Pub Date : 2020-06-08DOI: 10.2140/PJM.2021.310.375
Junfeng Li, Haixia Yu
In this paper, we determine the $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{gamma}(f,g)$ along a convex curve $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $frac{1}{p}+frac{1}{q}=frac{1}{r}$, and $r>frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{gamma}(f,g)$ along a convex curve $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$
{"title":"Bilinear Hilbert transforms and (sub)bilinear maximal functions along convex curves","authors":"Junfeng Li, Haixia Yu","doi":"10.2140/PJM.2021.310.375","DOIUrl":"https://doi.org/10.2140/PJM.2021.310.375","url":null,"abstract":"In this paper, we determine the $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{gamma}(f,g)$ along a convex curve $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $frac{1}{p}+frac{1}{q}=frac{1}{r}$, and $r>frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{gamma}(f,g)$ along a convex curve $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83418853","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 : 2020-06-05DOI: 10.1007/S11118-021-09903-6
Fabio Berra
{"title":"From A1 to $A_{infty }$: New Mixed Inequalities for Certain Maximal Operators","authors":"Fabio Berra","doi":"10.1007/S11118-021-09903-6","DOIUrl":"https://doi.org/10.1007/S11118-021-09903-6","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78096745","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 : 2020-06-04DOI: 10.1007/S12220-021-00666-5
Tongou Yang
{"title":"Uniform $$l^2$$-Decoupling in $$mathbb R^2$$ for Polynomials","authors":"Tongou Yang","doi":"10.1007/S12220-021-00666-5","DOIUrl":"https://doi.org/10.1007/S12220-021-00666-5","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89452755","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}
The aim of this paper is to provide a self-contained proof of a general case of the coarea inequality, also known as the Eilenberg inequality. The result is known, but we are not aware of any place that a proof would be written with all details. The known proof is based on a difficult result of Davies. Our proof is elementary and does not use Davies' theorem. Instead we use an elegant argument that we learned from Nazarov through MathOverflow. We also obtain some generalizations of the coarea inequality.
{"title":"The coarea inequality","authors":"Behnam Esmayli, P. Hajłasz","doi":"10.5186/aasfm.2021.4654","DOIUrl":"https://doi.org/10.5186/aasfm.2021.4654","url":null,"abstract":"The aim of this paper is to provide a self-contained proof of a general case of the coarea inequality, also known as the Eilenberg inequality. The result is known, but we are not aware of any place that a proof would be written with all details. The known proof is based on a difficult result of Davies. Our proof is elementary and does not use Davies' theorem. Instead we use an elegant argument that we learned from Nazarov through MathOverflow. We also obtain some generalizations of the coarea inequality.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"91 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81237511","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}
This paper investigates the existence of positive solutions for m-point p-Laplacian fractional boundary value problem involving Riemann Liouville fractional integral boundary conditions on the half line via the Leray-Schauder Nonlinear Alternative theorem and the use and some properties of the Green function. As an application, an example is presented to demonstrate our main result.
{"title":"Positive solutions for m-point p-Laplacian fractional boundary value problem involving Riemann Liouville fractional integral boundary conditions on the half line","authors":"D. Oz, I. Karaca","doi":"10.2298/FIL2009161O","DOIUrl":"https://doi.org/10.2298/FIL2009161O","url":null,"abstract":"This paper investigates the existence of positive solutions for m-point p-Laplacian fractional boundary value problem involving Riemann Liouville fractional integral boundary conditions on the half line via the Leray-Schauder Nonlinear Alternative theorem and the use and some properties of the Green function. As an application, an example is presented to demonstrate our main result.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76568307","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 : 2020-05-24DOI: 10.21494/iste.op.2021.0647
Pokou Nagacy, J. Feuto
We generalize Wiener amalgam spaces by using Dunkl translation instead of the classical one, and we give some relationship between these spaces, Dunkl-Lebesgue spaces and Dunkl-Morrey spaces. We prove that the Hardy-Litlewood maximal function associated with the Dunkl operator is bounded on these generalized Dunkl-Morrey spaces.
{"title":"Maximal operator in Dunkl-Fofana spaces","authors":"Pokou Nagacy, J. Feuto","doi":"10.21494/iste.op.2021.0647","DOIUrl":"https://doi.org/10.21494/iste.op.2021.0647","url":null,"abstract":"We generalize Wiener amalgam spaces by using Dunkl translation instead of the classical one, and we give some relationship between these spaces, Dunkl-Lebesgue spaces and Dunkl-Morrey spaces. We prove that the Hardy-Litlewood maximal function associated with the Dunkl operator is bounded on these generalized Dunkl-Morrey spaces.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78811943","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 describe all differentiable functions $varphi,psicolon Etomathbb{R}$ satisfying the functional-differential equation begin{equation*} [varphi(y) - varphi(x)]psi 'bigl(h(x,y)bigr) = [psi(y) - psi(x)]varphi 'bigl(h(x,y)bigr), end{equation*} for all $x,yin E$, $x
{"title":"On a functional-differential equation with quasi-arithmetic mean value","authors":"Shokhrukh Ibragimov","doi":"10.29229/uzmj.2020-2-6","DOIUrl":"https://doi.org/10.29229/uzmj.2020-2-6","url":null,"abstract":"In this paper we describe all differentiable functions $varphi,psicolon Etomathbb{R}$ satisfying the functional-differential equation begin{equation*} [varphi(y) - varphi(x)]psi 'bigl(h(x,y)bigr) = [psi(y) - psi(x)]varphi 'bigl(h(x,y)bigr), end{equation*} for all $x,yin E$, $x<y$, where $E subseteq mathbb{R}$ is a nonempty open interval, $h(cdot,cdot)$ is a quasi-arithmetic mean, i.e. $h(x,y)=H^{-1}(alpha H (x)+beta H (y))$, $x,yin E$, for some differentiable and strictly monotone function $Hcolon E to H(E)$ and fixed $alpha, betain (0,1)$ with $alpha+beta=1$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79197339","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 : 2020-05-16DOI: 10.18500/1816-9791-2021-21-2-151-161
S. Lukomskii, D. Lukomskii
In this article we consider the problem of approximative solution of linear differential equations $y'+p(x)y=q(x)$ with discontinuous coefficients $p$ and $q$. We assume that coefficients of such equation are Henstock integrable functions. To find the approximative solution we change the original Cauchy problem to another problem with piecewise-constant coefficients. The sharp solution of this new problems is the approximative solution of the original Cauchy problem. We find the degree approximation in terms of modulus of continuity $omega_delta (P), omega_delta (Q)$, where $P$ and $Q$ are $f$-primitive for coefficients $p$ and $q$.
{"title":"Numerical solution of linear differential equations with discontinuous coefficients and Henstock integral","authors":"S. Lukomskii, D. Lukomskii","doi":"10.18500/1816-9791-2021-21-2-151-161","DOIUrl":"https://doi.org/10.18500/1816-9791-2021-21-2-151-161","url":null,"abstract":"In this article we consider the problem of approximative solution of linear differential equations $y'+p(x)y=q(x)$ with discontinuous coefficients $p$ and $q$. We assume that coefficients of such equation are Henstock integrable functions. To find the approximative solution we change the original Cauchy problem to another problem with piecewise-constant coefficients. The sharp solution of this new problems is the approximative solution of the original Cauchy problem. We find the degree approximation in terms of modulus of continuity $omega_delta (P), omega_delta (Q)$, where $P$ and $Q$ are $f$-primitive for coefficients $p$ and $q$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"157 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84953780","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 homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{Omega}$ be the convolution singular integral operator with kernel $frac{Omega(x)}{|x|^n}$. For $bin{rm BMO}(mathbb{R}^n)$, let $T_{Omega,,b}$ be the commutator of $T_{Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $Llog L$ type for $T_{Omega,,b}$ when $Omegain L^q(S^{n-1})$ for some $qin (1,,infty]$.
{"title":"Weak type endpoint estimates for the commutators of rough singular integral operators","authors":"Jiacheng Lan, Xiangxing Tao, G. Hu","doi":"10.7153/mia-2020-23-91","DOIUrl":"https://doi.org/10.7153/mia-2020-23-91","url":null,"abstract":"Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{Omega}$ be the convolution singular integral operator with kernel $frac{Omega(x)}{|x|^n}$. For $bin{rm BMO}(mathbb{R}^n)$, let $T_{Omega,,b}$ be the commutator of $T_{Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $Llog L$ type for $T_{Omega,,b}$ when $Omegain L^q(S^{n-1})$ for some $qin (1,,infty]$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87484941","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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