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INTERMEDIATE ASSOUAD-LIKE DIMENSIONS FOR MEASURES 用于度量的中间类尺度
arXiv: Classical Analysis and ODEs Pub Date : 2020-04-10 DOI: 10.1142/s0218348x20501431
K. Hare, K. Hare
{"title":"INTERMEDIATE ASSOUAD-LIKE DIMENSIONS FOR MEASURES","authors":"K. Hare, K. Hare","doi":"10.1142/s0218348x20501431","DOIUrl":"https://doi.org/10.1142/s0218348x20501431","url":null,"abstract":"The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the `thickest' and `thinnest' parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, $theta $-Assouad spectrum, and $Phi $-dimensions. In this paper, we study the analogue of the upper and lower $Phi $-dimensions for measures. We give general properties of such dimensions, as well as more specific results for self-similar measures satisfying various separation properties and discrete measures.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"165 1-4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91479064","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}
引用次数: 5
On the dimensional weak-type (1,1) bound for Riesz transforms 关于Riesz变换的维度弱型(1,1)界
arXiv: Classical Analysis and ODEs Pub Date : 2020-04-07 DOI: 10.1142/s0219199720500728
Daniel Spector, Cody B. Stockdale
{"title":"On the dimensional weak-type (1,1) bound for Riesz transforms","authors":"Daniel Spector, Cody B. Stockdale","doi":"10.1142/s0219199720500728","DOIUrl":"https://doi.org/10.1142/s0219199720500728","url":null,"abstract":"Let $R_j$ denote the $j^{text{th}}$ Riesz transform on $mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that begin{align*} \u0000|{|R_jf|>lambda}|leq Cleft(frac{1}{lambda}|f|_{L^1(mathbb{R}^n)}+sup_{nu} |{|R_jnu|>lambda}|right) end{align*} for any $lambda>0$ and $f in L^1(mathbb{R}^n)$, where the above supremum is taken over measures of the form $nu=sum_{k=1}^Na_kdelta_{c_k}$ for $N in mathbb{N}$, $c_k in mathbb{R}^n$, and $a_k in mathbb{R}^+$ with $sum_{k=1}^N a_k leq 16|f|_{L^1(mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderon-Zygmund operators.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84774888","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}
引用次数: 0
An Introduction to the Gabor Wave Front Set 介绍Gabor波前集
arXiv: Classical Analysis and ODEs Pub Date : 2020-04-02 DOI: 10.1007/978-3-030-61346-4_17
L. Rodino, S. I. Trapasso
{"title":"An Introduction to the Gabor Wave Front Set","authors":"L. Rodino, S. I. Trapasso","doi":"10.1007/978-3-030-61346-4_17","DOIUrl":"https://doi.org/10.1007/978-3-030-61346-4_17","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82493850","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}
引用次数: 4
Characterization of multilinear multipliers in terms of Sobolev space regularity 基于Sobolev空间正则性的多线性乘法器的表征
arXiv: Classical Analysis and ODEs Pub Date : 2020-03-26 DOI: 10.1090/TRAN/8430
L. Grafakos, Bae Jun Park
{"title":"Characterization of multilinear multipliers in terms of Sobolev space regularity","authors":"L. Grafakos, Bae Jun Park","doi":"10.1090/TRAN/8430","DOIUrl":"https://doi.org/10.1090/TRAN/8430","url":null,"abstract":"We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in $L^r$-based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case $1 2$ cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous results of Miyachi, Nguyen, Tomita, and the first author, who only considered the case $r=2$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89612103","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}
引用次数: 7
Intrinsic rectifiability via flat cones in the Heisenberg group 海森堡群中平面锥的本征可整流性
arXiv: Classical Analysis and ODEs Pub Date : 2020-03-20 DOI: 10.2422/2036-2145.202005_012
A. Julia, Sebastiano Golo
{"title":"Intrinsic rectifiability via flat cones in the Heisenberg group","authors":"A. Julia, Sebastiano Golo","doi":"10.2422/2036-2145.202005_012","DOIUrl":"https://doi.org/10.2422/2036-2145.202005_012","url":null,"abstract":"We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76290913","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}
引用次数: 1
Volterra-Choquet nonlinear operators Volterra-Choquet非线性算子
arXiv: Classical Analysis and ODEs Pub Date : 2020-02-28 DOI: 10.12775/TMNA.2020.009
S. Gal
{"title":"Volterra-Choquet nonlinear operators","authors":"S. Gal","doi":"10.12775/TMNA.2020.009","DOIUrl":"https://doi.org/10.12775/TMNA.2020.009","url":null,"abstract":"In this paper we study to what extend some properties of the classical linear Volterra operators could be transferred to the nonlinear Volterra-Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclicity properties are studied.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79763622","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}
引用次数: 1
Self-improvement of weighted pointwise inequalities on open sets 开集上加权点型不等式的自我改进
arXiv: Classical Analysis and ODEs Pub Date : 2020-02-25 DOI: 10.1016/j.jfa.2020.108691
S. Eriksson-Bique, Juha Lehrbäck, Antti V. Vähäkangas
{"title":"Self-improvement of weighted pointwise inequalities on open sets","authors":"S. Eriksson-Bique, Juha Lehrbäck, Antti V. Vähäkangas","doi":"10.1016/j.jfa.2020.108691","DOIUrl":"https://doi.org/10.1016/j.jfa.2020.108691","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85384079","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}
引用次数: 1
Log-concavity results for a biparametric and an elliptic extension of the q-binomial coefficients 双参数和q-二项式系数的椭圆扩展的对数凹性结果
arXiv: Classical Analysis and ODEs Pub Date : 2020-02-18 DOI: 10.1142/s1793042120400187
M. Schlosser, K. Senapati, A. Uncu
{"title":"Log-concavity results for a biparametric and an elliptic extension of the q-binomial coefficients","authors":"M. Schlosser, K. Senapati, A. Uncu","doi":"10.1142/s1793042120400187","DOIUrl":"https://doi.org/10.1142/s1793042120400187","url":null,"abstract":"We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turan's inequality.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"475 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74561884","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}
引用次数: 4
On dimensions of frame spectral measures and their frame spectra 帧谱测度的维数及其帧谱
arXiv: Classical Analysis and ODEs Pub Date : 2020-02-10 DOI: 10.5186/aasfm.2021.4629
Ruxi Shi
{"title":"On dimensions of frame spectral measures and their frame spectra","authors":"Ruxi Shi","doi":"10.5186/aasfm.2021.4629","DOIUrl":"https://doi.org/10.5186/aasfm.2021.4629","url":null,"abstract":"In this paper, we prove that the entropy dimension of a frame spectral measure is superior than or equal to the Beurling dimension of its frame spectrum.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82770163","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}
引用次数: 9
Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems 关于Vilenkin系统fejsamr均值强收敛的一些新结果
arXiv: Classical Analysis and ODEs Pub Date : 2020-02-04 DOI: 10.37863/UMZH.V73I4.226
L. Persson, G. Tephnadze, G. Tutberidze, P. Wall
{"title":"Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems","authors":"L. Persson, G. Tephnadze, G. Tutberidze, P. Wall","doi":"10.37863/UMZH.V73I4.226","DOIUrl":"https://doi.org/10.37863/UMZH.V73I4.226","url":null,"abstract":"In this paper we discuss and prove some new strong convergence theorems for partial sums and Fejer means with respect to the Vilenkin system.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76084246","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}
引用次数: 5
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