N. H. Mohammed, E. A. Adegani, T. Bulboacă, N. Cho
{"title":"A family of holomorphic functions defined by differential inequality","authors":"N. H. Mohammed, E. A. Adegani, T. Bulboacă, N. Cho","doi":"10.7153/mia-2022-25-03","DOIUrl":"https://doi.org/10.7153/mia-2022-25-03","url":null,"abstract":"","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71205374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boundedness of Riemann-Liouville operator from weighted Sobolev space to weighted Lebesgue space for 1 < q < p < ∞","authors":"A. Kalybay, R. Oinarov","doi":"10.7153/mia-2022-25-02","DOIUrl":"https://doi.org/10.7153/mia-2022-25-02","url":null,"abstract":"","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71205361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We study the regularity properties for commutators of multilinear fractional maximal operators. More precisely, let m (cid:2) 1, 0 (cid:3) α < mn and (cid:2) b = ( b 1 ,..., b m ) with each b i belonging to the Lipschitz space Lip ( R ) , we denote by [ (cid:2) b , M α ] (resp., M α ,(cid:2) b ) the commutator of the multilinear fractional maximal operator M α with (cid:2) b (resp., the multilinear fractional maximal commutators). When α = 0, we denote [ (cid:2) b , M α ] = [ (cid:2) b , M ] and M α ,(cid:2) b = M (cid:2) b . We show that for 0 < s < 1, 1 < p 1 ,..., p m , p , q < ∞ , 1 / p = 1 / p 1 + ··· + 1 / p m , both [ (cid:2) b , M ] and M (cid:2) b are bounded and continuous from W s , p 1 ( R n ) ×···× W s , p m ( R n ) to W s , p ( R n ) , from F p 1 , q s ( R n ) × ···× F p m , q s ( R n ) to F p , q s ( R n ) and from B p 1 , q s ( R n ) ×···× B p m , q s ( R n ) to B p , q s ( R n ) . It was also shown that for 0 (cid:3) α < mn , 1 < p 1 ,..., p m , q < ∞ and 1 / q = 1 / p 1 + ··· + 1 / p m − α / n , both [ (cid:2) b , M ] and M (cid:2) b are W 1 , p 1 ( R n ) ×···× W 1 , p m ( R n ) to W 1 , q ( R n ) .
． 我们研究了最大多线型框架联合人员的监管属性。更多precisely,让m (cid 3: 2) 1、0 (cid)α< mn和(cid): 2) b = (b 1, ...b, m)和每b i ' belonging to Lipschitz太空嘴唇杂志》(R),我们denote由(cid:(2) b, mα](代表。, Mα(cid commutator》:2)b) multilinear最大限度fractional Mα与操作员(cid: 2) b(代表)。最大限度,《multilinear fractional commutators)。当α= 0,则我们denote (cid:(2) b, Mα]= [(cid: 2) b、M)和α(cid: 2) b = M (cid: 2) b。我们展示给0 < s < 1, 1 < p 1， ...p m, p, q <∞,1 / p = 1 / p p +···+ 1 / m,两者(cid:(2) b, m和m (cid): 2) b是bounded挑战从W s,睡意朦胧,p (n)×R···1×W s, R p m (n)到R W s, p (n),从F p 1, q R s (n ) × ···× F p m, p q R s (n)到F,从B p q R s (n)和1,q s (n)×R···×B p m, p q R s (n) to B, q R s (n)。那是还展示为0 (cid: 3)α< p < 1, ...哪里m, p, q <∞和p - q = 1 / 1 +···+ 1 / p m−α/ n, [(cid): 2) b, m和m (cid): 2) b是1,p (n)×R W·R·m·W×1,p (n)到R W 1, q (n)。
{"title":"Regularity of commutators of multilinear maximal operators with Lipschitz symbols","authors":"Ting Chen, Feng Liu","doi":"10.7153/mia-2022-25-08","DOIUrl":"https://doi.org/10.7153/mia-2022-25-08","url":null,"abstract":". We study the regularity properties for commutators of multilinear fractional maximal operators. More precisely, let m (cid:2) 1, 0 (cid:3) α < mn and (cid:2) b = ( b 1 ,..., b m ) with each b i belonging to the Lipschitz space Lip ( R ) , we denote by [ (cid:2) b , M α ] (resp., M α ,(cid:2) b ) the commutator of the multilinear fractional maximal operator M α with (cid:2) b (resp., the multilinear fractional maximal commutators). When α = 0, we denote [ (cid:2) b , M α ] = [ (cid:2) b , M ] and M α ,(cid:2) b = M (cid:2) b . We show that for 0 < s < 1, 1 < p 1 ,..., p m , p , q < ∞ , 1 / p = 1 / p 1 + ··· + 1 / p m , both [ (cid:2) b , M ] and M (cid:2) b are bounded and continuous from W s , p 1 ( R n ) ×···× W s , p m ( R n ) to W s , p ( R n ) , from F p 1 , q s ( R n ) × ···× F p m , q s ( R n ) to F p , q s ( R n ) and from B p 1 , q s ( R n ) ×···× B p m , q s ( R n ) to B p , q s ( R n ) . It was also shown that for 0 (cid:3) α < mn , 1 < p 1 ,..., p m , q < ∞ and 1 / q = 1 / p 1 + ··· + 1 / p m − α / n , both [ (cid:2) b , M ] and M (cid:2) b are W 1 , p 1 ( R n ) ×···× W 1 , p m ( R n ) to W 1 , q ( R n ) .","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71205319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the Hardy-Littlewood maximal and Calder´on-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
{"title":"Weighted boundedness of the Hardy-Littlewood maximal and Calderón-Zygmund operators on Orlicz-Morrey and weak Orlicz-Morrey spaces","authors":"Ryota Kawasumi, E. Nakai","doi":"10.7153/mia-2021-24-81","DOIUrl":"https://doi.org/10.7153/mia-2021-24-81","url":null,"abstract":"For the Hardy-Littlewood maximal and Calder´on-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41808271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The norm of C−I on l, where C is the Cesàro operator, is shown to be 1/(p − 1) when 1 < p ≤ 2. This verifies a recent conjecture of G. J. O. Jameson. The norm of C − I on l is also determined when 2 < p < ∞. The two parts together answer a question raised by G. Bennett in 1996. Operator norms in the continuous case, Hardy’s averaging operator minus identity, are already known. Norms in the discrete and continuous cases coincide. The Cesàro operator, C, maps a sequence (xn) to (yn), where yn = 1 n n
当1＜p≤2时，C−I在l上的范数（其中C是Cesàro算子）被证明是1/（p−1）。这证实了詹姆逊最近的一个猜想。当2＜p＜∞时，C−I在l上的范数也被确定。这两部分共同回答了G.Bennett在1996年提出的一个问题。连续情况下的算子范数，Hardy的平均算子减恒等式，已经为人所知。离散和连续情况下的规范是一致的。Cesàro算子C将序列（xn）映射到（yn），其中yn=1 n n
{"title":"Norm of the discrete Cesàaro operator minus identity","authors":"G. Sinnamon","doi":"10.7153/mia-2022-25-04","DOIUrl":"https://doi.org/10.7153/mia-2022-25-04","url":null,"abstract":"The norm of C−I on l, where C is the Cesàro operator, is shown to be 1/(p − 1) when 1 < p ≤ 2. This verifies a recent conjecture of G. J. O. Jameson. The norm of C − I on l is also determined when 2 < p < ∞. The two parts together answer a question raised by G. Bennett in 1996. Operator norms in the continuous case, Hardy’s averaging operator minus identity, are already known. Norms in the discrete and continuous cases coincide. The Cesàro operator, C, maps a sequence (xn) to (yn), where yn = 1 n n","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44549536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Hardy property of means has been extensively studied by Páles and Pasteczka since 2016. The core of this research is based on few of their properties: concavity, symmetry, monotonicity, repetition invariance and homogeneity (last axiom was recently omitted using some homogenizations techniques). In the present paper we deliver a study of possible omitting monotonicity and replacing repetition invariance by a weaker axiom. These results are then used to establish the Hardy constant for certain types of mixed means.
{"title":"On the Hardy property of mixed means","authors":"P. Pasteczka","doi":"10.7153/mia-2021-24-60","DOIUrl":"https://doi.org/10.7153/mia-2021-24-60","url":null,"abstract":". Hardy property of means has been extensively studied by Páles and Pasteczka since 2016. The core of this research is based on few of their properties: concavity, symmetry, monotonicity, repetition invariance and homogeneity (last axiom was recently omitted using some homogenizations techniques). In the present paper we deliver a study of possible omitting monotonicity and replacing repetition invariance by a weaker axiom. These results are then used to establish the Hardy constant for certain types of mixed means.","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49612642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A chain rule for power product is studied with fractional differential operators in the framework of Sobolev spaces. The fractional differential operators are defined by the Fourier multipliers. The chain rule is considered newly in the case where the order of differential operators is between one and two. The study is based on the analogy of the classical chain rule or Leibniz rule.
{"title":"Remark on the Chain rule of fractional derivative in the Sobolev framework","authors":"K. Fujiwara","doi":"10.7153/mia-2021-24-77","DOIUrl":"https://doi.org/10.7153/mia-2021-24-77","url":null,"abstract":"A chain rule for power product is studied with fractional differential operators in the framework of Sobolev spaces. The fractional differential operators are defined by the Fourier multipliers. The chain rule is considered newly in the case where the order of differential operators is between one and two. The study is based on the analogy of the classical chain rule or Leibniz rule.","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42643281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For every given real value of the ratio $mu:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $mathsf{P}(X/G_Xge v)$ and $mathsf{P}(X/G_Xle v)$ are obtained, in terms of $mu$ and $v$. In particular, these bounds imply that $X/G_Xto1$ in probability as $A_X/G_Xdownarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function $f(x) equiv -x^2$. As applications of the mentioned new results, improvements of the Markov, Bernstein--Chernoff, sub-Gaussian, and Bennett--Hoeffding probability inequalities are given.
{"title":"Exact converses to a reverse AM-GM inequality, with applications to sums of independent random variables and (super)martingales","authors":"I. Pinelis","doi":"10.7153/MIA-2021-24-40","DOIUrl":"https://doi.org/10.7153/MIA-2021-24-40","url":null,"abstract":"For every given real value of the ratio $mu:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $mathsf{P}(X/G_Xge v)$ and $mathsf{P}(X/G_Xle v)$ are obtained, in terms of $mu$ and $v$. In particular, these bounds imply that $X/G_Xto1$ in probability as $A_X/G_Xdownarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function $f(x) equiv -x^2$. As applications of the mentioned new results, improvements of the Markov, Bernstein--Chernoff, sub-Gaussian, and Bennett--Hoeffding probability inequalities are given.","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48610376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the Cesaro operator C, it is known that ||C-I||_2 = 1. Here we prove that ||C-I||_4 < 3^(1/4) and ||C^T-I||_4 = 3. Bounds for intermediate values of p are derived from the Riesz-Thorin interpolation theorem. An estimate for lower bounds is obtained.
对于Cesaro算子C，已知||C- i ||_2 = 1。我们证明| |我| | _4 < 3 ^(1/4)和| | C ^我| | _4 = 3。由Riesz-Thorin插值定理导出了p的中间值的边界。得到了下界的估计。
{"title":"The ℓ_p-norm of C-I, where C is the Cesàro operator","authors":"G. Jameson","doi":"10.7153/MIA-2021-24-38","DOIUrl":"https://doi.org/10.7153/MIA-2021-24-38","url":null,"abstract":"For the Cesaro operator C, it is known that ||C-I||_2 = 1. Here we prove that ||C-I||_4 < 3^(1/4) and ||C^T-I||_4 = 3. Bounds for intermediate values of p are derived from the Riesz-Thorin interpolation theorem. An estimate for lower bounds is obtained.","PeriodicalId":49868,"journal":{"name":"Mathematical Inequalities & Applications","volume":"1 1","pages":"551-557"},"PeriodicalIF":1.0,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42871635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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