inviscid liquid-gas two-phase flow model in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mspace></mspace><mo>(</mo><mi>d</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>. We investigate the global existence of strong solutions to the Cauchy problem in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> critical regularity framework by constructing local solution and the a priori estimate. Additionally, by establishing Lyapunov energy functional, we obtain the decay estimates of solutions in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-norms without additional smallness assumptions on the initial data.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129413"},"PeriodicalIF":1.2,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.jmaa.2025.129407
Gabriela Revelo-Silverio, Marco Calahorrano
This article demonstrates the existence of at least three solutions to the Ambrosetti-Prodi type problem with non-homogeneous Dirichlet boundary conditions. For the search for the first solution, which will also be minimal, we employ the sub-supersolutions method combined with a monotone iteration scheme. In addition, we utilize the variational method and the steepest descent technique, along with the Palais-Smale conditions and the Mountain Pass Theorem, to find the second and third solutions. It should be noted that the non-homogeneity of the boundary conditions generates a shift; therefore, it is important to impose some growth hypotheses on non-linearity. Furthermore, it is observed that the trivial function is not a solution to this problem.
{"title":"Ambrosetti-Prodi type problem with non-homogeneous Dirichlet boundary conditions","authors":"Gabriela Revelo-Silverio, Marco Calahorrano","doi":"10.1016/j.jmaa.2025.129407","DOIUrl":"10.1016/j.jmaa.2025.129407","url":null,"abstract":"<div><div>This article demonstrates the existence of at least three solutions to the Ambrosetti-Prodi type problem with non-homogeneous Dirichlet boundary conditions. For the search for the first solution, which will also be minimal, we employ the sub-supersolutions method combined with a monotone iteration scheme. In addition, we utilize the variational method and the steepest descent technique, along with the Palais-Smale conditions and the Mountain Pass Theorem, to find the second and third solutions. It should be noted that the non-homogeneity of the boundary conditions generates a shift; therefore, it is important to impose some growth hypotheses on non-linearity. Furthermore, it is observed that the trivial function is not a solution to this problem.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129407"},"PeriodicalIF":1.2,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.jmaa.2025.129406
Van Thanh Nguyen
<div><div>This paper investigates the asymptotic behavior of solutions <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> to <em>p</em>-Laplacian type equations as <em>p</em> goes to ∞, under a homogeneous Neumann boundary condition given by<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mrow><mi>div</mi></mrow><mrow><mo>(</mo><mfrac><mrow><mo>|</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd><mtd><mtext> in </mtext><mi>Ω</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>p</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>η</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mtd><mtd><mtext> on </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are space-dependent diffusion functions. Under suitable conditions on the diffusion functions <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, particularly the uniform convergence <span><math><munder><mi>lim</mi><mrow><mi>p</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> on <span><math><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover></math></span>, Mazon, Rossi and Toledo <span><span>[17]</span></span> show that, along a subsequence, the sequence of solutions <span><math><mo>{</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> converges uniformly to a limit function <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> and the sequence of gradients <span><math><mo>{</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> converges weakly to <span><math><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> in Lebesgue spaces as <em>p</em> goes to ∞. Among other results, the present paper proves that the sequence of gradients <span><math><mo>{</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> actually converges strongly on the so-called transport set as <em>p</em> goes to ∞. This strong conver
{"title":"Strong convergence for Neumann p-Laplacian problems with spatial dependence as p goes to infinity","authors":"Van Thanh Nguyen","doi":"10.1016/j.jmaa.2025.129406","DOIUrl":"10.1016/j.jmaa.2025.129406","url":null,"abstract":"<div><div>This paper investigates the asymptotic behavior of solutions <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> to <em>p</em>-Laplacian type equations as <em>p</em> goes to ∞, under a homogeneous Neumann boundary condition given by<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mrow><mi>div</mi></mrow><mrow><mo>(</mo><mfrac><mrow><mo>|</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd><mtd><mtext> in </mtext><mi>Ω</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>p</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>η</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mtd><mtd><mtext> on </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are space-dependent diffusion functions. Under suitable conditions on the diffusion functions <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, particularly the uniform convergence <span><math><munder><mi>lim</mi><mrow><mi>p</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><msub><mrow><mi>k</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> on <span><math><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover></math></span>, Mazon, Rossi and Toledo <span><span>[17]</span></span> show that, along a subsequence, the sequence of solutions <span><math><mo>{</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> converges uniformly to a limit function <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> and the sequence of gradients <span><math><mo>{</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> converges weakly to <span><math><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> in Lebesgue spaces as <em>p</em> goes to ∞. Among other results, the present paper proves that the sequence of gradients <span><math><mo>{</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> actually converges strongly on the so-called transport set as <em>p</em> goes to ∞. This strong conver","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129406"},"PeriodicalIF":1.2,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.jmaa.2025.129408
David Norrbo
We study the Hilbert matrix operator H and a related integral operator T acting on the standard weighted Bergman spaces . We obtain an upper bound for T, which yields the smallest currently known explicit upper bound for the norm of H for and . We also calculate the essential norm for all , extending a part of the main result in [Adv. Math. 408 (2022) 108598] to the standard unbounded weights. It is worth mentioning that except for an application of Minkowski's inequality, the norm estimates obtained for T are sharp.
{"title":"Norm estimates for the Hilbert matrix operator on weighted Bergman spaces","authors":"David Norrbo","doi":"10.1016/j.jmaa.2025.129408","DOIUrl":"10.1016/j.jmaa.2025.129408","url":null,"abstract":"<div><div>We study the Hilbert matrix operator <em>H</em> and a related integral operator <em>T</em> acting on the standard weighted Bergman spaces <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>. We obtain an upper bound for <em>T</em>, which yields the smallest currently known explicit upper bound for the norm of <em>H</em> for <span><math><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo><</mo><mn>0</mn></math></span> and <span><math><mn>2</mn><mo>+</mo><mi>α</mi><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mo>(</mo><mn>2</mn><mo>+</mo><mi>α</mi><mo>)</mo></math></span>. We also calculate the essential norm for all <span><math><mi>p</mi><mo>></mo><mn>2</mn><mo>+</mo><mi>α</mi><mo>></mo><mn>1</mn></math></span>, extending a part of the main result in [Adv. Math. 408 (2022) 108598] to the standard unbounded weights. It is worth mentioning that except for an application of Minkowski's inequality, the norm estimates obtained for <em>T</em> are sharp.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129408"},"PeriodicalIF":1.2,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.jmaa.2025.129405
Chao Zu, Yufeng Lu
<div><div>Let <span><math><mi>θ</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>φ</mi><mo>(</mo><mi>w</mi><mo>)</mo></math></span> be two nonconstant inner functions and <em>M</em> be a submodule in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> denote the composition operator on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> defined by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>θ</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>φ</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>)</mo></math></span>, and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> denote the submodule <span><math><mo>[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>M</mi><mo>]</mo></math></span>, that is, the smallest submodule containing <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>M</mi></math></span>. Let <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow><mrow><mi>M</mi></mrow></msubsup><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></mrow></msubsup><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> be the reproducing kernel of <em>M</em> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that<span><span><span><math><msup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></mrow></msup><mo>=</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>M</mi></mrow></msup><mo>∘</mo><mi>B</mi><mspace></mspace><mo>⋅</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mi>B</mi><mo>=</mo><mo>(</mo><mi>θ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>θ</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>‾</mo></mover><mi>θ</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>λ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>z</mi></mrow></mfrac><mfrac><mrow><mn>1</mn><mo>−</mo><mover><mrow>
{"title":"Hilbert-Schmidtness of the Mθ,φ-type submodules","authors":"Chao Zu, Yufeng Lu","doi":"10.1016/j.jmaa.2025.129405","DOIUrl":"10.1016/j.jmaa.2025.129405","url":null,"abstract":"<div><div>Let <span><math><mi>θ</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>φ</mi><mo>(</mo><mi>w</mi><mo>)</mo></math></span> be two nonconstant inner functions and <em>M</em> be a submodule in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> denote the composition operator on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> defined by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>θ</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>φ</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>)</mo></math></span>, and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span> denote the submodule <span><math><mo>[</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>M</mi><mo>]</mo></math></span>, that is, the smallest submodule containing <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub><mi>M</mi></math></span>. Let <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow><mrow><mi>M</mi></mrow></msubsup><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></mrow></msubsup><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span> be the reproducing kernel of <em>M</em> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></math></span>, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that<span><span><span><math><msup><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi><mo>,</mo><mi>φ</mi></mrow></msub></mrow></msup><mo>=</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>M</mi></mrow></msup><mo>∘</mo><mi>B</mi><mspace></mspace><mo>⋅</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mi>B</mi><mo>=</mo><mo>(</mo><mi>θ</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>θ</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>‾</mo></mover><mi>θ</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>λ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>z</mi></mrow></mfrac><mfrac><mrow><mn>1</mn><mo>−</mo><mover><mrow>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"548 2","pages":"Article 129405"},"PeriodicalIF":1.2,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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