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Heat Kernel Estimates of Fractional Schrödinger Operators with Hardy Potential on Half-line 半线上具有哈迪势的分数薛定谔算子的热核估计
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-09-14 DOI: 10.1007/s11118-024-10163-3
Tomasz Jakubowski, Paweł Maciocha
{"title":"Heat Kernel Estimates of Fractional Schrödinger Operators with Hardy Potential on Half-line","authors":"Tomasz Jakubowski, Paweł Maciocha","doi":"10.1007/s11118-024-10163-3","DOIUrl":"https://doi.org/10.1007/s11118-024-10163-3","url":null,"abstract":"<p>We provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249555","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}
引用次数: 0
Sharp Regularity Estimates for a Singular Inhomogeneous (m, p)-Laplacian Equation 奇异非均质 (m, p)- 拉普拉斯方程的尖锐正则估计值
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-08-26 DOI: 10.1007/s11118-024-10164-2
Pêdra D. S. Andrade, João Vitor da Silva, Giane C. Rampasso, Makson S. Santos
{"title":"Sharp Regularity Estimates for a Singular Inhomogeneous (m, p)-Laplacian Equation","authors":"Pêdra D. S. Andrade, João Vitor da Silva, Giane C. Rampasso, Makson S. Santos","doi":"10.1007/s11118-024-10164-2","DOIUrl":"https://doi.org/10.1007/s11118-024-10164-2","url":null,"abstract":"<p>In this paper, we investigate a class of doubly nonlinear evolutions PDEs. We establish sharp regularity for the solutions in Hölder spaces. The proof is based on the geometric tangential method and intrinsic scaling technique. Our findings extend and recover the results in the context of the classical evolution PDEs with singular signature via a unified treatment in the slow, normal and fast diffusion regimes. In addition, we provide some applications to certain nonlinear evolution models, which may have their own mathematical interest.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183194","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}
引用次数: 0
Differentiability of the Nonlocal-to-local Transition in Fractional Poisson Problems 分数泊松问题中的非局部到局部转变的可微分性
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-08-21 DOI: 10.1007/s11118-024-10162-4
Sven Jarohs, Alberto Saldaña, Tobias Weth
{"title":"Differentiability of the Nonlocal-to-local Transition in Fractional Poisson Problems","authors":"Sven Jarohs, Alberto Saldaña, Tobias Weth","doi":"10.1007/s11118-024-10162-4","DOIUrl":"https://doi.org/10.1007/s11118-024-10162-4","url":null,"abstract":"<p>Let <span>(u_{s})</span> denote a solution of the fractional Poisson problem </p><span>$$begin{aligned} (-Delta )^{s} u_{s} = fquad text { in }Omega ,qquad u_{s}=0quad text { on }{mathbb {R}}^{N}setminus Omega , end{aligned}$$</span><p>where <span>(Nge 2)</span> and <span>(Omega subset {mathbb {R}}^{N})</span> is a bounded domain of class <span>(C^{2})</span>. We show that the solution mapping <span>(smapsto u_{s})</span> is differentiable in <span>(L^infty (Omega ))</span> at <i>s</i> = 1, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative <span>(partial _{s} u_{s})</span> as the solution to a boundary value problem. This complements the previously known differentiability results for <i>s</i> in the open interval (0, 1). Our proofs are based on an asymptotic analysis to describe the collapse of the nonlocality of the fractional Laplacian as <i>s</i> approaches 1. We also provide a new representation of <span>(partial _{s} u_{s})</span> for <i>s</i> <span>(in (0,1))</span> which allows us to refine previously obtained Green function estimates.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142183195","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}
引用次数: 0
Heat Kernel Asymptotics for Scaling Limits of Isoradial Graphs 等轴图缩放极限的热核渐近法
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-31 DOI: 10.1007/s11118-024-10161-5
Simon Schwarz, Anja Sturm, Max Wardetzky
{"title":"Heat Kernel Asymptotics for Scaling Limits of Isoradial Graphs","authors":"Simon Schwarz, Anja Sturm, Max Wardetzky","doi":"10.1007/s11118-024-10161-5","DOIUrl":"https://doi.org/10.1007/s11118-024-10161-5","url":null,"abstract":"<p>We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"105 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871526","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}
引用次数: 0
Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures 关于加权高斯度量的索波列夫规范的等价性
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-31 DOI: 10.1007/s11118-024-10155-3
D. Addona
{"title":"Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures","authors":"D. Addona","doi":"10.1007/s11118-024-10155-3","DOIUrl":"https://doi.org/10.1007/s11118-024-10155-3","url":null,"abstract":"<p>We consider the spaces <span>({text {L}}^p(X,nu ;V))</span>, where <i>X</i> is a separable Banach space, <span>(mu )</span> is a centred non-degenerate Gaussian measure, <span>(nu :=Ke^{-U}mu )</span> with normalizing factor <i>K</i> and <i>V</i> is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions <span>(Fin W^{1,p}(X,nu ;V))</span>, which allows us to show that for every <span>(pin (1,infty ))</span> and every <span>(kin mathbb {N})</span> the norm in <span>(W^{k,p}(X,nu ))</span> is equivalent to the graph norm of <span>(D_H^{k})</span> (the <i>k</i>-th Malliavin derivative) in <span>({text {L}}^p(X,nu ))</span>. To conclude, we show exponential decay estimates for the <i>V</i>-valued perturbed Ornstein-Uhlenbeck semigroup <span>((T^V(t))_{tge 0})</span>, defined in Section 2.6, as <i>t</i> goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck <span>((T(t))_{tge 0})</span>, and pointwise estimates for <span>(|D_HT(t)f|_H^p)</span> by means of both <span>(T(t)|D_Hf|^p_H)</span> and <span>(T(t)|f|^p)</span>.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871523","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}
引用次数: 0
Calderón-Zygmund Decomposition, Hardy Spaces Associated with Operators and Weak Type Estimates 卡尔德龙-齐格蒙分解、与算子和弱类型估计相关的哈代空间
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-27 DOI: 10.1007/s11118-024-10158-0
The Anh Bui, Xuan Thinh Duong
{"title":"Calderón-Zygmund Decomposition, Hardy Spaces Associated with Operators and Weak Type Estimates","authors":"The Anh Bui, Xuan Thinh Duong","doi":"10.1007/s11118-024-10158-0","DOIUrl":"https://doi.org/10.1007/s11118-024-10158-0","url":null,"abstract":"<p>Let <span>((X, d, mu ))</span> be a metric space with a metric <i>d</i> and a doubling measure <span>(mu )</span>. Assume that the operator <i>L</i> generates a bounded holomorphic semigroup <span>(e^{-tL})</span> on <span>(L^2(X))</span> whose semigroup kernel satisfies the Gaussian upper bound. Also assume that <i>L</i> has a bounded holomorphic functional calculus on <span>(L^2(X))</span>. Then the Hardy spaces <span>(H^p_L(X))</span> associated with the operator <i>L</i> can be defined for <span>(0 &lt; p le 1)</span>. In this paper, we revisit the Calderón-Zygmund decomposition and show that a function <span>(f in L^1(X)cap L^2(X))</span> can be decomposed into a good part which is an <span>(L^{infty })</span> function and a bad part which is in <span>(H^p_L(X))</span> for some <span>(0&lt; p &lt;1)</span>. An important result of our variants of Calderón-Zygmund decompositions is that if a sub-linear operator <i>T</i> is bounded from <span>(L^r(X))</span> to <span>(L^r(X))</span> for some <span>(r &gt; 1)</span> and also bounded from <span>(H^p_L(X))</span> to <span>(L^p(X))</span> for some <span>(0&lt; p &lt; 1)</span>, then <i>T</i> is of weak type (1, 1) and bounded from <span>(L^q(X))</span> to <span>(L^q(X))</span> for all <span>(1&lt; q &lt;r)</span>.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"22 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782727","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}
引用次数: 0
Global Weighted Lorentz Estimates of Oblique Tangential Derivative Problems for Weakly Convex Fully Nonlinear Operators 弱凸全非线性算子斜切向衍生问题的全局加权洛伦兹估计值
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-23 DOI: 10.1007/s11118-024-10156-2
Junior da S. Bessa, Gleydson C. Ricarte
{"title":"Global Weighted Lorentz Estimates of Oblique Tangential Derivative Problems for Weakly Convex Fully Nonlinear Operators","authors":"Junior da S. Bessa, Gleydson C. Ricarte","doi":"10.1007/s11118-024-10156-2","DOIUrl":"https://doi.org/10.1007/s11118-024-10156-2","url":null,"abstract":"<p>In this work, we develop weighted Lorentz-Sobolev estimates for viscosity solutions of fully nonlinear elliptic equations with oblique boundary condition under weakened convexity conditions in the following configuration: </p><span>$$left{ begin{array}{rclcl} F(D^2u,Du,u,x) &amp; =&amp; f(x)&amp; text {in} &amp; Omega beta cdot Du + gamma u&amp; =&amp; g &amp; text {on}&amp; partial Omega ,end{array}right. $$</span><p>where <span>(Omega )</span> is a bounded domain in <span>(mathbb {R}^{n})</span> (<span>(nge 2)</span>), under suitable assumptions on the source term <i>f</i>, data <span>(beta , gamma )</span> and <i>g</i>. In addition, we obtain Lorentz-Sobolev estimates for solutions to the obstacle problem and others applications.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"42 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782888","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}
引用次数: 0
Mean Exit Times from Submanifolds with Bounded Mean Curvature 有界平均曲率子曼形体的平均出口时间
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-22 DOI: 10.1007/s11118-024-10160-6
G. Pacelli Bessa, Steen Markvorsen, Leandro F. Pessoa
{"title":"Mean Exit Times from Submanifolds with Bounded Mean Curvature","authors":"G. Pacelli Bessa, Steen Markvorsen, Leandro F. Pessoa","doi":"10.1007/s11118-024-10160-6","DOIUrl":"https://doi.org/10.1007/s11118-024-10160-6","url":null,"abstract":"<p>We show that submanifolds with infinite mean exit time can not be isometrically and minimally immersed into cylinders, horocylinders, cones, and wedges of some product spaces. Our approach is not based on the weak maximum principle at infinity, and thus it permits us to generalize previous results concerning non-immersibility of stochastically complete submanifolds. We also produce estimates for the complete tower of moments for submanifolds with small mean curvature immersed into cylinders.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"42 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782728","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}
引用次数: 0
Boundary Harnack Principle on Uniform Domains 均匀域上的边界哈纳克原理
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-22 DOI: 10.1007/s11118-024-10154-4
Aobo Chen
{"title":"Boundary Harnack Principle on Uniform Domains","authors":"Aobo Chen","doi":"10.1007/s11118-024-10154-4","DOIUrl":"https://doi.org/10.1007/s11118-024-10154-4","url":null,"abstract":"<p>We present a proof of scale-invariant boundary Harnack principle for uniform domains when the underlying space satisfies a scale-invariant elliptic Harnack inequality. Our approach does not assume the underlying space to be geodesic. Additionally, the existence of Green functions is also not assumed beforehand and is ensured by a recent result from M. T. Barlow, Z.-Q. Chen and M. Murugan.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"41 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782754","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}
引用次数: 0
Extremal Functions for a Trudinger-Moser Inequality with a Sign-Changing Weight 带有符号变化权重的特鲁丁格-莫泽尔不等式的极值函数
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-07-20 DOI: 10.1007/s11118-024-10159-z
Pengxiu Yu, Xiaobao Zhu
{"title":"Extremal Functions for a Trudinger-Moser Inequality with a Sign-Changing Weight","authors":"Pengxiu Yu, Xiaobao Zhu","doi":"10.1007/s11118-024-10159-z","DOIUrl":"https://doi.org/10.1007/s11118-024-10159-z","url":null,"abstract":"<p>Let <span>((Sigma ,g))</span> be a closed Riemann surface, <span>(lambda _1(Sigma ))</span> be the first eigenvalue of the Laplace-Beltrami operator. Assume <span>(h:Sigma rightarrow mathbb {R})</span> is some smooth sign-changing function. Using blow-up analysis, we prove that for any <span>(alpha &lt;lambda _1(Sigma ))</span>, the supremum </p><span>$$sup _{int _Sigma |nabla _gu|^2dv_g-alpha int _Sigma u^2dv_gle 1,,int _Sigma udv_g=0}int _Sigma he^{4pi u^2}dv_g$$</span><p>is attained by some admissible function <span>(u_alpha )</span>. This generalizes earlier results of Yang (J. Differential Equations 2015) and Hou (J. Math. ineq. 2018). Our result resembles existence of solutions to the mean field equations </p><span>$$Delta _gu=8pi left( frac{he^u}{int _Sigma he^udv_g}-frac{1}{|Sigma |}right) ,$$</span><p>where <i>h</i> is a smooth sign-changing function. Such problems were extensively studied by L. Sun and J. Y. Zhu (Cal. Var. 2021; arXiv: 2012.12840).</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"93 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744583","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}
引用次数: 0
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