Journal of Mathematical Analysis and Applications最新文献

{"title":"Global solvability for the heat equations in two half spaces and an interface","authors":"Hajime Koba","doi":"10.1016/j.jmaa.2025.130050","DOIUrl":"10.1016/j.jmaa.2025.130050","url":null,"abstract":"<div><div>This paper considers the existence of a global-in-time strong solution to the heat equations in the two half spaces <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>)</mo></math></span>, <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>−</mo></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>)</mo></math></span>, and the interface <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>(</mo><mo>≅</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. We introduce and study some function spaces in the two half spaces and the interface. We apply our function spaces and the maximal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-regularity for Hilbert space-valued functions to show the existence of a local-in-time strong solution to our heat equations. By using an energy equality of our heat system, we prove the existence of a unique global-in-time strong solution to the system with large initial data. The key idea of constructing strong solutions to our system is to make use of nice properties of the heat semigroups and kernels for <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span>, <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>−</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span>, and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. In Appendix, we derive our heat equations in the two half spaces and the interface from an energetic point of view.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130050"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099885","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}

{"title":"Kernels for composition of positive linear operators","authors":"Ulrich Abel ,&nbsp;Ana Maria Acu ,&nbsp;Margareta Heilmann ,&nbsp;Ioan Raşa","doi":"10.1016/j.jmaa.2025.130052","DOIUrl":"10.1016/j.jmaa.2025.130052","url":null,"abstract":"<div><div>This paper investigates the composition of Bernstein–Durrmeyer operators and Szász–Mirakjan–Durrmeyer operators, focusing on the structure and properties of the associated kernel functions. In the case of the Bernstein–Durrmeyer operators, we establish new identities for the kernel arising from the composition of two and three operators. Like the well-known representation in terms of Legendre polynomials, they show the commutativity of these operators naturally. While this Legendre representation contains all possible products <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>j</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, of Bernstein basis polynomials, the new representation has the beautiful property to contain only products <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mn>0</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>, where <em>n</em> is the smallest degree of the Bernstein–Durrmeyer polynomials involved. This fact immediately implies that the composition can be written as a linear combination of the operators themselves. Building on the eigenstructure of the Bernstein–Durrmeyer operator <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, we obtain a representation of its <em>r</em>-th iterate as a linear combination of the operators <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>. We also address the composition of Szász–Mirakjan–Durrmeyer operators and revisit a known result giving an elementary proof.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 2","pages":"Article 130052"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145098647","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}

{"title":"Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions","authors":"Nicolas E. Angelini ,&nbsp;Ursula M. Molter ,&nbsp;Jose M. Tejada","doi":"10.1016/j.jmaa.2025.130039","DOIUrl":"10.1016/j.jmaa.2025.130039","url":null,"abstract":"<div><div>In this paper, we define a family of dimensions for Borel measures that lie between the Hausdorff and Minkowski dimensions for measures, analogous to the intermediate dimensions of sets. Previously, Hare et al. in <span><span>[11]</span></span> defined families of dimensions that interpolate between the Minkowski and Assouad dimensions for measures. Additionally, Fraser, in <span><span>[8]</span></span> introduced an additional family of dimensions that interpolate between the Fourier and Sobolev dimensions of measures. Our results address a “gap” in the study of dimension interpolation for measures, almost completing the spectrum of intermediate dimensions for measures: from Hausdorff to Assouad dimensions. Furthermore, <span><span>Theorem 3.13</span></span> can be interpreted as a “reverse Frostman” lemma for intermediate dimensions. We also obtain a capacity-theoretic definition that enables us to estimate the intermediate dimensions of pushforward measures by projections.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130039"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099888","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}

{"title":"Some properties of a new mean measure","authors":"Yu Liu ,&nbsp;Bilel Selmi ,&nbsp;Zhiming Li","doi":"10.1016/j.jmaa.2025.130051","DOIUrl":"10.1016/j.jmaa.2025.130051","url":null,"abstract":"<div><div>The aim of this paper is to introduce the concept of the mean multifractal measure, which is a multifractal-measure-like quantity on an infinite dimensional space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, and also a dynamical-like quantity with respect to the natural shift. We first focus on the existence of sets with finite mean multifractal measure. Then properties concerning the density of the mean multifractal measure are explored.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130051"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099887","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}

{"title":"Perspectives on the ρ-operator radius","authors":"Pintu Bhunia ,&nbsp;Mohammad Sal Moslehian ,&nbsp;Ali Zamani","doi":"10.1016/j.jmaa.2025.130049","DOIUrl":"10.1016/j.jmaa.2025.130049","url":null,"abstract":"<div><div>Let <span><math><mi>ρ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> and let <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the <em>ρ</em>-operator radius of a Hilbert space operator <em>X</em>. Using techniques involving the Kronecker product, it is shown that<span><span><span><math><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><mo>|</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo>|</mo></mrow><mrow><mi>ρ</mi></mrow></mfrac><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>ρ</mi></mrow></mfrac><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo></mrow></math></span></span></span> where <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the numerical radius of <em>X</em>. These bounds for <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> are sharper than those presented by J. A. R. Holbrook. Furthermore, the cases of equality are investigated. We prove that the <em>ρ</em>-operator radius exposes certain operators as projections. We establish new inequalities for the <em>ρ</em>-operator radius, focusing on the sum and product of operators. For the generalized Aluthge transform <span><math><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow></msub></math></span> of an operator <em>X</em>, we prove the inequality:<span><span><span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>w</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ρ</mi></mrow></mfrac><mo>‖</mo><mi>X</mi><mo>‖</mo><mo>,</mo><mspace></mspace><mtext>for all</mtext><mspace></mspace><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>.</mo></mrow></math></span></span></span> The derived inequalities extend and generalize several well-known results for the classical operator norm and numerical radius.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130049"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099886","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}

{"title":"On new approximations of a functional equation having monomials","authors":"Hamid Khodaei","doi":"10.1016/j.jmaa.2025.130045","DOIUrl":"10.1016/j.jmaa.2025.130045","url":null,"abstract":"<div><div>Using a different direct method from the previous studies and the Banach fixed point theorem, we investigate the stability problem of a functional equation having monomials. The results of this paper improve the main results of <span><span>[3]</span></span>, <span><span>[12]</span></span>, <span><span>[18]</span></span>, <span><span>[23]</span></span>, <span><span>[24]</span></span>. Some examples are included for comparison with previous studies.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130045"},"PeriodicalIF":1.2,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010871","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}

{"title":"Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations","authors":"Y.-X. Lin ,&nbsp;Y.-G. Wang","doi":"10.1016/j.jmaa.2025.130042","DOIUrl":"10.1016/j.jmaa.2025.130042","url":null,"abstract":"<div><div>In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. The generalized Navier-Stokes equations are obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian <span><math><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup></math></span> with <span><math><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in <span><math><msup><mrow><mover><mrow><mi>H</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> with <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>α</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130042"},"PeriodicalIF":1.2,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049183","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}

{"title":"On critical logarithmic double phase problems with locally defined perturbation","authors":"Yino B. Cueva Carranza ,&nbsp;Marcos T.O. Pimenta ,&nbsp;Francesca Vetro ,&nbsp;Patrick Winkert","doi":"10.1016/j.jmaa.2025.130047","DOIUrl":"10.1016/j.jmaa.2025.130047","url":null,"abstract":"<div><div>This paper deals with critical logarithmic double phase problems of the form<span><span><span><math><mrow><mo>−</mo><mi>div</mi><mspace></mspace><mi>K</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><mtext>in </mtext><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mtext>on </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mrow></math></span></span></span> where <span><math><mi>div</mi><mspace></mspace><mi>K</mi></math></span> is the logarithmic double phase operator defined by<span><span><span><math><mrow><mi>div</mi><mspace></mspace><mrow><mo>(</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>+</mo><mi>μ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>e</mi><mo>+</mo><mo>|</mo><mi>∇</mi><mi>u</mi><mo>|</mo><mo>)</mo><mo>+</mo><mfrac><mrow><mo>|</mo><mi>∇</mi><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>(</mo><mi>e</mi><mo>+</mo><mo>|</mo><mi>∇</mi><mi>u</mi><mo>|</mo><mo>)</mo></mrow></mfrac><mo>)</mo></mrow><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span> <em>e</em> is Euler's number, <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>N</mi><mo>≥</mo><mn>2</mn></math></span>, is a bounded domain with Lipschitz boundary ∂Ω, <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mi>N</mi></math></span>, <span><math><mi>p</mi><mo>&lt;</mo><mi>q</mi><mo>&lt;</mo><msup><mrow><mi>p</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mfrac><mrow><mi>N</mi><mi>p</mi></mrow><mrow><mi>N</mi><mo>−</mo><mi>p</mi></mrow></mfrac></math></span>, <span><math><mn>0</mn><mo>≤</mo><mi>μ</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> and <span><math><mi>g</mi><mo>:</mo><mi>Ω</mi><mo>×</mo><mo>[</mo><mo>−</mo><mi>ξ</mi><mo>,</mo><mi>ξ</mi><mo>]</mo><mo>→</mo><mi>R</mi></math></span> for <span><math><mi>ξ</mi><mo>&gt;</mo><mn>0</mn></math></span> is a locally defined Carathéodory function satisfying a certain behavior near the origin. Based on appropriate truncation techniques and a suitable auxiliary problem, we prove the existence of a whole sequence of sign-changing solutions of the problem above which converges to 0 in the logarithmic Musielak-Orlicz Sobolev space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>log</mi></mrow></msub></mrow></msubsup><mo>(</mo><mi>Ω</m","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130047"},"PeriodicalIF":1.2,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049180","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}

{"title":"Upper and lower bounds of the value function for optimal control in the Wasserstein space","authors":"Yurii Averboukh,&nbsp;Aleksei Volkov","doi":"10.1016/j.jmaa.2025.130043","DOIUrl":"10.1016/j.jmaa.2025.130043","url":null,"abstract":"<div><div>This paper explores the application of nonsmooth analysis in the Wasserstein space to finite-horizon optimal control problems for nonlocal continuity equations. We characterize the value function as a strict viscosity solution of the corresponding Bellman equation using the notions of <em>ε</em>-subdifferentials and <em>ε</em>-superdifferentials. The main paper's result is the fact that continuous subsolutions and supersolutions of this Bellman equation yield lower and upper bounds for the value function. These estimates rely on proximal calculus in the space of probability measures and the Moreau–Yosida regularization. Furthermore, the upper estimates provide a family of approximately optimal feedback strategies that realize the concept of proximal aiming.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130043"},"PeriodicalIF":1.2,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020224","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}

{"title":"On the Hausdorff measure of self-similar sets in Qpd","authors":"Mamateli Kadir","doi":"10.1016/j.jmaa.2025.130046","DOIUrl":"10.1016/j.jmaa.2025.130046","url":null,"abstract":"<div><div>The study of self-similar sets and their fractal dimensions has been a central topic in geometric measure theory and fractal geometry. In this paper, we extend classical results on self-similar sets to the <em>p</em>-adic settings. Specifically, we investigate the Hausdorff measure and dimension of self-similar sets in <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, the <em>d</em>-dimensional vector space over the <em>p</em>-adic numbers. Our main results provide necessary and sufficient conditions for the open set condition (OSC) and strong open set condition (SOSC) in <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> and establish the relationship between these conditions and the Hausdorff measure of the attractor.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"555 1","pages":"Article 130046"},"PeriodicalIF":1.2,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049177","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}