Let $mu$ be a positive measure supported on a domain $Omega$. We consider the behavior of the balayage measure $nu:=mathrm{Bal}(mu,partial Omega)$ near a point $z_{0}in partial Omega$ at which $Omega$ has an outward-pointing cusp. Assuming that the order and coefficient of tangency of the cusp are $d>0$ and $a>0$, respectively, and that $dmu(z) asymp |z-z_{0}|^{2b-2}d^{2}z$ as $zto z_0$ for some $b > 0$, we obtain the leading order term of $nu$ near $z_{0}$. This leading term is universal in the sense that it only depends on $d$, $a$, and $b$. We also treat the case when the domain has multiple corners and cusps at the same point. Finally, we obtain an explicit expression for the balayage of the uniform measure on the tacnodal region between two osculating circles, and we give an application of this result to two-dimensional Coulomb gases.
{"title":"Balayage of measures: behavior near a cusp","authors":"Christophe Charlier, Jonatan Lenells","doi":"arxiv-2408.05487","DOIUrl":"https://doi.org/arxiv-2408.05487","url":null,"abstract":"Let $mu$ be a positive measure supported on a domain $Omega$. We consider\u0000the behavior of the balayage measure $nu:=mathrm{Bal}(mu,partial Omega)$\u0000near a point $z_{0}in partial Omega$ at which $Omega$ has an\u0000outward-pointing cusp. Assuming that the order and coefficient of tangency of\u0000the cusp are $d>0$ and $a>0$, respectively, and that $dmu(z) asymp\u0000|z-z_{0}|^{2b-2}d^{2}z$ as $zto z_0$ for some $b > 0$, we obtain the leading\u0000order term of $nu$ near $z_{0}$. This leading term is universal in the sense\u0000that it only depends on $d$, $a$, and $b$. We also treat the case when the\u0000domain has multiple corners and cusps at the same point. Finally, we obtain an\u0000explicit expression for the balayage of the uniform measure on the tacnodal\u0000region between two osculating circles, and we give an application of this\u0000result to two-dimensional Coulomb gases.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209771","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}
Yoshikazu Giga, Ayato Kubo, Hirotoshi Kuroda, Jun Okamoto, Koya Sakakibara
We consider a total variation type energy which measures the jump discontinuities different from usual total variation energy. Such a type of energy is obtained as a singular limit of the Kobayashi-Warren-Carter energy with minimization with respect to the order parameter. We consider the Rudin-Osher-Fatemi type energy by replacing relaxation term by this type of total variation energy. We show that all minimizers are piecewise constant if the data is continuous in one-dimensional setting. Moreover, the number of jumps is bounded by an explicit constant involving a constant related to the fidelity. This is quite different from conventional Rudin-Osher-Fatemi energy where a minimizer must have no jump if the data has no jumps. The existence of a minimizer is guaranteed in multi-dimensional setting when the data is bounded.
{"title":"Piecewise constant profiles minimizing total variation energies of Kobayashi-Warren-Carter type with fidelity","authors":"Yoshikazu Giga, Ayato Kubo, Hirotoshi Kuroda, Jun Okamoto, Koya Sakakibara","doi":"arxiv-2408.04228","DOIUrl":"https://doi.org/arxiv-2408.04228","url":null,"abstract":"We consider a total variation type energy which measures the jump\u0000discontinuities different from usual total variation energy. Such a type of\u0000energy is obtained as a singular limit of the Kobayashi-Warren-Carter energy\u0000with minimization with respect to the order parameter. We consider the\u0000Rudin-Osher-Fatemi type energy by replacing relaxation term by this type of\u0000total variation energy. We show that all minimizers are piecewise constant if\u0000the data is continuous in one-dimensional setting. Moreover, the number of\u0000jumps is bounded by an explicit constant involving a constant related to the\u0000fidelity. This is quite different from conventional Rudin-Osher-Fatemi energy\u0000where a minimizer must have no jump if the data has no jumps. The existence of\u0000a minimizer is guaranteed in multi-dimensional setting when the data is\u0000bounded.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930352","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 article we study the generalized Fourier dimension of the set of Liouville numbers $mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as control for the Fourier transform of (Rajchman) measures supported on $mathbb{L}$. We give an almost complete characterization of admissible decays for this set in terms of comparison to power-like functions. This work can be seen as the ``Fourier side'' of the analysis made by Olsen and Renfro regarding the generalized Hausdorff dimension using gauge functions. We also provide an approach to deal with the problem of classifying oscillating candidates for a Fourier decay for $mathbb{L}$ relying on its translation invariance property.
{"title":"The exact dimension of Liouville numbers: The Fourier side","authors":"Iván Polasek, Ezequiel Rela","doi":"arxiv-2408.04148","DOIUrl":"https://doi.org/arxiv-2408.04148","url":null,"abstract":"In this article we study the generalized Fourier dimension of the set of\u0000Liouville numbers $mathbb{L}$. Being a set of zero Hausdorff dimension, the\u0000analysis has to be done at the level of functions with a slow decay at infinity\u0000acting as control for the Fourier transform of (Rajchman) measures supported on\u0000$mathbb{L}$. We give an almost complete characterization of admissible decays\u0000for this set in terms of comparison to power-like functions. This work can be\u0000seen as the ``Fourier side'' of the analysis made by Olsen and Renfro regarding\u0000the generalized Hausdorff dimension using gauge functions. We also provide an\u0000approach to deal with the problem of classifying oscillating candidates for a\u0000Fourier decay for $mathbb{L}$ relying on its translation invariance property.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930349","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 $P$ denote the $3$-dimensional paraboloid over a finite field of odd characteristic in which $-1$ is not a square. We show that the associated Fourier extension operator maps $L^2$ to $L^{r}$ for $r > frac{24}{7} approx 3.428$. Previously this was known (in the case of prime order fields) for $r > frac{188}{53} approx 3.547$. In contrast with much of the recent progress on this problem, our argument does not use state-of-the-art incidence estimates but rather proceeds by obtaining estimates on a related bilinear operator. These estimates are based on a geometric result that, roughly speaking, states that a set of points in the finite plane $F^2$ can be decomposed as a union of sets each of which either contains a controlled number of rectangles or a controlled number of trapezoids.
{"title":"A bilinear approach to the finite field restriction problem","authors":"Mark Lewko","doi":"arxiv-2408.03514","DOIUrl":"https://doi.org/arxiv-2408.03514","url":null,"abstract":"Let $P$ denote the $3$-dimensional paraboloid over a finite field of odd\u0000characteristic in which $-1$ is not a square. We show that the associated\u0000Fourier extension operator maps $L^2$ to $L^{r}$ for $r > frac{24}{7} approx\u00003.428$. Previously this was known (in the case of prime order fields) for $r >\u0000frac{188}{53} approx 3.547$. In contrast with much of the recent progress on\u0000this problem, our argument does not use state-of-the-art incidence estimates\u0000but rather proceeds by obtaining estimates on a related bilinear operator.\u0000These estimates are based on a geometric result that, roughly speaking, states\u0000that a set of points in the finite plane $F^2$ can be decomposed as a union of\u0000sets each of which either contains a controlled number of rectangles or a\u0000controlled number of trapezoids.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930353","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}
Given a non-negative real sequence ${c_n}_n$ such that the series $sum_{n=1}^{infty}c_n$ diverges, it is known that the size of an infinite subset $Asubsetmathbb{N}$ can be measured in terms of the linear density such that the sub-series $sum_{nin A}c_n$ either (a) converges or (b) still diverges. The purpose of this research is to study these convergence/divergence questions by measuring the size of the set $Asubsetmathbb{N}$ in a more precise way in terms of the recently introduced asymptotic $psi$-density. The convergence of the associated sub-signed series $sum_{n=1 }^{infty}m_nc_n$ is also discussed, where ${m_n}_n$ is a real sequence with values restricted to the set ${-1, 0, 1}$.
{"title":"Convergence of sub-series' and sub-signed series' in terms of the asymptotic $ψ$-density","authors":"Janne Heittokangas, Zinelaabidine Latreuch","doi":"arxiv-2408.03973","DOIUrl":"https://doi.org/arxiv-2408.03973","url":null,"abstract":"Given a non-negative real sequence ${c_n}_n$ such that the series\u0000$sum_{n=1}^{infty}c_n$ diverges, it is known that the size of an infinite\u0000subset $Asubsetmathbb{N}$ can be measured in terms of the linear density such\u0000that the sub-series $sum_{nin A}c_n$ either (a) converges or (b) still\u0000diverges. The purpose of this research is to study these convergence/divergence\u0000questions by measuring the size of the set $Asubsetmathbb{N}$ in a more\u0000precise way in terms of the recently introduced asymptotic $psi$-density. The\u0000convergence of the associated sub-signed series $sum_{n=1 }^{infty}m_nc_n$ is\u0000also discussed, where ${m_n}_n$ is a real sequence with values restricted to\u0000the set ${-1, 0, 1}$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930350","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}
Existence of global mild solutions to the infinite dimensional Redner--ben-Avraham--Kahng cluster system is shown without growth or structure condition on the kinetic coefficients, thereby extending previous results in the literature. The key idea is to exploit the dissipative features of the system to derive a control on the tails of the infinite sums involved in the reaction terms. Classical solutions are also constructed for a suitable class of kinetic coefficients and initial conditions.
{"title":"The Redner-ben-Avraham-Kahng cluster system without growth condition on the kinetic coefficients","authors":"Philippe LaurençotLAMA","doi":"arxiv-2408.02465","DOIUrl":"https://doi.org/arxiv-2408.02465","url":null,"abstract":"Existence of global mild solutions to the infinite dimensional\u0000Redner--ben-Avraham--Kahng cluster system is shown without growth or structure\u0000condition on the kinetic coefficients, thereby extending previous results in\u0000the literature. The key idea is to exploit the dissipative features of the\u0000system to derive a control on the tails of the infinite sums involved in the\u0000reaction terms. Classical solutions are also constructed for a suitable class\u0000of kinetic coefficients and initial conditions.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930354","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 construct a compact set in R2 of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a range of $p$, $q$. Moreover, our construction can be generalised from parabolas to a family of curves with cinematic curvature.
{"title":"Construction of a curved Kakeya set","authors":"Tongou Yang, Yue Zhong","doi":"arxiv-2408.01917","DOIUrl":"https://doi.org/arxiv-2408.01917","url":null,"abstract":"We construct a compact set in R2 of measure 0 containing a piece of a\u0000parabola of every aperture between 1 and 2. As a consequence, we improve lower\u0000bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a\u0000range of $p$, $q$. Moreover, our construction can be generalised from parabolas\u0000to a family of curves with cinematic curvature.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930355","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 extend a recently derived optimal Hardy inequality in integral form on finite intervals by Dimitrov, Gadjev, and Ismail cite{DGI24} to the case of additional power weights and then derive an optimal power-weighted Hardy inequality in differential form on finite intervals, noting that the optimal constant of the latter inequality differs from the former. We also derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on spherical shell domains.
{"title":"Optimal power-weighted Hardy inequalities on finite intervals","authors":"Fritz Gesztesy, Michael M. H. Pang","doi":"arxiv-2408.01884","DOIUrl":"https://doi.org/arxiv-2408.01884","url":null,"abstract":"We extend a recently derived optimal Hardy inequality in integral form on\u0000finite intervals by Dimitrov, Gadjev, and Ismail cite{DGI24} to the case of\u0000additional power weights and then derive an optimal power-weighted Hardy\u0000inequality in differential form on finite intervals, noting that the optimal\u0000constant of the latter inequality differs from the former. We also derive an\u0000optimal multi-dimensional version of the power-weighted Hardy inequality in\u0000differential form on spherical shell domains.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930356","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}
Elementary, but very useful lemma due to Biernacki and Krzy.{z} (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their corresponding coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficient conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficient conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity.
{"title":"Unimodality preservation by ratios of functional series and integral transforms","authors":"Dmitrii Karp, Anna Vishnyakova, Yi Zhang","doi":"arxiv-2408.01755","DOIUrl":"https://doi.org/arxiv-2408.01755","url":null,"abstract":"Elementary, but very useful lemma due to Biernacki and Krzy.{z} (1955)\u0000asserts that the ratio of two power series inherits monotonicity from that of\u0000the sequence of ratios of their corresponding coefficients. Over the last two\u0000decades it has been realized that, under some additional assumptions, similar\u0000claims hold for more general series ratios as well as for unimodality in place\u0000of monotonicity. This paper continues this line of research: we consider ratios\u0000of general functional series and integral transforms and furnish natural\u0000sufficient conditions for preservation of unimodality by such ratios. Numerous\u0000series and integral transforms appearing in applications satisfy our sufficient\u0000conditions, including Dirichlet, factorial and inverse factorial series,\u0000Laplace, Mellin and generalized Stieltjes transforms, among many others.\u0000Finally, we illustrate our general results by exhibiting certain statements on\u0000monotonicity patterns for ratios of some special functions. The key role in our\u0000considerations is played by the notion of sign regularity.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"85 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930357","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}
P. K. Barik, F. P. da Costa, J. T. Pinto, R. Sasportes
We study a discrete model for generalized exchange-driven growth in which the particle exchanged between two clusters is not limited to be of size one. This set of models include as special cases the usual exchange-driven growth system and the coagulation-fragmentation system with binary fragmentation. Under reasonable general condition on the rate coefficients we establish the existence of admissible solutions, meaning solutions that are obtained as appropriate limit of solutions to a finite-dimensional truncation of the infinite-dimensional ODE. For these solutions we prove that, in the class of models we call isolated both the total number of particles and the total mass are conserved, whereas in those models we can non-isolated only the mass is conserved. Additionally, under more restrictive growth conditions for the rate equations we obtain uniqueness of solutions to the initial value problems.
{"title":"The discrete generalized exchange-driven system","authors":"P. K. Barik, F. P. da Costa, J. T. Pinto, R. Sasportes","doi":"arxiv-2408.00345","DOIUrl":"https://doi.org/arxiv-2408.00345","url":null,"abstract":"We study a discrete model for generalized exchange-driven growth in which the\u0000particle exchanged between two clusters is not limited to be of size one. This\u0000set of models include as special cases the usual exchange-driven growth system\u0000and the coagulation-fragmentation system with binary fragmentation. Under\u0000reasonable general condition on the rate coefficients we establish the\u0000existence of admissible solutions, meaning solutions that are obtained as\u0000appropriate limit of solutions to a finite-dimensional truncation of the\u0000infinite-dimensional ODE. For these solutions we prove that, in the class of\u0000models we call isolated both the total number of particles and the total mass\u0000are conserved, whereas in those models we can non-isolated only the mass is\u0000conserved. Additionally, under more restrictive growth conditions for the rate\u0000equations we obtain uniqueness of solutions to the initial value problems.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882717","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：