In this article, we provide two-weight inequalities for the heat flow on the whole space by applying the sparse domination. For power weights, such inequalities were given by several authors. Owing to the sparse domination, we can treat general weights in Muckenhoupt classes. As a application, we present the local and global existence results for the Hardy-H'enon parabolic equation, which has a potential belonging to a Muckenhoupt class.
{"title":"Two-weight inequality for the heat flow and solvability of Hardy-Hénon parabolic equation","authors":"Yohei Tsutsui","doi":"arxiv-2407.17704","DOIUrl":"https://doi.org/arxiv-2407.17704","url":null,"abstract":"In this article, we provide two-weight inequalities for the heat flow on the\u0000whole space by applying the sparse domination. For power weights, such\u0000inequalities were given by several authors. Owing to the sparse domination, we\u0000can treat general weights in Muckenhoupt classes. As a application, we present\u0000the local and global existence results for the Hardy-H'enon parabolic\u0000equation, which has a potential belonging to a Muckenhoupt class.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781067","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}
The signature of a $p$-weakly geometric rough path summarises a path up to a generalised notion of reparameterisation. The quotient space of equivalence classes on which the signature is constant yields unparameterised path space. The study of topologies on unparameterised path space, initiated in [CT24b] for paths of bounded variation, has practical bearing on the use of signature based methods in a variety applications. This note extends the majority of results from [CT24b] to unparameterised weakly geometric rough path space. We study three classes of topologies: metrisable topologies for which the quotient map is continuous; the quotient topology derived from the underlying path space; and an explicit metric between the tree-reduced representatives of each equivalence class. We prove that topologies of the first type (under an additional assumption) are separable and Lusin, but not locally compact or completely metrisable. The quotient topology is Hausdorff but not metrisable, while the metric generating the third topology is not complete and its topology is not locally compact. We also show that the third topology is Polish when $p=1$.
{"title":"Topologies on unparameterised rough path space","authors":"Thomas Cass, William F. Turner","doi":"arxiv-2407.17828","DOIUrl":"https://doi.org/arxiv-2407.17828","url":null,"abstract":"The signature of a $p$-weakly geometric rough path summarises a path up to a\u0000generalised notion of reparameterisation. The quotient space of equivalence\u0000classes on which the signature is constant yields unparameterised path space.\u0000The study of topologies on unparameterised path space, initiated in [CT24b] for\u0000paths of bounded variation, has practical bearing on the use of signature based\u0000methods in a variety applications. This note extends the majority of results\u0000from [CT24b] to unparameterised weakly geometric rough path space. We study\u0000three classes of topologies: metrisable topologies for which the quotient map\u0000is continuous; the quotient topology derived from the underlying path space;\u0000and an explicit metric between the tree-reduced representatives of each\u0000equivalence class. We prove that topologies of the first type (under an\u0000additional assumption) are separable and Lusin, but not locally compact or\u0000completely metrisable. The quotient topology is Hausdorff but not metrisable,\u0000while the metric generating the third topology is not complete and its topology\u0000is not locally compact. We also show that the third topology is Polish when\u0000$p=1$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781073","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}
We discuss the notion of resonance, as well as the existence and uniqueness of periodic solutions for a forced simple harmonic oscillator. While this topic is elementary, and well-studied for sinusoidal forcing, this does not seem to be the case when the forcing function is general (perhaps discontinuous). Clear statements of theorems and proofs do not readily appear in standard textbooks or online. For that reason, we provide a characterization of resonant solutions, written in terms of the relationship between the forcing and natural frequencies, as well as a condition on a particular Fourier mode. While our discussions involve some notions from $L^2$-spaces, our proofs are elementary, using this the variation of parameters formula; the main theorem and its proof should be readable by students who have completed a differential equations course and have some experience with analysis. We provide several examples, and give various constructions of resonant solutions. Additionally, we connect our discussion to notions of resonance in systems of partial differential equations, including fluid-structure interactions and partially damped systems.
{"title":"Resonance and Periodic Solutions for Harmonic Oscillators with General Forcing","authors":"Isaac Benson, Justin T. Webster","doi":"arxiv-2407.17144","DOIUrl":"https://doi.org/arxiv-2407.17144","url":null,"abstract":"We discuss the notion of resonance, as well as the existence and uniqueness\u0000of periodic solutions for a forced simple harmonic oscillator. While this topic\u0000is elementary, and well-studied for sinusoidal forcing, this does not seem to\u0000be the case when the forcing function is general (perhaps discontinuous). Clear\u0000statements of theorems and proofs do not readily appear in standard textbooks\u0000or online. For that reason, we provide a characterization of resonant\u0000solutions, written in terms of the relationship between the forcing and natural\u0000frequencies, as well as a condition on a particular Fourier mode. While our\u0000discussions involve some notions from $L^2$-spaces, our proofs are elementary,\u0000using this the variation of parameters formula; the main theorem and its proof\u0000should be readable by students who have completed a differential equations\u0000course and have some experience with analysis. We provide several examples, and\u0000give various constructions of resonant solutions. Additionally, we connect our\u0000discussion to notions of resonance in systems of partial differential\u0000equations, including fluid-structure interactions and partially damped systems.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781069","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}
We put forward a conical principle and a degeneracy locating principle of decoupling. The former generalises the Pramanik-Seeger argument used in the proof of decoupling for the light cone. The latter locates the degenerate part of a manifold and effectively reduces the decoupling problem to two extremes: non-degenerate case and totally degenerate case. Both principles aim to provide a new algebraic approach of reducing decoupling for new manifolds to decoupling for known manifolds.
{"title":"Two principles of decoupling","authors":"Jianhui Li, Tongou Yang","doi":"arxiv-2407.16108","DOIUrl":"https://doi.org/arxiv-2407.16108","url":null,"abstract":"We put forward a conical principle and a degeneracy locating principle of\u0000decoupling. The former generalises the Pramanik-Seeger argument used in the\u0000proof of decoupling for the light cone. The latter locates the degenerate part\u0000of a manifold and effectively reduces the decoupling problem to two extremes:\u0000non-degenerate case and totally degenerate case. Both principles aim to provide\u0000a new algebraic approach of reducing decoupling for new manifolds to decoupling\u0000for known manifolds.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785886","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}
We study the relative asymptotics of two sequences of multiple orthogonal polynomials corresponding to two Nikishin systems of measures on the real line, the second one of which is obtained from the first one perturbing the generating measures with non-negative integrable functions. Each Nikishin system consists of two measures.
{"title":"Relative asymptotics of multiple orthogonal polynomials for Nikishin systems of two measures","authors":"Abey López-García, Guillermo López Lagomasino","doi":"arxiv-2407.16738","DOIUrl":"https://doi.org/arxiv-2407.16738","url":null,"abstract":"We study the relative asymptotics of two sequences of multiple orthogonal\u0000polynomials corresponding to two Nikishin systems of measures on the real line,\u0000the second one of which is obtained from the first one perturbing the\u0000generating measures with non-negative integrable functions. Each Nikishin\u0000system consists of two measures.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781070","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}
A paper of Beckner, Carbery, Semmes, and Soria proved that the Fourier extension operator associated to the sphere cannot be weak-type bounded at the restriction endpoint $q = 2d/(d-1)$. We generalize their approach to prove that the extension operator associated with any $n$-dimensional quadratic manifold in $mathbb{R}^d$ cannot be weak-type bounded at $q = 2d/n$. The key step in generalizing the proof of Beckner, Carbery, Semmes, and Soria will be replacing Kakeya sets with what we will call $mathcal{N}$-Kakeya sets, where $mathcal{N}$ denotes a closed subset of the Grassmannian $text{Gr}(d-n,d)$. We define $mathcal{N}$-Kakeya sets to be subsets of $mathbb{R}^d$ containing a translate of every $d-n$-plane segment in $mathcal{N}$. We will prove that if $mathcal{N}$ is closed and $n$-dimensional, then there exists compact, measure zero $mathcal{N}$-Kakeya sets, generalizing the same result for standard Kakeya sets.
{"title":"Failure of weak-type endpoint restriction estimates for quadratic manifolds","authors":"Sam Craig","doi":"arxiv-2407.15034","DOIUrl":"https://doi.org/arxiv-2407.15034","url":null,"abstract":"A paper of Beckner, Carbery, Semmes, and Soria proved that the Fourier\u0000extension operator associated to the sphere cannot be weak-type bounded at the\u0000restriction endpoint $q = 2d/(d-1)$. We generalize their approach to prove that\u0000the extension operator associated with any $n$-dimensional quadratic manifold\u0000in $mathbb{R}^d$ cannot be weak-type bounded at $q = 2d/n$. The key step in\u0000generalizing the proof of Beckner, Carbery, Semmes, and Soria will be replacing\u0000Kakeya sets with what we will call $mathcal{N}$-Kakeya sets, where\u0000$mathcal{N}$ denotes a closed subset of the Grassmannian $text{Gr}(d-n,d)$.\u0000We define $mathcal{N}$-Kakeya sets to be subsets of $mathbb{R}^d$ containing\u0000a translate of every $d-n$-plane segment in $mathcal{N}$. We will prove that\u0000if $mathcal{N}$ is closed and $n$-dimensional, then there exists compact,\u0000measure zero $mathcal{N}$-Kakeya sets, generalizing the same result for\u0000standard Kakeya sets.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781072","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}
In this note we prove that the discrete Riesz potential $I_{alpha}$ defined on $mathbb{Z}^n$ is a bounded operator $H^p (mathbb{Z}^n) to ell^q (mathbb{Z}^n)$ for $0 < p leq 1$ and $frac{1}{q} = frac{1}{p} - frac{alpha}{n}$, where $0 < alpha < n$.
{"title":"A note about discrete Riesz potential on $mathbb{Z}^n$","authors":"Pablo Rocha","doi":"arxiv-2407.15262","DOIUrl":"https://doi.org/arxiv-2407.15262","url":null,"abstract":"In this note we prove that the discrete Riesz potential $I_{alpha}$ defined\u0000on $mathbb{Z}^n$ is a bounded operator $H^p (mathbb{Z}^n) to ell^q\u0000(mathbb{Z}^n)$ for $0 < p leq 1$ and $frac{1}{q} = frac{1}{p} -\u0000frac{alpha}{n}$, where $0 < alpha < n$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781155","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}
The aim of this paper is to give the answer to the problem of characterization of acting conditions (necessary as well as sufficient) for composition operators in some sequence spaces. We also characterize their boundedness and local boundedness. We focus on composition operators acting to or from the space $bv_p(E)$ of all sequences of $p$-bounded variation; here $pgeq 1$ and $E$ is a normed space.
{"title":"Composition operators in $bv_p$-spaces, part I: acting conditions and boundedness","authors":"Daria Bugajewska, Piotr Kasprzak","doi":"arxiv-2407.14176","DOIUrl":"https://doi.org/arxiv-2407.14176","url":null,"abstract":"The aim of this paper is to give the answer to the problem of\u0000characterization of acting conditions (necessary as well as sufficient) for\u0000composition operators in some sequence spaces. We also characterize their\u0000boundedness and local boundedness. We focus on composition operators acting to\u0000or from the space $bv_p(E)$ of all sequences of $p$-bounded variation; here\u0000$pgeq 1$ and $E$ is a normed space.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745018","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}
We address the Poincar'e-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order $nge 3$, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.
{"title":"Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions","authors":"Harold Bustos, Pablo Figueroa, Manuel Pinto","doi":"arxiv-2407.14444","DOIUrl":"https://doi.org/arxiv-2407.14444","url":null,"abstract":"We address the Poincar'e-Perron's classical problem of approximation for\u0000high order linear differential equations in the class of almost periodic type\u0000functions, extending the results for a second order linear differential\u0000equation in [23]. We obtain explicit formulae for solutions of these equations,\u0000for any fixed order $nge 3$, by studying a Riccati type equation associated\u0000with the logarithmic derivative of a solution. Moreover, we provide sufficient\u0000conditions to ensure the existence of a fundamental system of solutions. The\u0000fixed point Banach argument allows us to find almost periodic and\u0000asymptotically almost periodic solutions to this Riccati type equation. A\u0000decomposition property of the perturbations induces a decomposition on the\u0000Riccati type equation and its solutions. In particular, by using this\u0000decomposition we obtain asymptotically almost periodic and also $p$-almost\u0000periodic solutions to the Riccati type equation. We illustrate our results for\u0000a third order linear differential equation.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745016","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}
We find necessary and sufficient conditions on the function $Phi$ for the inequality $$Big|int_Omega Phi(K*f)Big|lesssim |f|_{L_1(mathbb{R}^d)}^p$$ to be true. Here $K$ is a positively homogeneous of order $alpha - d$, possibly vector valued, kernel, $Phi$ is a $p$-homogeneous function, and $p=d/(d-alpha)$. The domain $Omegasubset mathbb{R}^d$ is either bounded with $C^{1,beta}$ smooth boundary for some $beta > 0$ or a halfspace in $mathbb{R}^d$. As a corollary, we describe the positively homogeneous of order $d/(d-1)$ functions $Phicolon mathbb{R}^d to mathbb{R}$ that are suitable for the bound $$Big|int_Omega Phi(nabla u)Big|lesssim int_Omega |Delta u|.$$
{"title":"Maz'ya's $Φ$-inequalities on domains","authors":"Dmitriy Stolyarov","doi":"arxiv-2407.14052","DOIUrl":"https://doi.org/arxiv-2407.14052","url":null,"abstract":"We find necessary and sufficient conditions on the function $Phi$ for the\u0000inequality $$Big|int_Omega Phi(K*f)Big|lesssim\u0000|f|_{L_1(mathbb{R}^d)}^p$$ to be true. Here $K$ is a positively homogeneous\u0000of order $alpha - d$, possibly vector valued, kernel, $Phi$ is a\u0000$p$-homogeneous function, and $p=d/(d-alpha)$. The domain $Omegasubset\u0000mathbb{R}^d$ is either bounded with $C^{1,beta}$ smooth boundary for some\u0000$beta > 0$ or a halfspace in $mathbb{R}^d$. As a corollary, we describe the\u0000positively homogeneous of order $d/(d-1)$ functions $Phicolon mathbb{R}^d\u0000to mathbb{R}$ that are suitable for the bound $$Big|int_Omega Phi(nabla\u0000u)Big|lesssim int_Omega |Delta u|.$$","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745020","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}
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