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Proof that a Form of Rubio de Francia’s Conjectured Littlewood-Paley Type Inequality for $$A_{1}left( {mathbb {R}}right) $$ -Weighted $$L^{2}left( {mathbb {R}}right) $$ is Valid for Every Even $$A_{1}left( {mathbb {R}}right) $$ Weight 证明鲁比奥-德-弗朗西亚猜想的利特尔伍德-佩利式不等式中的 $$A_{1}left( {mathbb {R}right) $$ -加权 $$L^{2}left( {mathbb {R}right) $$ 对于每个偶数 $$A_{1}left( {mathbb {R}right) $$ 加权都是有效的
The Journal of Geometric Analysis Pub Date : 2024-08-24 DOI: 10.1007/s12220-024-01762-y
Earl Berkson
{"title":"Proof that a Form of Rubio de Francia’s Conjectured Littlewood-Paley Type Inequality for $$A_{1}left( {mathbb {R}}right) $$ -Weighted $$L^{2}left( {mathbb {R}}right) $$ is Valid for Every Even $$A_{1}left( {mathbb {R}}right) $$ Weight","authors":"Earl Berkson","doi":"10.1007/s12220-024-01762-y","DOIUrl":"https://doi.org/10.1007/s12220-024-01762-y","url":null,"abstract":"<p>It is demonstrated that a form of Rubio de Francia’s hitherto unresolved Littlewood-Paley Type Conjecture from the year 1985 is valid for the weighted-<span>(L^{2}left( {mathbb {R}}right) )</span> space corresponding to any even <span>(A_{1}left( {mathbb {R}}right) )</span> weight. Otherwise expressed, we show that if <span>(omega )</span> is any even <span>(A_{1}left( {mathbb {R}}right) )</span> weight, <i>C</i> is an <span>(A_{1}left( {mathbb {R}}right) )</span> weight constant for <span>(omega )</span>, <span>( fin )</span> <span>(L^{2}left( {mathbb {R}},omega left( tright) dtright) )</span>, and <span>(left{ J_{k}right} _{kge 1})</span> is any finite or infinite sequence of disjoint intervals of <span>({mathbb {R}})</span>, then the following estimate holds for the corresponding Littlewood-Paley Type square function defined by <span>(left{ S_{J_{k}}left( fright) right} _{kge 1})</span>(where the symbol <span>(S_{_{J_{k}} })</span> denotes the indicated partial sum projection for the context of <span>({mathbb {R}})</span>): </p><span>$$begin{aligned} left| left{ sum limits _{kge 1}left| S_{J_{k}}left( fright) right| ^{2}right} ^{1/2}right| _{L^{2}left( {mathbb {R}},omega left( tright) dtright) }le 2^{5}C^{1/2}left| fright| _{L^{2}left( {mathbb {R}},omega ^*left( tright) dtright) }, end{aligned}$$</span><p>where <span>(omega ^*)</span> is the decreasing rearrangement of <span>(omega )</span>. A corollary of this even <span>(A_{1}left( {mathbb {R}}right) )</span>-weighted theorem is obtained which provides a related variant thereof in the setting of any (not necessarily even) <span>(A_{1}left( {mathbb {R}}right) )</span> weight.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224863","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
Normalized Solutions to at Least Mass Critical Problems: Singular Polyharmonic Equations and Related Curl–Curl Problems 至少质量临界问题的归一化解：奇异多谐方程及相关的曲线-曲线问题
The Journal of Geometric Analysis Pub Date : 2024-08-23 DOI: 10.1007/s12220-024-01770-y
Bartosz Bieganowski, Jarosław Mederski, Jacopo Schino
{"title":"Normalized Solutions to at Least Mass Critical Problems: Singular Polyharmonic Equations and Related Curl–Curl Problems","authors":"Bartosz Bieganowski, Jarosław Mederski, Jacopo Schino","doi":"10.1007/s12220-024-01770-y","DOIUrl":"https://doi.org/10.1007/s12220-024-01770-y","url":null,"abstract":"<p>We are interested in the existence of normalized solutions to the problem </p><span>$$begin{aligned} {left{ begin{array}{ll} (-Delta )^m u+frac{mu }{|y|^{2m}}u + lambda u = g(u), quad x = (y,z) in mathbb {R}^K times mathbb {R}^{N-K}, int _{mathbb {R}^N} |u|^2 , dx = rho &gt; 0, end{array}right. } end{aligned}$$</span><p>in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the <span>(L^2)</span>-ball. Moreover, we find also a solution to the related curl–curl problem </p><span>$$begin{aligned} {left{ begin{array}{ll} nabla times nabla times textbf{U}+lambda textbf{U}=f(textbf{U}), quad x in mathbb {R}^N, int _{mathbb {R}^N}|textbf{U}|^2,dx=rho , end{array}right. } end{aligned}$$</span><p>which arises from the system of Maxwell equations and is of great importance in nonlinear optics.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190136","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 Polynomial Carleson Operators Along Quadratic Hypersurfaces 论沿二次超曲面的多项式卡莱森算子
The Journal of Geometric Analysis Pub Date : 2024-08-23 DOI: 10.1007/s12220-024-01676-9
Theresa C. Anderson, Dominique Maldague, Lillian B. Pierce, Po-Lam Yung
{"title":"On Polynomial Carleson Operators Along Quadratic Hypersurfaces","authors":"Theresa C. Anderson, Dominique Maldague, Lillian B. Pierce, Po-Lam Yung","doi":"10.1007/s12220-024-01676-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01676-9","url":null,"abstract":"<p>We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by <span>((y,Q(y))subseteq mathbb {R}^{n+1})</span>, for an arbitrary non-degenerate quadratic form <i>Q</i>, admits an <i>a priori</i> bound on <span>(L^p)</span> for all <span>(1&lt;p&lt;infty )</span>, for each <span>(n ge 2)</span>. This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of <span>({p_2,ldots ,p_d})</span> for any set of fixed real-valued polynomials <span>(p_j)</span> such that <span>(p_j)</span> is homogeneous of degree <i>j</i>, and <span>(p_2)</span> is not a multiple of <i>Q</i>(<i>y</i>). The general method developed in this work applies to quadratic forms of arbitrary signature, while previous work considered only the special positive definite case <span>(Q(y)=|y|^2)</span>.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190137","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
Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds 一些紧凑赫尔墨斯漫场上的退化复蒙哥-安培方程
The Journal of Geometric Analysis Pub Date : 2024-08-21 DOI: 10.1007/s12220-024-01772-w
Omar Alehyane, Chinh H. Lu, Mohammed Salouf
{"title":"Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds","authors":"Omar Alehyane, Chinh H. Lu, Mohammed Salouf","doi":"10.1007/s12220-024-01772-w","DOIUrl":"https://doi.org/10.1007/s12220-024-01772-w","url":null,"abstract":"<p>Let <i>X</i> be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan–Li. Given a semipositive form <span>(theta )</span> with positive volume, we define the Monge–Ampère operator for unbounded <span>(theta )</span>-psh functions and prove that it is continuous with respect to convergence in capacity. We then develop pluripotential tools to study degenerate complex Monge–Ampère equations in this context, extending recent results of Tosatti–Weinkove, Kolodziej–Nguyen, Guedj–Lu and many others who treat bounded solutions.\u0000</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190139","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
Minimal Networks on Balls and Spheres for Almost Standard Metrics 球和球上几乎标准度量的最小网络
The Journal of Geometric Analysis Pub Date : 2024-08-20 DOI: 10.1007/s12220-024-01765-9
Luciano Sciaraffia
{"title":"Minimal Networks on Balls and Spheres for Almost Standard Metrics","authors":"Luciano Sciaraffia","doi":"10.1007/s12220-024-01765-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01765-9","url":null,"abstract":"<p>We study the existence of minimal networks in the unit sphere <span>({textbf{S}}^d)</span> and the unit ball <span>({textbf{B}}^d)</span> of <span>({textbf{R}}^d)</span> endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of <span>(theta )</span>-networks in <span>({textbf{S}}^d)</span> and triods in <span>({textbf{B}}^d)</span>, jointly with the Lusternik–Schnirelmann category.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190146","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
BV Functions and Nonlocal Functionals in Metric Measure Spaces 公度量空间中的 BV 函数和非局部函数
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01766-8
Panu Lahti, Andrea Pinamonti, Xiaodan Zhou
{"title":"BV Functions and Nonlocal Functionals in Metric Measure Spaces","authors":"Panu Lahti, Andrea Pinamonti, Xiaodan Zhou","doi":"10.1007/s12220-024-01766-8","DOIUrl":"https://doi.org/10.1007/s12220-024-01766-8","url":null,"abstract":"<p>We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that the limits of these nonlocal functionals are comparable to the total variation <span>(Vert DfVert (Omega ))</span> or the Sobolev semi-norm <span>(int _Omega g_f^p, dmu )</span>, which extends Euclidean results to metric measure spaces. In contrast to the classical setting, we also give an example to show that the limits are not always equal to the corresponding total variation even for Lipschitz functions.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190176","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
Rado’s Theorem for CR Functions on Hypersurfaces 超曲面上 CR 函数的拉多定理
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01763-x
S. Berhanu, Xiaoshan Li
{"title":"Rado’s Theorem for CR Functions on Hypersurfaces","authors":"S. Berhanu, Xiaoshan Li","doi":"10.1007/s12220-024-01763-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01763-x","url":null,"abstract":"<p>We prove a generalization of a well-known theorem of Rado for continuous CR functions on a class of bihololomorphically invariant hypersurfaces that are considerably larger than convex ones of finite type and strictly pseudoconvex hypersurfaces.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190142","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
Normalized Solutions to N-Laplacian Equations in $${mathbb {R}}^N$$ with Exponential Critical Growth 具有指数临界增长的 $${mathbb {R}}^N$ 中 N 拉普拉斯方程的归一化解
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01771-x
Jingbo Dou, Ling Huang, Xuexiu Zhong
{"title":"Normalized Solutions to N-Laplacian Equations in $${mathbb {R}}^N$$ with Exponential Critical Growth","authors":"Jingbo Dou, Ling Huang, Xuexiu Zhong","doi":"10.1007/s12220-024-01771-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01771-x","url":null,"abstract":"<p>In this paper, we are concerned with normalized solutions <span>((u,lambda )in W^{1,N}(mathbb {R}^N)times mathbb {R}^+)</span> to the following <i>N</i>-Laplacian problem </p><span>$$begin{aligned} -{text {div}}(|nabla u|^{N-2} nabla u)+lambda |u|^{N-2} u=f(u) text{ in } mathbb {R}^N,~N ge 2, end{aligned}$$</span><p>satisfying the normalization constraint <span>(int _{mathbb {R}^N}|u|^Ntextrm{d}x=c^N)</span>. The nonlinearity <i>f</i>(<i>s</i>) is an exponential critical growth function, i.e., behaves like <span>(exp (alpha |s|^{N /(N-1)}))</span> for some <span>(alpha &gt;0)</span> as <span>(|s| rightarrow infty )</span>. Under some mild conditions, we show the existence of normalized mountain pass type solution via the variational method. We also emphasize the normalized ground state solution has a mountain pass characterization under some further assumption. Our existence results in present paper also solve a Soave’s type open problem (J Funct Anal 279(6):108610, 2020) on the nonlinearities having an exponential critical growth.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190145","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
Large Steklov Eigenvalues Under Volume Constraints 体积约束下的大斯特克洛夫特征值
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01768-6
Alexandre Girouard, Panagiotis Polymerakis
{"title":"Large Steklov Eigenvalues Under Volume Constraints","authors":"Alexandre Girouard, Panagiotis Polymerakis","doi":"10.1007/s12220-024-01768-6","DOIUrl":"https://doi.org/10.1007/s12220-024-01768-6","url":null,"abstract":"<p>In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold with connected boundary diffeomorphic to a product admits a family of Riemannian metrics which coincide on the boundary, have fixed volume and arbitrarily large first non-zero Steklov eigenvalue. These are the first examples of Riemannian metrics with these properties on three-dimensional manifolds.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190143","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
Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula 各向异性亚历山德罗夫-芬切尔式不等式和熊-闵科夫斯基公式
The Journal of Geometric Analysis Pub Date : 2024-08-13 DOI: 10.1007/s12220-024-01759-7
Jinyu Gao, Guanghan Li
{"title":"Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula","authors":"Jinyu Gao, Guanghan Li","doi":"10.1007/s12220-024-01759-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01759-7","url":null,"abstract":"<p>In this paper, we introduce an anisotropic geometric quantity <span>(mathbb {W}_{p,q;k} )</span> which involves the weighted integral of <i>k</i>-th elementary symmetric function. We first show the monotonicity of <span>({mathbb {W}}_{p,1;k})</span> and <span>({mathbb {W}}_{0,q;k})</span> along a class of inverse anisotropic curvature flows, and then prove the generalization of anisotropic Alexandrov–Fenchel type inequalities. On the other hand, an extension of anisotropic Hsiung–Minkowski formula is derived. Therefore, we at last obtain an extension of the Alexandrov–Fenchel type inequality, which involve the general <span>(mathbb {W}_{p,q;k})</span>. In terms of the above inequalities, we have also demonstrated some other meaningful conclusions on convex body geometry, such as generalized <span>(L^p)</span>-Minkowski inequality and estimates of anisotropic <i>p</i>-affine surface area.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190151","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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