Pub Date : 2020-08-10DOI: 10.1016/J.JMAA.2021.125235
Andrei Sipoş
{"title":"Rates of metastability for iterations on the unit interval","authors":"Andrei Sipoş","doi":"10.1016/J.JMAA.2021.125235","DOIUrl":"https://doi.org/10.1016/J.JMAA.2021.125235","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89759692","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}
Pub Date : 2020-08-05DOI: 10.1016/J.AIM.2021.107831
Philip T. Gressman
{"title":"L-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality","authors":"Philip T. Gressman","doi":"10.1016/J.AIM.2021.107831","DOIUrl":"https://doi.org/10.1016/J.AIM.2021.107831","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88174475","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 behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy's inequalities $$ sum_{k=1}^{n}Big(frac{1}{k}sum_{j=1}^{k}a_jBig)^2leq d_n,sum_{k=1}^{n}a_k^2, qquad (a_1,ldots,a_n) in mathbb{R}^n $$ and $$ int_{0}^{infty}Bigg(frac{1}{x}intlimits_{0}^{x}f(t),dtBigg)^2 dx leq c_n int_{0}^{infty} f^2(x),dx, fin mathcal{H}_n, $$ for the finite dimensional spaces $mathbb{R}^n$ and $mathcal{H}_n:={f,:, int_0^x f(t) dt =e^{-x/2},p(x) : pin mathcal{P}_n, p(0)=0}$, where $mathcal{P}_n$ is the set of real-valued algebraic polynomials of degree not exceeding $n$. The constants $d_n$ and $c_n$ are identified as the smallest eigenvalues of certain Jacobi matrices and the two-sided estimates for $d_n$ and $c_n$ of the form $$ 4-frac{c}{ln n} 0, $$ are established.
{"title":"Hardy’s inequalities in finite dimensional Hilbert spaces","authors":"D. Dimitrov, I. Gadjev, G. Nikolov, R. Uluchev","doi":"10.1090/PROC/15467","DOIUrl":"https://doi.org/10.1090/PROC/15467","url":null,"abstract":"We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy's inequalities $$ sum_{k=1}^{n}Big(frac{1}{k}sum_{j=1}^{k}a_jBig)^2leq d_n,sum_{k=1}^{n}a_k^2, qquad (a_1,ldots,a_n) in mathbb{R}^n $$ and $$ int_{0}^{infty}Bigg(frac{1}{x}intlimits_{0}^{x}f(t),dtBigg)^2 dx leq c_n int_{0}^{infty} f^2(x),dx, fin mathcal{H}_n, $$ for the finite dimensional spaces $mathbb{R}^n$ and $mathcal{H}_n:={f,:, int_0^x f(t) dt =e^{-x/2},p(x) : pin mathcal{P}_n, p(0)=0}$, where $mathcal{P}_n$ is the set of real-valued algebraic polynomials of degree not exceeding $n$. The constants $d_n$ and $c_n$ are identified as the smallest eigenvalues of certain Jacobi matrices and the two-sided estimates for $d_n$ and $c_n$ of the form $$ 4-frac{c}{ln n} 0, $$ are established.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88725846","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}
Pub Date : 2020-07-14DOI: 10.1016/J.MATPUR.2021.01.009
B. Bruce, D. O. Silva, Betsy Stovall
{"title":"Restriction inequalities for the hyperbolic hyperboloid","authors":"B. Bruce, D. O. Silva, Betsy Stovall","doi":"10.1016/J.MATPUR.2021.01.009","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.01.009","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86678681","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 the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painleve I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painleve equations. In particular, Painleve equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the "time variable" may depend on the type. We describe unified treatments for several possible "time variables".
{"title":"From Heun Class Equations to Painlevé Equations","authors":"J. Derezi'nski, A. Ishkhanyan, Adam Latosi'nski","doi":"10.3842/SIGMA.2021.056","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.056","url":null,"abstract":"In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painleve I-VI equations. \u0000Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painleve equations. In particular, Painleve equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the \"time variable\" may depend on the type. We describe unified treatments for several possible \"time variables\".","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81457493","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}
Pub Date : 2020-07-05DOI: 10.1142/S0219199721500048
Yangyang Zhang, Dachun Yang, Wen Yuan
Recently, both the bilinear decompositions $h^1(mathbb{R}^n)times mathrm{,bmo}(mathbb{R}^n) subset L^1 (mathbb{R}^n)+h_ast^Phi(mathbb{R}^n)$ and $h^1(mathbb{R}^n) times mathrm{bmo}(mathbb{R}^n) subset L^1 (mathbb{R}^n) + h^{log}(mathbb{R}^n)$ were established. In this article, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains the variant $h_ast^Phi(mathbb{R}^n)$ of the local Orlicz Hardy space introduced by A. Bonami and J. Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms and various maximal functions, which are new even for $h_ast^{Phi}(mathbb R^n)$. The relationships $h_ast^Phi(mathbb{R}^n) subsetneqq h^{log}(mathbb{R}^n)$ is also clarified.
{"title":"Real-variable characterizations of local Orlicz-slice Hardy spaces with application to bilinear decompositions","authors":"Yangyang Zhang, Dachun Yang, Wen Yuan","doi":"10.1142/S0219199721500048","DOIUrl":"https://doi.org/10.1142/S0219199721500048","url":null,"abstract":"Recently, both the bilinear decompositions $h^1(mathbb{R}^n)times mathrm{,bmo}(mathbb{R}^n) subset L^1 (mathbb{R}^n)+h_ast^Phi(mathbb{R}^n)$ and $h^1(mathbb{R}^n) times mathrm{bmo}(mathbb{R}^n) subset L^1 (mathbb{R}^n) + h^{log}(mathbb{R}^n)$ were established. In this article, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains the variant $h_ast^Phi(mathbb{R}^n)$ of the local Orlicz Hardy space introduced by A. Bonami and J. Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms and various maximal functions, which are new even for $h_ast^{Phi}(mathbb R^n)$. The relationships $h_ast^Phi(mathbb{R}^n) subsetneqq h^{log}(mathbb{R}^n)$ is also clarified.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87974617","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 consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305, 1971) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate.
{"title":"On the necessity of the constant rank condition forLpestimates","authors":"André Guerra, Bogdan Raiță","doi":"10.5802/crmath.105","DOIUrl":"https://doi.org/10.5802/crmath.105","url":null,"abstract":"We consider a generalization of the elliptic $L^p$-estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. (4) 88: 229-305, 1971) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83296733","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}
Let $Omega$ be a domain in $mathbb{R}^{d+1}$, $d geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $Omega$ satisfies a corkscrew condition and if $partial Omega$ is $d$-Ahlfors regular, i.e. Hausdorff measure $mathcal{H}^d(B(x,r) cap partial Omega) sim r^d$ for all $x in partial Omega$ and $0 < r < {rm diam}(partial Omega)$, then $partial Omega$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $Omega$ or (b) an $varepsilon$-approximation property for all $0 < varepsilon <1$ for every such function. Here we explore (a) and (b) when $partial Omega$ is not required to be Ahlfors regular. We first prove that (a) and (b) hold for any domain $Omega$ for which there exists a domain $widetilde Omega subset Omega$ such that $partial Omega subset partial widetilde Omega$ and $partial widetilde Omega$ is uniformly rectifiable. We next assume $Omega$ satisfies a corkscrew condition and $partial Omega$ satisfies a capacity density condition. Under these assumptions we prove conversely that the existence of such $widetilde Omega$ implies (a) and (b) hold on $Omega$ and give further characterizations of domains for which (a) or (b) holds. One is that harmonic measure satisfies a Carleson packing condition for diameters similar to the corona decompositionm proved equivalent to uniform rectifiability in [GMT]. The second characterization is reminiscent of the Carleson measure description of $H^{infty}$ interpolating sequences in the unit disc.
{"title":"Carleson measure estimates and $varepsilon$-approximation for bounded harmonic functions, without Ahlfors regularity assumptions","authors":"J. Garnett","doi":"10.4171/RMI/1288","DOIUrl":"https://doi.org/10.4171/RMI/1288","url":null,"abstract":"Let $Omega$ be a domain in $mathbb{R}^{d+1}$, $d geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $Omega$ satisfies a corkscrew condition and if $partial Omega$ is $d$-Ahlfors regular, i.e. Hausdorff measure $mathcal{H}^d(B(x,r) cap partial Omega) sim r^d$ for all $x in partial Omega$ and $0 < r < {rm diam}(partial Omega)$, then $partial Omega$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $Omega$ or (b) an $varepsilon$-approximation property for all $0 < varepsilon <1$ for every such function. Here we explore (a) and (b) when $partial Omega$ is not required to be Ahlfors regular. We first prove that (a) and (b) hold for any domain $Omega$ for which there exists a domain $widetilde Omega subset Omega$ such that $partial Omega subset partial widetilde Omega$ and $partial widetilde Omega$ is uniformly rectifiable. We next assume $Omega$ satisfies a corkscrew condition and $partial Omega$ satisfies a capacity density condition. Under these assumptions we prove conversely that the existence of such $widetilde Omega$ implies (a) and (b) hold on $Omega$ and give further characterizations of domains for which (a) or (b) holds. One is that harmonic measure satisfies a Carleson packing condition for diameters similar to the corona decompositionm proved equivalent to uniform rectifiability in [GMT]. The second characterization is reminiscent of the Carleson measure description of $H^{infty}$ interpolating sequences in the unit disc.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87367487","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}
Recently renewed interest in the Lobachevsky-type integrals and interesting identities involving the cardinal sine motivate an extension of the classical Parseval formula involving both periodic and non-periodic functions. We develop a version of the Parseval formula that is often more practical in applications and illustrate its use by extending recent results on Lobachevsky-type integrals. Some previously known, interesting identities are re-proved in a more transparent manner and new formulas for integrals involving cardinal sine and Bessel functions are given.
{"title":"Lobachevsky-type formulas via Fourier analysis","authors":"Runze Cai, Horst Hohberger, Mian Li","doi":"10.4171/EM/448","DOIUrl":"https://doi.org/10.4171/EM/448","url":null,"abstract":"Recently renewed interest in the Lobachevsky-type integrals and interesting identities involving the cardinal sine motivate an extension of the classical Parseval formula involving both periodic and non-periodic functions. We develop a version of the Parseval formula that is often more practical in applications and illustrate its use by extending recent results on Lobachevsky-type integrals. Some previously known, interesting identities are re-proved in a more transparent manner and new formulas for integrals involving cardinal sine and Bessel functions are given.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83809002","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}
Pub Date : 2020-06-10DOI: 10.1016/J.JFA.2021.109190
X. Duong, Ji Li, E. Sawyer, Manasa N. Vempati, B. Wick, Dongyong Yang
{"title":"A two weight inequality for Calder'{o}n-Zygmund operators on spaces of homogeneous type with applications","authors":"X. Duong, Ji Li, E. Sawyer, Manasa N. Vempati, B. Wick, Dongyong Yang","doi":"10.1016/J.JFA.2021.109190","DOIUrl":"https://doi.org/10.1016/J.JFA.2021.109190","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90890285","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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