The Journal of Geometric Analysis最新文献

{"title":"Bilinear Operators on Ball Banach Function Spaces","authors":"Kwok-Pun Ho","doi":"10.1007/s12220-024-01786-4","DOIUrl":"https://doi.org/10.1007/s12220-024-01786-4","url":null,"abstract":"<p>This paper establishes the mapping properties of the bilinear operators on the ball Banach function spaces. The main result of this paper yields the mapping properties of the bilinear Fourier multipliers, the rough bilinear singular integrals and the bilinear Calderón–Zygmund operators on the ball Banach function spaces. As applications of the main result, we have the mapping properties of the bilinear Fourier multipliers, the rough bilinear singular integrals and the bilinear Calderón–Zygmund operators on the Morrey spaces and the Herz spaces.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190398","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}

{"title":"The Classical Boundary Blow-Up Solutions for a Class of Gaussian Curvature Equations","authors":"Haitao Wan","doi":"10.1007/s12220-024-01785-5","DOIUrl":"https://doi.org/10.1007/s12220-024-01785-5","url":null,"abstract":"<p>In this article, we consider the Gaussian curvature problem </p><span>$$begin{aligned} frac{hbox {det}(D^{2}u)}{(1+|nabla u|^{2})^{frac{N+2}{2}}}=b(x)f(u)g(|nabla u|);hbox {in};Omega ,,u=+infty ;hbox {on};partial Omega , end{aligned}$$</span><p>where <span>(Omega )</span> is a bounded smooth uniformly convex domain in <span>({mathbb {R}}^{N})</span> with <span>(Nge 2)</span>, <span>(bin mathrm C^{infty }(Omega ))</span> is positive in <span>(Omega )</span> and may be singular or vanish on <span>(partial Omega )</span>, <span>(fin C^{infty }[0, +infty ))</span> (or <span>(fin C^{infty }({mathbb {R}}))</span>) is positive and increasing on <span>([0, +infty ))</span> <span>((hbox {or } {mathbb {R}}))</span>, <span>(gin C^{infty }[0, +infty ))</span> is positive on <span>([0, +infty ))</span>. We first establish the existence and global estimates of <span>(C^{infty })</span>-strictly convex solutions to this problem by constructing some fine coupling (limit) structures on <i>f</i> and <i>g</i>. Our results (Theorems 2.1–2.3) clarify the influence of properties of <i>b</i> (on the boundary <span>(partial Omega )</span>) on the existence and global estimates, and reveal the relationship between the solutions of the above problem and some corresponding problems (some details see page 6–7). Then, the nonexistence of convex solutions and strictly convex solutions are also obtained (see Theorems 2.4 and C). Finally, we study the principal expansions of strictly convex solutions near <span>(partial Omega )</span> by analyzing some coupling structure and using the Karamata regular and rapid variation theories.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190144","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}

{"title":"Extension and Embedding of Triebel–Lizorkin-Type Spaces Built on Ball Quasi-Banach Spaces","authors":"Zongze Zeng, Dachun Yang, Wen Yuan","doi":"10.1007/s12220-024-01761-z","DOIUrl":"https://doi.org/10.1007/s12220-024-01761-z","url":null,"abstract":"<p>Let <span>(Omega subset mathbb {R}^n)</span> be a domain and <i>X</i> be a ball quasi-Banach function space with some extra mild assumptions. In this article, the authors establish the extension theorem about inhomogeneous <i>X</i>-based Triebel–Lizorkin-type spaces <span>(F^s_{X,q}(Omega ))</span> for any <span>(sin (0,1))</span> and <span>(qin (0,infty ))</span> and prove that <span>(Omega )</span> is an <span>(F^s_{X,q}(Omega ))</span>-extension domain if and only if <span>(Omega )</span> satisfies the measure density condition. The authors also establish the Sobolev embedding about <span>(F^s_{X,q}(Omega ))</span> with an extra mild assumption, that is, <i>X</i> satisfies the extra <span>(beta )</span>-doubling condition. These extension results when <i>X</i> is the Lebesgue space coincide with the known best ones of the fractional Sobolev space and the Triebel–Lizorkin space. Moreover, all these results have a wide range of applications and, particularly, even when they are applied, respectively, to weighted Lebesgue spaces, Morrey spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, mixed-norm Lebesgue spaces, and Lorentz spaces, the obtained results are also new. The main novelty of this article exists in that the authors use the boundedness of the Hardy–Littlewood maximal operator and the extrapolation about <i>X</i> to overcome those essential difficulties caused by the deficiency of the explicit expression of the norm of <i>X</i>.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190402","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}

{"title":"Homogeneous Spaces in Hartree–Fock–Bogoliubov Theory","authors":"Claudia D. Alvarado, Eduardo Chiumiento","doi":"10.1007/s12220-024-01776-6","DOIUrl":"https://doi.org/10.1007/s12220-024-01776-6","url":null,"abstract":"<p>We study the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree–Fock–Bogoliubov theory. We show that the orbits of this action are reductive homogeneous spaces, and we give several equivalences that characterize when they are embedded submanifolds of natural ambient spaces. We use Lie theoretic arguments to prove that these orbits admit an invariant symplectic form. If, in addition, the operators in the orbits have finite spectrum, then we obtain that the orbits are actually Kähler homogeneous spaces.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190403","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}