Pub Date : 2020-04-10DOI: 10.1142/s0218348x20501431
K. Hare, K. Hare
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.
{"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}
Pub Date : 2020-04-07DOI: 10.1142/s0219199720500728
Daniel Spector, Cody B. Stockdale
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*} |{|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.
设$R_j$表示$mathbb{R}^n$上的$j^{text{th}}$ Riesz变换。我们证明了存在一个绝对常数$C>0$，使得$lambda>0$和$f in L^1(mathbb{R}^n)$的begin{align*} |{|R_jf|>lambda}|leq Cleft(frac{1}{lambda}|f|_{L^1(mathbb{R}^n)}+sup_{nu} |{|R_jnu|>lambda}|right) end{align*}，其中上述至上被$N in mathbb{N}$、$c_k in mathbb{R}^n$和$a_k in mathbb{R}^+$的$nu=sum_{k=1}^Na_kdelta_{c_k}$形式的措施取代为$sum_{k=1}^N a_k leq 16|f|_{L^1(mathbb{R}^n)}$。这表明，为了建立Riesz变换的弱型$(1,1)$不等式的量纲估计，研究应用于Dirac质量有限线性组合的Riesz变换的相应弱型不等式就足够了。我们利用这一事实给出了最著名的维数上界的一个新的证明，同时我们的约简结果也适用于一类更一般的Calderon-Zygmund算子。
{"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}
Pub Date : 2020-04-02DOI: 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}
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 $r=2$.
{"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}
Pub Date : 2020-03-20DOI: 10.2422/2036-2145.202005_012
A. Julia, Sebastiano Golo
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.
用平面锥代替开锥，给出了第一Heisenberg群拓扑曲面为本征Lipschitz图的几何判据。
{"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}
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.
{"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}
Pub Date : 2020-02-25DOI: 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}
Pub Date : 2020-02-18DOI: 10.1142/s1793042120400187
M. Schlosser, K. Senapati, A. Uncu
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.
{"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}
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.
本文证明了帧谱测度的熵维数优于或等于其帧谱的伯林维数。
{"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}
In this paper we discuss and prove some new strong convergence theorems for partial sums and Fejer means with respect to the Vilenkin system.
本文讨论并证明了关于Vilenkin系统的部分和和Fejer均值的一些新的强收敛性定理。
{"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}
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