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Fourier Transform for $$L^p$$ -Functions with a Vector Measure on a Homogeneous Space of Compact Groups 紧凑群同质空间上具有向量量度的 $$L^p$$ 函数的傅里叶变换
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-04-11 DOI: 10.1007/s00041-024-10077-z
Sorravit Phonrakkhet, Keng Wiboonton
{"title":"Fourier Transform for $$L^p$$ -Functions with a Vector Measure on a Homogeneous Space of Compact Groups","authors":"Sorravit Phonrakkhet, Keng Wiboonton","doi":"10.1007/s00041-024-10077-z","DOIUrl":"https://doi.org/10.1007/s00041-024-10077-z","url":null,"abstract":"<p>Let <i>G</i> be a compact group and <i>G</i>/<i>H</i> a homogeneous space where <i>H</i> is a closed subgroup of <i>G</i>. Define an operator <span>(T_H:C(G) rightarrow C(G/H))</span> by <span>(T_Hf(tH)=int _H f(th) , dh)</span> for each <span>(tH in G/H)</span>. In this paper, we extend <span>(T_H)</span> to a norm-decreasing operator between <span>(L^p)</span>-spaces with a vector measure for each <span>(1 le p &lt;infty )</span>. This extension will be used to derive properties of invariant vector measures on <i>G</i>/<i>H</i>. Moreover, a definition of the Fourier transform for <span>(L^p)</span>-functions with a vector measure is introduced on <i>G</i>/<i>H</i>. We also prove the uniqueness theorem and the Riemann–Lebesgue lemma.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571295","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
An $$L^p$$ -Spectral Multiplier Theorem with Sharp p-Specific Regularity Bound on Heisenberg Type Groups 海森堡类型群上具有 p 特定锐正则约束的 $$L^p$$ 谱乘数定理
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-04-04 DOI: 10.1007/s00041-024-10075-1
{"title":"An $$L^p$$ -Spectral Multiplier Theorem with Sharp p-Specific Regularity Bound on Heisenberg Type Groups","authors":"","doi":"10.1007/s00041-024-10075-1","DOIUrl":"https://doi.org/10.1007/s00041-024-10075-1","url":null,"abstract":"<h3>Abstract</h3> <p>We prove an <span> <span>(L^p)</span> </span>-spectral multiplier theorem for sub-Laplacians on Heisenberg type groups under the sharp regularity condition <span> <span>(s&gt;dleft| 1/p-1/2right| )</span> </span>, where <em>d</em> is the topological dimension of the underlying group. Our approach relies on restriction type estimates where the multiplier is additionally truncated along the spectrum of the Laplacian on the center of the group.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571286","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
Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames 通过紧密 Gabor 框架对磁伪微分算子进行矩阵表示
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-04-04 DOI: 10.1007/s00041-024-10072-4
Horia D. Cornean, Bernard Helffer, Radu Purice
{"title":"Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames","authors":"Horia D. Cornean, Bernard Helffer, Radu Purice","doi":"10.1007/s00041-024-10072-4","DOIUrl":"https://doi.org/10.1007/s00041-024-10072-4","url":null,"abstract":"<p>In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on <span>(mathbb {R}^d)</span> (<span>(dge 1)</span>), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571291","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
Spherical Analysis Attached to Some m-Step Nilpotent Lie Group 附属于某些 m 阶无势李群的球面分析
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-04-02 DOI: 10.1007/s00041-024-10076-0
Silvina Campos, José García, Linda Saal
{"title":"Spherical Analysis Attached to Some m-Step Nilpotent Lie Group","authors":"Silvina Campos, José García, Linda Saal","doi":"10.1007/s00041-024-10076-0","DOIUrl":"https://doi.org/10.1007/s00041-024-10076-0","url":null,"abstract":"<p>We introduce a family of generalized Gelfand pairs <span>((K_m,N_m))</span> where <span>(N_m)</span> is an <span>(m+2)</span>-step nilpotent Lie group and <span>(K_m)</span> is isomorphic to the 3-dimensional Heisenberg group. We develop the associated spherical analysis computing the set of the spherical distributions and we obtain some results on the algebra of <span>(K_m)</span>-invariant and left invariant differential operators on <span>(N_m)</span>.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571293","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
Ideals in the Convolution Algebra of Periodic Distributions 周期分布卷积代数中的理想值
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-03-27 DOI: 10.1007/s00041-024-10078-y
Amol Sasane
{"title":"Ideals in the Convolution Algebra of Periodic Distributions","authors":"Amol Sasane","doi":"10.1007/s00041-024-10078-y","DOIUrl":"https://doi.org/10.1007/s00041-024-10078-y","url":null,"abstract":"<p>The ring of periodic distributions on <span>(mathbb {R}^{texttt {d}})</span> with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring <span>(mathcal {S}'(mathbb {Z}^{texttt {d}}))</span> of all maps <span>(f:mathbb {Z}^{texttt {d}}rightarrow mathbb {C})</span> of at most polynomial growth (that is, there exist a real number <span>(M&gt;0)</span> and an integer <span>(texttt {m}ge 0)</span> such that <span>( |f(varvec{n})|le M(1+|texttt{n}_1|+cdots +|texttt {n}_{texttt {d}}|)^{texttt {m}})</span> for all <span>(varvec{n}=(texttt{n}_1,cdots , texttt {n}_{texttt {d}})in mathbb {Z}^{texttt {d}})</span>), with pointwise operations. It is shown that finitely generated ideals in <span>(mathcal {S}'(mathbb {Z}^{texttt {d}}))</span> are principal, and ideal membership is characterised analytically. Calling an ideal in <span>(mathcal {S}'(mathbb {Z}^texttt{d}))</span> fixed if there is a common index <span>(varvec{n}in mathbb {Z}^{texttt {d}})</span> where each member vanishes, the fixed maximal ideals are described, and it is shown that not all maximal ideals are fixed. It is shown that finitely generated (hence principal) prime ideals in <span>(mathcal {S}'(mathbb {Z}^{texttt {d}}))</span> are fixed maximal ideals. The Krull dimension of <span>(mathcal {S}'(mathbb {Z}^{texttt {d}}))</span> is proved to be infinite, while the weak Krull dimension is shown to be equal to 1.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315028","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
Product of Sets on Varieties in Finite Fields 有限域中变量上的集合积
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-03-27 DOI: 10.1007/s00041-024-10079-x
{"title":"Product of Sets on Varieties in Finite Fields","authors":"","doi":"10.1007/s00041-024-10079-x","DOIUrl":"https://doi.org/10.1007/s00041-024-10079-x","url":null,"abstract":"<h3>Abstract</h3> <p>Let <em>V</em> be a variety in <span> <span>(mathbb {F}_q^d)</span> </span> and <span> <span>(Esubset V)</span> </span>. It is known that if any line passing through the origin contains a bounded number of points from <em>E</em>, then <span> <span>(left| prod (E) right| =|{xcdot y:x, yin E}|gg q)</span> </span> whenever <span> <span>(|E|gg q^{frac{d}{2}})</span> </span>. In this paper, we show that the barrier <span> <span>(frac{d}{2})</span> </span> can be broken when <em>V</em> is a paraboloid in some specific dimensions. The main novelty in our approach is to link this question to the distance problem in one lower dimensional vector space, allowing us to use recent developments in this area to obtain improvements.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315029","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
Universal Spectra in $$Gtimes {mathbb {Z}}_p$$ $$Gtimes {mathbb {Z}}_p$$ 中的通用光谱
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-03-21 DOI: 10.1007/s00041-024-10074-2
Weiqi Zhou
{"title":"Universal Spectra in $$Gtimes {mathbb {Z}}_p$$","authors":"Weiqi Zhou","doi":"10.1007/s00041-024-10074-2","DOIUrl":"https://doi.org/10.1007/s00041-024-10074-2","url":null,"abstract":"<p>Let <i>G</i> be an additive and finite Abelian group, and <i>p</i> a prime number that does not divide the order of <i>G</i>. We show that if <i>G</i> has the universal spectrum property, then so does <span>(Gtimes {mathbb {Z}}_p)</span>. This is similar to and extends a previous result for cyclic groups using the same dilation trick but on non-cyclic groups as well. Inductively applying this statement on the known list of permissible <i>G</i> one can replace <i>p</i> with any square-free number that does not divide the order of <i>G</i>, and produce new tiling to spectral results in finite Abelian groups generated by at most two elements.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201026","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
A Restriction Estimate with a Log-Concavity Assumption 对数凹假定下的限制估计值
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-03-14 DOI: 10.1007/s00041-024-10073-3
Kyoungtae Moon
{"title":"A Restriction Estimate with a Log-Concavity Assumption","authors":"Kyoungtae Moon","doi":"10.1007/s00041-024-10073-3","DOIUrl":"https://doi.org/10.1007/s00041-024-10073-3","url":null,"abstract":"<p>The purpose of this paper is to prove an optimal restriction estimate for a class of flat curves in <span>({mathbb {R}} ^d)</span>, <span>(dge 3)</span>. Namely, we consider the problem of determining all the pairs (<i>p</i>, <i>q</i>) for which the <span>(L^p-L^q)</span> estimate holds (or a suitable Lorentz norm substitute at the endpoint, where the <span>(L^p-L^q)</span> estimate fails) for the extension operator associated to <span>(gamma (t) = (t, {frac{t^2}{2!}}, ldots , {frac{t^{d-1}}{(d-1)!}}, phi (t)))</span>, <span>(0le tle 1)</span>, with respect to the affine arclength measure. In particular, we are interested in the flat case, i.e. when <span>(phi (t))</span> satisfies <span>(phi ^{(d)}(0) = 0)</span> for all integers <span>(dge 1)</span>. A prototypical example is given by <span>(phi (t) = e^{-1/t})</span>. The paper (Bak et al., J. Aust. Math. Soc. 85:1–28, 2008) addressed precisely this problem. The examples in Bak et al. (2008) are defined recursively in terms of an integral, and they represent progressively flatter curves. Although these include arbitrarily flat curves, it is not clear if they cover, for instance, the prototypical case <span>(phi (t) = e^{-1/t})</span>. We will show that the desired estimate does hold for that example and indeed for a class of examples satisfying some hypotheses involving a log-concavity condition.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147117","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
One-Dimensional Discrete Hardy and Rellich Inequalities on Integers 整数上的一维离散哈代不等式和雷利克不等式
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-03-08 DOI: 10.1007/s00041-024-10070-6
Shubham Gupta
{"title":"One-Dimensional Discrete Hardy and Rellich Inequalities on Integers","authors":"Shubham Gupta","doi":"10.1007/s00041-024-10070-6","DOIUrl":"https://doi.org/10.1007/s00041-024-10070-6","url":null,"abstract":"<p>In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form <span>(n^alpha )</span>. We prove the inequality when <span>(alpha )</span> is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities(with weights <span>(n^alpha )</span>) which are asymptotically sharp as <span>(alpha rightarrow infty )</span>. As a by-product of this work we derive a combinatorial identity using purely analytic methods, which suggests a plausible correlation between combinatorial and functional identities.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140073644","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
Shape Holomorphy of Boundary Integral Operators on Multiple Open Arcs 多个开放弧线上边界积分算子的形状全貌
IF 1.2 3区 数学
Journal of Fourier Analysis and Applications Pub Date : 2024-02-27 DOI: 10.1007/s00041-024-10071-5
José Pinto, Fernando Henríquez, Carlos Jerez-Hanckes
{"title":"Shape Holomorphy of Boundary Integral Operators on Multiple Open Arcs","authors":"José Pinto, Fernando Henríquez, Carlos Jerez-Hanckes","doi":"10.1007/s00041-024-10071-5","DOIUrl":"https://doi.org/10.1007/s00041-024-10071-5","url":null,"abstract":"<p>We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their solutions depend holomorphically upon perturbations of the arcs’ parametrizations. These results are key to prove the shape (domain) holomorphy of domain-to-solution maps associated to boundary integral equations appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140010656","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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