Abstract In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.
摘要在这篇解释性文章中，我们讨论了Volterra算子的各种函数的算子范数的评估和估计。
{"title":"Volterra operator norms : a brief survey","authors":"T. Ransford","doi":"10.2478/mjpaa-2023-0018","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0018","url":null,"abstract":"Abstract In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42142474","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}
Abstract This note is dedicated to recalling the virtues and the important contributions in mathematics of mohamed zarabi who passed a way on mid december 2021.
{"title":"Mohamed Zarrabi 1964-2021","authors":"Omar El Fallah, K. Kellay, Ouhabaz El Maati","doi":"10.2478/mjpaa-2023-0011","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0011","url":null,"abstract":"Abstract This note is dedicated to recalling the virtues and the important contributions in mathematics of mohamed zarabi who passed a way on mid december 2021.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47843094","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}
Abstract Zarrabi proved in 1993 that if the spectrum of a contraction T on a Banach space is a countable subset of the unit circle 𝕋, and if limn→+∞log(‖ T−n ‖)n=0 {lim _{n to + infty }}{{log left( {left| {{T^{ - n}}} right|} right)} over {sqrt n }} = 0 , then T is an isometry, so that ‖Tn‖ = 1 for every n ∈ ℤ. It is also known that if C is the usual triadic Cantor set then every contraction T on a Banach space such that Spec(T ) ⊂ 𝒞 satisfying lim supn→+∞log(‖ T−n ‖)nα<+∞ lim ,su{p_{n to + infty }}{{log left( {left| {{T^{ - n}}} right|} right)} over {{n^alpha }}} < + infty for some α0 ‖T−n ‖ ≤ K, where K < + ∞ denotes the “AA+-constant” of E (closed countanble subsets of 𝕋 and the triadic Cantor set are strong AA+-sets of constant 1).
{"title":"Negative Powers of Contractions Having a Strong AA+ Spectrum","authors":"J. Esterle","doi":"10.2478/mjpaa-2023-0015","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0015","url":null,"abstract":"Abstract Zarrabi proved in 1993 that if the spectrum of a contraction T on a Banach space is a countable subset of the unit circle 𝕋, and if limn→+∞log(‖ T−n ‖)n=0 {lim _{n to + infty }}{{log left( {left| {{T^{ - n}}} right|} right)} over {sqrt n }} = 0 , then T is an isometry, so that ‖Tn‖ = 1 for every n ∈ ℤ. It is also known that if C is the usual triadic Cantor set then every contraction T on a Banach space such that Spec(T ) ⊂ 𝒞 satisfying lim supn→+∞log(‖ T−n ‖)nα<+∞ lim ,su{p_{n to + infty }}{{log left( {left| {{T^{ - n}}} right|} right)} over {{n^alpha }}} < + infty for some α<log(3)−log(2)2 log(3)−log(2) alpha < {{log left( 3 right) - log left( 2 right)} over {2,log left( 3 right) - log left( 2 right)}} is an isometry. In the other direction an easy refinement of known results shows that if a closed E ⊂ 𝕋 is not a “strong AA+-set” then for every sequence (un)n≥1 of positive real numbers such that lim infn→+∞un = + ∞ there exists a contraction T on some Banach space such that Spec(T )⊂ E, ‖T−n‖ = O(un) as n → + ∞ and supn≥1 ‖T−n‖ = + ∞. We show conversely that if E ⊂ 𝕋 is a strong AA+-set then there exists a nondecreasing unbounded sequence (un)n≥1 such that for every contraction T on a Banach space satsfying Spec(T) ⊂ E and ‖T−n ‖ = O(un) as n → + ∞ we have supn>0 ‖T−n ‖ ≤ K, where K < + ∞ denotes the “AA+-constant” of E (closed countanble subsets of 𝕋 and the triadic Cantor set are strong AA+-sets of constant 1).","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42518556","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}
Abstract The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice X, including weighted ℓp spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 𝔻X∞ mathbb{D}_X^infty . Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 𝔻X∞,|ℂN=𝔻N mathbb{D}_X^infty ,|{mathbb{C}^N} = {mathbb{D}^N} and the ℓp-unit balls 𝔻X∞,|ℂN=𝔹pN mathbb{D}_X^infty ,|{mathbb{C}^N} = mathbb{B}_p^N , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as X=ℓp(ℤ+N,(1+α)s) X = {ell ^p}left( {_ + ^N,{{left( {1 + alpha } right)}^s}} right)) , s = (s1, s2, … ) and X=ℓp(ℤ+N, (α!| α |!)t(1+| α |)s) X = {ell ^p}left( {mathbb{Z}_ + ^N,,,{{left( {{{alpha !} over {left| alpha right|!}}} right)}^t}{{left( {1 + left| alpha right|} right)}^s}} right) , s,t ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.
{"title":"On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras","authors":"N. Nikolski","doi":"10.2478/mjpaa-2023-0017","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0017","url":null,"abstract":"Abstract The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice X, including weighted ℓp spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain 𝔻X∞ mathbb{D}_X^infty . Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs 𝔻X∞,|ℂN=𝔻N mathbb{D}_X^infty ,|{mathbb{C}^N} = {mathbb{D}^N} and the ℓp-unit balls 𝔻X∞,|ℂN=𝔹pN mathbb{D}_X^infty ,|{mathbb{C}^N} = mathbb{B}_p^N , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as X=ℓp(ℤ+N,(1+α)s) X = {ell ^p}left( {_ + ^N,{{left( {1 + alpha } right)}^s}} right)) , s = (s1, s2, … ) and X=ℓp(ℤ+N, (α!| α |!)t(1+| α |)s) X = {ell ^p}left( {mathbb{Z}_ + ^N,,,{{left( {{{alpha !} over {left| alpha right|!}}} right)}^t}{{left( {1 + left| alpha right|} right)}^s}} right) , s,t ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48971112","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}
Abstract We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.
{"title":"H∞ interpolation constrained by Beurling–Sobolev norms","authors":"A. Baranov, R. Zarouf","doi":"10.2478/mjpaa-2023-0012","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0012","url":null,"abstract":"Abstract We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49548913","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}
Abstract There exist two real valued periodic functions on the real line such that, for every x ∈ ℝ, f1(x) + f2(x) = x, but it is impossible to find two real valued periodic functions on the real line such that, for every x ∈ ℝ, f1(x) + f2(x) = x2. The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.
{"title":"Sums and products of periodic functions","authors":"R. Deville","doi":"10.2478/mjpaa-2023-0014","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0014","url":null,"abstract":"Abstract There exist two real valued periodic functions on the real line such that, for every x ∈ ℝ, f1(x) + f2(x) = x, but it is impossible to find two real valued periodic functions on the real line such that, for every x ∈ ℝ, f1(x) + f2(x) = x2. The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43906572","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}
Abstract For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.
{"title":"The Maximum Locus of the Bloch Norm","authors":"Youssfi El Hassan","doi":"10.2478/mjpaa-2023-0019","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0019","url":null,"abstract":"Abstract For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46562742","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}
Abstract We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.
摘要我们将Carleson公式推广到径向多项式加权的Dirichlet空间。
{"title":"Carleson’s formula for some weighted Dirichlet spaces","authors":"B. Bouya, A. Hartmann","doi":"10.2478/mjpaa-2023-0013","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0013","url":null,"abstract":"Abstract We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48058388","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}
Abstract New copulas, based on perturbation theory, are introduced to clarify a symmetrization procedure for asymmetric copulas. We give also some properties of the symmetrized copula mainly conservation of concordance. Finally, we examine some copulas with a prescribed symmetrized part. The start point of the treatment is the independence copula and the last one will be an arbitrary member of Farlie-Gumbel-Morgenstein family. By the way, we study topologically, the set of all symmetric copulas and give some of its classical and new properties.
{"title":"New asymmetric perturbations of FGM bivariate copulas and concordance preserving problems","authors":"Mohamed El maazouz, A. Sani","doi":"10.2478/mjpaa-2023-0008","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0008","url":null,"abstract":"Abstract New copulas, based on perturbation theory, are introduced to clarify a symmetrization procedure for asymmetric copulas. We give also some properties of the symmetrized copula mainly conservation of concordance. Finally, we examine some copulas with a prescribed symmetrized part. The start point of the treatment is the independence copula and the last one will be an arbitrary member of Farlie-Gumbel-Morgenstein family. By the way, we study topologically, the set of all symmetric copulas and give some of its classical and new properties.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48830060","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}
M. Ammi, Ibrahim Dahi, A. E. Hachimi, Delfim F. M. Torres
Abstract We study the existence and uniqueness of a bounded weak solution for a triply nonlinear thermistor problem in Sobolev spaces. Furthermore, we prove the existence of an absorbing set and, consequently, the universal attractor.
{"title":"Existence result of the global attractor for a triply nonlinear thermistor problem","authors":"M. Ammi, Ibrahim Dahi, A. E. Hachimi, Delfim F. M. Torres","doi":"10.2478/mjpaa-2023-0002","DOIUrl":"https://doi.org/10.2478/mjpaa-2023-0002","url":null,"abstract":"Abstract We study the existence and uniqueness of a bounded weak solution for a triply nonlinear thermistor problem in Sobolev spaces. Furthermore, we prove the existence of an absorbing set and, consequently, the universal attractor.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44177797","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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