The Journal of Geometric Analysis最新文献

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A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum 关于沿切线曲线几乎处处收敛于薛定谔方程初始基的说明
The Journal of Geometric Analysis Pub Date : 2024-09-09 DOI: 10.1007/s12220-024-01755-x
Javier Minguillón
{"title":"A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum","authors":"Javier Minguillón","doi":"10.1007/s12220-024-01755-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01755-x","url":null,"abstract":"<p>In this short note, we give an easy proof of the following result: for <span>( nge 2, )</span> <span>(underset{trightarrow 0}{lim } ,e^{itDelta }fleft( x+gamma (t)right) = f(x) )</span> almost everywhere whenever <span>( gamma )</span> is an <span>( alpha )</span>-Hölder curve with <span>( frac{1}{2}le alpha le 1 )</span> and <span>( fin H^s({mathbb {R}}^n) )</span>, with <span>( s &gt; frac{n}{2(n+1)} )</span>. This is the optimal range of regularity up to the endpoint.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Weighted Compactness of Commutators of Stein’s Square Functions Associated with Bochner-Riesz means 论与波赫纳-里兹手段相关的斯坦因平方函数换元的加权紧凑性
The Journal of Geometric Analysis Pub Date : 2024-09-05 DOI: 10.1007/s12220-024-01775-7
Qingying Xue, Chunmei Zhang
{"title":"On Weighted Compactness of Commutators of Stein’s Square Functions Associated with Bochner-Riesz means","authors":"Qingying Xue, Chunmei Zhang","doi":"10.1007/s12220-024-01775-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01775-7","url":null,"abstract":"<p>In this paper, our object of investigation is the commutators of the Stein’s square functions asssoicated with the Bochner-Riesz means of order <span>({uplambda })</span> defined by </p><span>$$begin{aligned} G_{b,m}^{uplambda }f(x)=Big (int _0^infty Big |int _{{mathbb {R}}^n}(b(x)-b(y))^mK_t^{uplambda }(x-y)f(y)dy Big |^2frac{dt}{t}Big )^{frac{1}{2}}, end{aligned}$$</span><p>where <span>(widehat{K_t^{uplambda }}({upxi })=frac{|{upxi }|^2}{t^2}Big (1-frac{|{upxi }|^2}{t^2}Big )_+^{{uplambda }-1})</span> and <span>(bin mathrm BMO(mathbb {R}^n))</span>. We show that <span>(G_{b,m}^{uplambda })</span> is a compact operator from <span>(L^p(w))</span> to <span>(L^p(w))</span> for <span>(1&lt;p&lt;infty )</span> and <span>({uplambda }&gt;frac{n+1}{2})</span> whenever <span>(bin mathrm CMO({mathbb {R}^n}))</span>, where <span>(textrm{CMO}(mathbb {R}^n))</span> is the closure of <span>(mathcal {C}_c^infty (mathbb {R}^n))</span> in the <span>(textrm{BMO}(mathbb {R}^n))</span> topology.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bilinear Decompositions for Products of Orlicz–Hardy and Orlicz–Campanato Spaces 奥利兹-哈代和奥利兹-坎帕纳托空间乘积的双线性分解
The Journal of Geometric Analysis Pub Date : 2024-09-04 DOI: 10.1007/s12220-024-01777-5
Chenglong Fang, Liguang Liu
{"title":"Bilinear Decompositions for Products of Orlicz–Hardy and Orlicz–Campanato Spaces","authors":"Chenglong Fang, Liguang Liu","doi":"10.1007/s12220-024-01777-5","DOIUrl":"https://doi.org/10.1007/s12220-024-01777-5","url":null,"abstract":"<p>For an Orlicz function <span>(varphi )</span> with critical lower type <span>(i(varphi )in (0, 1))</span> and upper type <span>(I(varphi )in (0,1))</span>, set <span>(m(varphi )=lfloor n(1/i(varphi )-1)rfloor )</span>. In this paper, the authors establish bilinear decomposition for the product of the Orlicz–Hardy space <span>(H^{varphi }({mathbb {R}}^{n}))</span> and its dual space—the Orlicz–Campanato space <span>({mathfrak {L}}_{varphi }({mathbb {R}}^{n}))</span>. In particular, the authors prove that the product (in the sense of distributions) of <span>(fin H^{varphi }({mathbb {R}}^{n}))</span> and <span>(gin {mathfrak {L}}_{varphi }({mathbb {R}}^{n}))</span> can be decomposed into the sum of <i>S</i>(<i>f</i>, <i>g</i>) and <i>T</i>(<i>f</i>, <i>g</i>), where <i>S</i> is a bilinear operator bounded from <span>(H^{varphi }({mathbb {R}}^{n})times {mathfrak {L}}_{varphi }({mathbb {R}}^{n}))</span> to <span>(L^{1}({mathbb {R}}^{n}))</span> and <i>T</i> is another bilinear operator bounded from <span>(H^{varphi }({mathbb {R}}^{n})times {mathfrak {L}}_{varphi }({mathbb {R}}^{n}))</span> to the Musielak–Orlicz–Hardy space <span>(H^{Phi }({mathbb {R}}^{n}))</span>, with <span>(Phi )</span> being a Musielak–Orlicz function determined by <span>(varphi )</span>. The bilinear decomposition is sharp in the following sense: any vector space <span>({mathcal {Y}}subset H^{Phi }({mathbb {R}}^{n}))</span> that adapted to the above bilinear decomposition should satisfy <span>( L^infty ({mathbb {R}}^{n})cap {mathcal {Y}}^{*}=L^infty ({mathbb {R}}^{n})cap (H^{Phi }({mathbb {R}}^{n}))^{*} )</span>. Indeed, <span>(L^infty ({mathbb {R}}^{n})cap (H^{Phi }({mathbb {R}}^{n}))^{*})</span> is just the multiplier space of <span>({mathfrak {L}}_{varphi }({mathbb {R}}^{n}))</span>. As applications, the authors obtain not only a priori estimate of the div-curl product involving the space <span>(H^{Phi }({mathbb {R}}^{n}))</span>, but also the boundedness of the Calderón–Zygmund commutator [<i>b</i>, <i>T</i>] from the Hardy type space <span>(H^{varphi }_{b}({mathbb {R}}^{n}))</span> to <span>(L^{1}({mathbb {R}}^{n}))</span> or <span>(H^{1}({mathbb {R}}^{n}))</span> under <span>(bin {mathfrak {L}}_{varphi }({mathbb {R}}^{n}))</span>, <span>(m(varphi )=0)</span> and suitable cancellation conditions of <i>T</i>.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of Normalized Solutions for Mass Super-Critical Quasilinear Schrödinger Equation with Potentials 带势能的质量超临界准薛定谔方程的归一化解的存在性
The Journal of Geometric Analysis Pub Date : 2024-09-04 DOI: 10.1007/s12220-024-01779-3
Fengshuang Gao, Yuxia Guo
{"title":"Existence of Normalized Solutions for Mass Super-Critical Quasilinear Schrödinger Equation with Potentials","authors":"Fengshuang Gao, Yuxia Guo","doi":"10.1007/s12220-024-01779-3","DOIUrl":"https://doi.org/10.1007/s12220-024-01779-3","url":null,"abstract":"<p>This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation: </p><span>$$begin{aligned} left{ begin{array}{ll} -Delta u-uDelta u^2+V(x)u+lambda u=g(u),hbox { in }{mathbb {R}}^N, uge 0, end{array}right. end{aligned}$$</span>(0.1)<p>satisfying the constraint <span>(int _{{mathbb {R}}^N}u^2=a)</span>. We will investigate how the potential and the nonlinearity effect the existence of the normalized solution. As a consequence, under a smallness assumption on <i>V</i>(<i>x</i>) and a relatively strict growth condition on <i>g</i>, we obtain a normalized solution for <span>(N=2)</span>, 3. Moreover, when <i>V</i>(<i>x</i>) is not too small in some sense, we show the existence of a normalized solution for <span>(Nge 2)</span> and <span>(g(u)={u}^{q-2}u)</span> with <span>(4+frac{4}{N}&lt;q&lt;2cdot 2^*)</span>.\u0000</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability of Membranes 薄膜的稳定性
The Journal of Geometric Analysis Pub Date : 2024-09-04 DOI: 10.1007/s12220-024-01767-7
Bennett Palmer, Álvaro Pámpano
{"title":"Stability of Membranes","authors":"Bennett Palmer, Álvaro Pámpano","doi":"10.1007/s12220-024-01767-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01767-7","url":null,"abstract":"<p>In Palmer and Pámpano (Calc Var Partial Differ Equ 61:79, 2022), the authors studied a particular class of equilibrium solutions of the Helfrich energy which satisfy a second order condition called the reduced membrane equation. In this paper we develop and apply a second variation formula for the Helfrich energy for this class of surfaces. The reduced membrane equation also arises as the Euler–Lagrange equation for the area of surfaces under the action of gravity in the three dimensional hyperbolic space. We study the second variation of this functional for a particular example.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$$L^p$$ -Improving Bounds of Maximal Functions Along Planar Curves $L^p$$ -沿平面曲线最大函数边界的改进
The Journal of Geometric Analysis Pub Date : 2024-09-04 DOI: 10.1007/s12220-024-01783-7
Naijia Liu, Haixia Yu
{"title":"$$L^p$$ -Improving Bounds of Maximal Functions Along Planar Curves","authors":"Naijia Liu, Haixia Yu","doi":"10.1007/s12220-024-01783-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01783-7","url":null,"abstract":"<p>In this paper, we study the <span>(L^p({mathbb {R}}^2))</span>-improving bounds, i.e., <span>(L^p({mathbb {R}}^2)rightarrow L^q({mathbb {R}}^2))</span> estimates, of the maximal function <span>(M_{gamma })</span> along a plane curve <span>((t,gamma (t)))</span>, where </p><span>$$begin{aligned} M_{gamma }f(x_1,x_2):=sup _{uin [1,2]}left| int _{0}^{1}f(x_1-ut,x_2-u gamma (t)),text {d}tright| , end{aligned}$$</span><p>and <span>(gamma )</span> is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain <span>(M_{gamma }: L^p({mathbb {R}}^2)rightarrow L^q({mathbb {R}}^2))</span> if <span>(left( frac{1}{p},frac{1}{q}right) in Delta cup {(0,0)})</span> and <span>(left( frac{1}{p},frac{1}{q}right) )</span> satisfying <span>(1+(1 +omega )left( frac{1}{q}-frac{1}{p}right) &gt;0)</span>, where <span>(Delta :=left{ left( frac{1}{p},frac{1}{q}right) : frac{1}{2p}&lt;frac{1}{q}le frac{1}{p}, frac{1}{q}&gt;frac{3}{p}-1 right} )</span> and <span>(omega :=limsup _{trightarrow 0^{+}}frac{ln |gamma (t)|}{ln t})</span>. This result is sharp except for some borderline cases.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Curvature, Diameter and Signs of Graphs 图形的曲率、直径和符号
The Journal of Geometric Analysis Pub Date : 2024-08-31 DOI: 10.1007/s12220-024-01774-8
Wei Chen, Shiping Liu
{"title":"Curvature, Diameter and Signs of Graphs","authors":"Wei Chen, Shiping Liu","doi":"10.1007/s12220-024-01774-8","DOIUrl":"https://doi.org/10.1007/s12220-024-01774-8","url":null,"abstract":"<p>We prove a Li-Yau type eigenvalue-diameter estimate for signed graphs. That is, the nonzero eigenvalues of the Laplacian of a non-negatively curved signed graph are lower bounded by <span>(1/D^2)</span> up to a constant, where <i>D</i> stands for the diameter. This leads to several interesting applications, including a volume estimate for non-negatively curved signed graphs in terms of frustration index and diameter, and a two-sided Li-Yau estimate for triangle-free graphs. Our proof is built upon a combination of Chung-Lin-Yau type gradient estimate and a new trick involving strong nodal domain walks of signed graphs. We further discuss extensions of part of our results to nonlinear Laplacians on signed graphs.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New Expanding Ricci Solitons Starting in Dimension Four 始于四维的新扩展里奇孤子
The Journal of Geometric Analysis Pub Date : 2024-08-31 DOI: 10.1007/s12220-024-01778-4
Jan Nienhaus, Matthias Wink
{"title":"New Expanding Ricci Solitons Starting in Dimension Four","authors":"Jan Nienhaus, Matthias Wink","doi":"10.1007/s12220-024-01778-4","DOIUrl":"https://doi.org/10.1007/s12220-024-01778-4","url":null,"abstract":"<p>We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on <span>(mathbb {R}^3 times S^1.)</span> More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric 论发散复几何积分的有限部分及其与赫米蒂公设选择的关系
The Journal of Geometric Analysis Pub Date : 2024-08-29 DOI: 10.1007/s12220-024-01773-9
Ludvig Svensson
{"title":"On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric","authors":"Ludvig Svensson","doi":"10.1007/s12220-024-01773-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01773-9","url":null,"abstract":"<p>Let <i>X</i> be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on <i>X</i> that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle <span>(E rightarrow X)</span>. Given a choice of Hermitian metric on <i>E</i> we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.\u0000</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unramified Riemann Domains Satisfying the Oka–Grauert Principle over a Stein Manifold 满足 Stein Manifold 上奥卡-格劳尔特原理的非ramified 黎曼域
The Journal of Geometric Analysis Pub Date : 2024-08-27 DOI: 10.1007/s12220-024-01756-w
Makoto Abe, Shun Sugiyama
{"title":"Unramified Riemann Domains Satisfying the Oka–Grauert Principle over a Stein Manifold","authors":"Makoto Abe, Shun Sugiyama","doi":"10.1007/s12220-024-01756-w","DOIUrl":"https://doi.org/10.1007/s12220-024-01756-w","url":null,"abstract":"<p>Let <span>((D, pi ))</span> be an unramified Riemann domain over a Stein manifold of dimension <i>n</i>. Assume that <span>(H^k(D,mathscr {O}) = 0)</span> for <span>(2 le k le n - 1)</span> and there exists a complex Lie group <i>G</i> of positive dimension such that all differentiably trivial holomorphic principal <i>G</i>-bundles on <i>D</i> are holomorphically trivial. Then, we prove that <i>D</i> is Stein.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"2011 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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