Pub Date : 2019-09-18DOI: 10.2422/2036-2145.201910_008
J. Canto, Kangwei Li, L. Roncal, Olli Tapiola
We consider Coifman--Fefferman inequalities for rough homogeneous singular integrals $T_Omega$ and $C_p$ weights. It was recently shown by Li-Perez-Rivera-Rios-Roncal that $$ |T_Omega |_{L^p(w)} le C_{p,T,w} |Mf|_{L^p(w)} $$ for every $0 max{1,p}$ without using extrapolation theory. Although the bounds we prove are new even in a qualitative sense, we also give the quantitative bound with respect to the $C_q$ characteristic. Our techniques rely on recent advances in sparse domination theory and we actually prove most of our estimates for sparse forms. Our second goal is to continue the structural analysis of $C_p$ classes. We consider some weak self-improving properties of $C_p$ weights and weak and dyadic $C_p$ classes. We also revisit and generalize a counterexample by Kahanpaa and Mejlbro who showed that $C_p setminus bigcup_{q > p} C_q neq emptyset$. We combine their construction with techniques of Lerner to define an explicit weight class $widetilde{C}_p$ such that $bigcup_{q > p} C_q subsetneq widetilde{C}_p subsetneq C_p$ and every $w in widetilde{C}_p$ satisfies Muckenhoupt's conjecture. In particular, we give a different, self-contained proof for the fact that the $C_{p+varepsilon}$ condition is not necessary for the Coifman--Fefferman inequality and our ideas allow us to consider also dimensions higher than $1$.
{"title":"$C_p$ estimates for rough homogeneous singular integrals and sparse forms","authors":"J. Canto, Kangwei Li, L. Roncal, Olli Tapiola","doi":"10.2422/2036-2145.201910_008","DOIUrl":"https://doi.org/10.2422/2036-2145.201910_008","url":null,"abstract":"We consider Coifman--Fefferman inequalities for rough homogeneous singular integrals $T_Omega$ and $C_p$ weights. It was recently shown by Li-Perez-Rivera-Rios-Roncal that $$ \u0000|T_Omega |_{L^p(w)} le C_{p,T,w} |Mf|_{L^p(w)} $$ for every $0 max{1,p}$ without using extrapolation theory. Although the bounds we prove are new even in a qualitative sense, we also give the quantitative bound with respect to the $C_q$ characteristic. Our techniques rely on recent advances in sparse domination theory and we actually prove most of our estimates for sparse forms. \u0000Our second goal is to continue the structural analysis of $C_p$ classes. We consider some weak self-improving properties of $C_p$ weights and weak and dyadic $C_p$ classes. We also revisit and generalize a counterexample by Kahanpaa and Mejlbro who showed that $C_p setminus bigcup_{q > p} C_q neq emptyset$. We combine their construction with techniques of Lerner to define an explicit weight class $widetilde{C}_p$ such that $bigcup_{q > p} C_q subsetneq widetilde{C}_p subsetneq C_p$ and every $w in widetilde{C}_p$ satisfies Muckenhoupt's conjecture. In particular, we give a different, self-contained proof for the fact that the $C_{p+varepsilon}$ condition is not necessary for the Coifman--Fefferman inequality and our ideas allow us to consider also dimensions higher than $1$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80412492","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 : 2019-09-10DOI: 10.1016/j.jfa.2020.108532
M. Strzelecki
{"title":"Hardy's operator minus identity and power weights","authors":"M. Strzelecki","doi":"10.1016/j.jfa.2020.108532","DOIUrl":"https://doi.org/10.1016/j.jfa.2020.108532","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73193184","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 : 2019-09-10DOI: 10.1016/j.aim.2020.107144
João P. G. Ramos, Olli Saari, Julian Weigt
{"title":"Weak differentiability for fractional maximal functions of general L functions on domains","authors":"João P. G. Ramos, Olli Saari, Julian Weigt","doi":"10.1016/j.aim.2020.107144","DOIUrl":"https://doi.org/10.1016/j.aim.2020.107144","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86542633","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 : 2019-08-28DOI: 10.1142/s0218348x2050108x
L. Brown, Giovanni Ferrer, Gamal Mograby, Luke G. Rogers, K. Sangam
We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $mu$ we show there is a full $mu$-measure set on which continuity of the Laplacian implies existence of the gradient $nabla u$, and that this set is not all of $SG_N$. We also show there is a class of non-uniform measures on the usual Sierpinski Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere, in sharp contrast to the case with the standard measure.
{"title":"HARMONIC GRADIENTS ON HIGHER-DIMENSIONAL SIERPIŃSKI GASKETS","authors":"L. Brown, Giovanni Ferrer, Gamal Mograby, Luke G. Rogers, K. Sangam","doi":"10.1142/s0218348x2050108x","DOIUrl":"https://doi.org/10.1142/s0218348x2050108x","url":null,"abstract":"We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $mu$ we show there is a full $mu$-measure set on which continuity of the Laplacian implies existence of the gradient $nabla u$, and that this set is not all of $SG_N$. We also show there is a class of non-uniform measures on the usual Sierpinski Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere, in sharp contrast to the case with the standard measure.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83085281","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 : 2019-08-26DOI: 10.22075/IJNAA.2020.20005.2117
Pallavi U. Shikhare, Kishor D. Kucche, J. Sousa
In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $epsilon$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.
{"title":"On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations.","authors":"Pallavi U. Shikhare, Kishor D. Kucche, J. Sousa","doi":"10.22075/IJNAA.2020.20005.2117","DOIUrl":"https://doi.org/10.22075/IJNAA.2020.20005.2117","url":null,"abstract":"In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $epsilon$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80317934","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 $B^p_{sigma}$, $1le p 0$, denote the space of all $fin L^p(mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-sigma,sigma]$. The classical sampling theorem states that each $fin B^p_{sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points ${pi m/sigma}_{minmathbb{Z}} $ spaced by $pi /sigma$. Reconstruction is also possible from sample values at sampling points ${pi theta m/sigma}_m $ with certain $1< thetale 2$ if we know $f(thetapi m/sigma) $ and $f'(thetapi m/sigma)$, $minmathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.
{"title":"On a sampling expansion with partial derivatives for functions of several variables","authors":"S. Norvidas","doi":"10.2298/FIL2010339N","DOIUrl":"https://doi.org/10.2298/FIL2010339N","url":null,"abstract":"Let $B^p_{sigma}$, $1le p 0$, denote the space of all $fin L^p(mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-sigma,sigma]$. The classical sampling theorem states that each $fin B^p_{sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points ${pi m/sigma}_{minmathbb{Z}} $ spaced by $pi /sigma$. Reconstruction is also possible from sample values at sampling points ${pi theta m/sigma}_m $ with certain $1< thetale 2$ if we know $f(thetapi m/sigma) $ and $f'(thetapi m/sigma)$, $minmathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80166623","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 show that two-weight $L^2$ bounds for sparse square functions, uniformly with respect to the sparseness constant of the underlying sparse family, and in both directions, do not imply a two-weight $L^2$ bound for the Hilbert transform. We present an explicit example, making use of the construction due to Reguera--Thiele from [18]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions of the involved weights for $p=2$ (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of $Llog L$ bumps by Orlicz bumps (for Young functions satisfying an appropriate integrability condition) observed by Treil--Volberg in [20].
{"title":"Two-weight estimates for sparse square functions and the separated bump conjecture","authors":"S. Kakaroumpas","doi":"10.1090/tran/8524","DOIUrl":"https://doi.org/10.1090/tran/8524","url":null,"abstract":"We show that two-weight $L^2$ bounds for sparse square functions, uniformly with respect to the sparseness constant of the underlying sparse family, and in both directions, do not imply a two-weight $L^2$ bound for the Hilbert transform. We present an explicit example, making use of the construction due to Reguera--Thiele from [18]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions of the involved weights for $p=2$ (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of $Llog L$ bumps by Orlicz bumps (for Young functions satisfying an appropriate integrability condition) observed by Treil--Volberg in [20].","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75446854","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 generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{rm div}(Anabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_infty$ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition then $omega_Lin A_infty$ if and only if $omega_{L^top}in A_infty$. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $L$ to the class $A_infty$ yields that the domain is indeed a chord-arc domain.
{"title":"Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates","authors":"J. Cavero, S. Hofmann, J. M. Martell, T. Toro","doi":"10.1090/tran/8148","DOIUrl":"https://doi.org/10.1090/tran/8148","url":null,"abstract":"We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{rm div}(Anabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_infty$ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition then $omega_Lin A_infty$ if and only if $omega_{L^top}in A_infty$. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $L$ to the class $A_infty$ yields that the domain is indeed a chord-arc domain.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81016140","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 : 2019-07-21DOI: 10.1016/J.JFA.2021.109076
S. Steinerberger
{"title":"A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds","authors":"S. Steinerberger","doi":"10.1016/J.JFA.2021.109076","DOIUrl":"https://doi.org/10.1016/J.JFA.2021.109076","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84252449","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 : 2019-07-12DOI: 10.2140/tunis.2021.3.517
R. Han, M. Lacey, Fan Yang
Let $fin ell^2(mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=frac{1}{N}sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ ell ^{p}$-improving estimate, for $ 3/2 < p leq 2$: begin{equation*} N ^{-2/p'} lVert A_N f rVert _{ p'} lesssim N ^{-2/p} lVert frVert _{ell ^{p}}, end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = sup _{Ngeq 1} |A_Nf| $ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $ A$ follow. A critical step in the proof requires the control of a logarithmic average over $ q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$.
让$fin ell^2(mathbb Z)$。定义$ f$除以平方整数的平均值$ A_N f(x):=frac{1}{N}sum_{k=1}^N f(x+k^2) $。我们证明$ A_N$满足一个局部无标度$ ell ^{p}$ -改进估计，对于$ 3/2 < p leq 2$: begin{equation*} N ^{-2/p'} lVert A_N f rVert _{ p'} lesssim N ^{-2/p} lVert frVert _{ell ^{p}}, end{equation*}，假设$ f$在长度为$ N ^2 $的某个区间内被支持，并且$ p' =frac{p} {p-1}$是共轭指标。上面的不等式对于$ 1< p < 3/2$无效。极大函数$ A f = sup _{Ngeq 1} |A_Nf| $满足类似的稀疏界。接下来是$ A$的新的加权和向量值不等式。证明中的一个关键步骤需要控制一个函数$G(q,x)$对$ q$的对数平均值(计算$x$ mod $q$的平方根的个数)。在$x$中需要一个估计制服。
{"title":"Averages along the Square Integers: $ell^p$ improving and Sparse Inequalities","authors":"R. Han, M. Lacey, Fan Yang","doi":"10.2140/tunis.2021.3.517","DOIUrl":"https://doi.org/10.2140/tunis.2021.3.517","url":null,"abstract":"Let $fin ell^2(mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=frac{1}{N}sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ ell ^{p}$-improving estimate, for $ 3/2 < p leq 2$: begin{equation*} \u0000N ^{-2/p'} lVert A_N f rVert _{ p'} lesssim N ^{-2/p} lVert frVert _{ell ^{p}}, end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = sup _{Ngeq 1} |A_Nf| $ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $ A$ follow. A critical step in the proof requires the control of a logarithmic average over $ q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83939301","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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