Abstract In this work, we handle a time-dependent Navier-Stokes problem in dimension three with a mixed boundary conditions. The variational formulation is written considering three independent unknowns: vorticity, velocity, and pressure. We use the backward Euler scheme for time discretization and the spectral method for space discretization. We present a complete numerical analysis linked to this variational formulation, which leads us to a priori error estimate.
{"title":"Spectral discretization of the time-dependent Navier-Stokes problem with mixed boundary conditions","authors":"M. Abdelwahed, N. Chorfi","doi":"10.1515/anona-2022-0253","DOIUrl":"https://doi.org/10.1515/anona-2022-0253","url":null,"abstract":"Abstract In this work, we handle a time-dependent Navier-Stokes problem in dimension three with a mixed boundary conditions. The variational formulation is written considering three independent unknowns: vorticity, velocity, and pressure. We use the backward Euler scheme for time discretization and the spectral method for space discretization. We present a complete numerical analysis linked to this variational formulation, which leads us to a priori error estimate.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1447 - 1465"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45793108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article deals with the degenerate fractional Kirchhoff wave equation with structural damping or strong damping. The well-posedness and the existence of global attractor in the natural energy space by virtue of the Faedo-Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration (or even negativity) of the stiffness coefficient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be infinite by using Z 2 {{mathbb{Z}}}_{2} index theory.
{"title":"Global attractors of the degenerate fractional Kirchhoff wave equation with structural damping or strong damping","authors":"Wenhua Yang, Jun Zhou","doi":"10.1515/anona-2022-0226","DOIUrl":"https://doi.org/10.1515/anona-2022-0226","url":null,"abstract":"Abstract This article deals with the degenerate fractional Kirchhoff wave equation with structural damping or strong damping. The well-posedness and the existence of global attractor in the natural energy space by virtue of the Faedo-Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration (or even negativity) of the stiffness coefficient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be infinite by using Z 2 {{mathbb{Z}}}_{2} index theory.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"993 - 1029"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49500865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Harbir Antil, Shodai Kubota, K. Shirakawa, N. Yamazaki
Abstract In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.
{"title":"Constrained optimization problems governed by PDE models of grain boundary motions","authors":"Harbir Antil, Shodai Kubota, K. Shirakawa, N. Yamazaki","doi":"10.1515/anona-2022-0242","DOIUrl":"https://doi.org/10.1515/anona-2022-0242","url":null,"abstract":"Abstract In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1249 - 1286"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45301104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: ( − Δ ) s u + V ( x ) u = ∫ R N A ( ε y ) ∣ u ( y ) ∣ p ∣ x − y ∣ μ d y A ( ε x ) ∣ u ( x ) ∣ p − 2 u ( x ) , x ∈ R N , {left(-Delta )}^{s}u+Vleft(x)u=left(mathop{int }limits_{{{mathbb{R}}}^{N}}frac{Aleft(varepsilon y)| u(y){| }^{p}}{| x-y{| }^{mu }}{rm{d}}yright)Aleft(varepsilon x)| uleft(x){| }^{p-2}uleft(x),hspace{1em}xin {{mathbb{R}}}^{N}, where s ∈ ( 0 , 1 ) sin left(0,1) , N > 2 s Ngt 2s , 0 < μ < 2 s 0lt mu lt 2s , 2 < p < 2 N − 2 μ N − 2 s 2lt plt frac{2N-2mu }{N-2s} , and ε varepsilon is a positive parameter. Under some natural hypotheses on the potentials V V and A A , using the generalized Nehari manifold method, we obtain the existence of ground-state solutions. Moreover, we investigate the concentration behavior of ground-state solutions that concentrate at global maximum points of A A as ε → 0 varepsilon to 0 .
摘要本文研究了具有不定势的分数阶Choquard方程基态解的存在性和集中性：（−Δ）su+V（x）u=ŞR N A（εy）Şu（y）Ş^{s}u+Vleft（x）u=left^{p-2}uleft（x）， hspace{1em}x在｛mathbb｛R｝｝^｛N｝中，其中s∈（0，1）sinleft（0，0），N>2s Ngt 2s，0<μ<2s 0ltmult 2s，2
{"title":"Existence and concentration of ground-states for fractional Choquard equation with indefinite potential","authors":"Wen Zhang, Shuai Yuan, Lixi Wen","doi":"10.1515/anona-2022-0255","DOIUrl":"https://doi.org/10.1515/anona-2022-0255","url":null,"abstract":"Abstract This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: ( − Δ ) s u + V ( x ) u = ∫ R N A ( ε y ) ∣ u ( y ) ∣ p ∣ x − y ∣ μ d y A ( ε x ) ∣ u ( x ) ∣ p − 2 u ( x ) , x ∈ R N , {left(-Delta )}^{s}u+Vleft(x)u=left(mathop{int }limits_{{{mathbb{R}}}^{N}}frac{Aleft(varepsilon y)| u(y){| }^{p}}{| x-y{| }^{mu }}{rm{d}}yright)Aleft(varepsilon x)| uleft(x){| }^{p-2}uleft(x),hspace{1em}xin {{mathbb{R}}}^{N}, where s ∈ ( 0 , 1 ) sin left(0,1) , N > 2 s Ngt 2s , 0 < μ < 2 s 0lt mu lt 2s , 2 < p < 2 N − 2 μ N − 2 s 2lt plt frac{2N-2mu }{N-2s} , and ε varepsilon is a positive parameter. Under some natural hypotheses on the potentials V V and A A , using the generalized Nehari manifold method, we obtain the existence of ground-state solutions. Moreover, we investigate the concentration behavior of ground-state solutions that concentrate at global maximum points of A A as ε → 0 varepsilon to 0 .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1552 - 1578"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49274879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , left(a+bmathop{int }limits_{{{mathbb{R}}}^{N}}| {left(-Delta )}^{tfrac{s}{2}}uhspace{-0.25em}{| }^{2}{rm{d}}xright){left(-Delta )}^{s}u=left(1+varepsilon Kleft(x)){u}^{{2}_{s}^{ast }-1},hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, where a , b > 0 a,bgt 0 are given constants, ε varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{ast }=frac{2N}{N-2s} with 0 < s < 1 0lt slt 1 and N ≥ 4 s Nge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N > 4 s Ngt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε varepsilon small.
摘要本文讨论了临界指数为a+bŞRNŞ（−Δ）s2 uŞ2 d x（−Δleft（-Delta）｝^{s}u=left（1+varepsilon Kleft（x））｛u｝^{{2}_{s} ^｛ast｝-1｝， hspace｛1.0em｝ hspace｝0.1em｝text｛in｝ tspace｛0.1em} hspace{0.33em｝｛mathbb｛R｝｝｝｝^｛N｝，其中a，b>0 a，bgt 0是给定的常数，εvarepsilon是一个小参数，2s*=2 N−2s{2}_{s} ^｛ast｝= frac｛2N｝｛N-2s｝，其中0＜s＜1 0lt slt 1且N≥4 s N ge 4s。当ε=0 varepsilon=0时，我们首先证明了正解的非一般性。特别地，我们证明了维数N>4sNgt 4s的唯一性分解，即，我们证明存在两个非退化正解，这两个解似乎与分数阶薛定谔方程或低维分数阶基尔霍夫方程的结果完全不同。利用有限维约简方法和扰动变元，我们还得到了εvarepsilon small奇异扰动问题正解的存在性。
{"title":"On the singularly perturbation fractional Kirchhoff equations: Critical case","authors":"Guangze Gu, Zhipeng Yang","doi":"10.1515/anona-2022-0234","DOIUrl":"https://doi.org/10.1515/anona-2022-0234","url":null,"abstract":"Abstract This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , left(a+bmathop{int }limits_{{{mathbb{R}}}^{N}}| {left(-Delta )}^{tfrac{s}{2}}uhspace{-0.25em}{| }^{2}{rm{d}}xright){left(-Delta )}^{s}u=left(1+varepsilon Kleft(x)){u}^{{2}_{s}^{ast }-1},hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, where a , b > 0 a,bgt 0 are given constants, ε varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{ast }=frac{2N}{N-2s} with 0 < s < 1 0lt slt 1 and N ≥ 4 s Nge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N > 4 s Ngt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε varepsilon small.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1097 - 1116"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45264734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(ℝ+n+1,μ) L^q (mathbb{R}_ + ^{n + 1} ,mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new Lp–capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the Lp–capacity, including its dual form, the Lp–capacity of fractional parabolic balls, strong and weak type inequalities, are established.
{"title":"Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for Lq(ℝ+n+1,μ) L^q (mathbb{R}_ + ^{n + 1} ,mu ) −Extension","authors":"Pengtao Li, Zhichun Zhai","doi":"10.1515/anona-2021-0232","DOIUrl":"https://doi.org/10.1515/anona-2021-0232","url":null,"abstract":"Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(ℝ+n+1,μ) L^q (mathbb{R}_ + ^{n + 1} ,mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new Lp–capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the Lp–capacity, including its dual form, the Lp–capacity of fractional parabolic balls, strong and weak type inequalities, are established.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"850 - 887"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42858395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered and the results are illustrated by selected examples.
{"title":"Lower and upper estimates of semi-global and global solutions to mixed-type functional differential equations","authors":"J. Diblík, G. Vážanová","doi":"10.1515/anona-2021-0218","DOIUrl":"https://doi.org/10.1515/anona-2021-0218","url":null,"abstract":"Abstract In the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered and the results are illustrated by selected examples.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"757 - 784"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44595205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we study a large class of evolutionary variational–hemivariational inequalities of hyperbolic type without damping terms, in which the functional framework is considered in an evolution triple of spaces. The inequalities contain both a convex potential and a locally Lipschitz superpotential. The results on existence, uniqueness, and regularity of solution to the inequality problem are provided through the Rothe method.
{"title":"A class of hyperbolic variational–hemivariational inequalities without damping terms","authors":"Shengda Zeng, S. Migórski, V. T. Nguyen","doi":"10.1515/anona-2022-0237","DOIUrl":"https://doi.org/10.1515/anona-2022-0237","url":null,"abstract":"Abstract In this article, we study a large class of evolutionary variational–hemivariational inequalities of hyperbolic type without damping terms, in which the functional framework is considered in an evolution triple of spaces. The inequalities contain both a convex potential and a locally Lipschitz superpotential. The results on existence, uniqueness, and regularity of solution to the inequality problem are provided through the Rothe method.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1287 - 1306"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42179705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H5 × H4 × H4 framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 {L^2} - {rm{rate}},{(1 + t)^{- {{11} over 4}}} , which is same as one of the heat equation, and particularly faster than the L2-rate (1+t)-54 {L^2} - {rm{rate}},{(1 + t)^{- {5 over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L2-rate (1+t)-94 {L^2} - {rm{rate}},{(1 + t)^{- {9 over 4}}} , in Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Second, we prove that fifth–order spatial derivative of density ρ converges to zero at the L2-rate (1+t)-134 {L^2} - {rm{rate}},{(1 + t)^{- {{13} over 4}}} , which is same as that of the heat equation, and particularly faster than ones of Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity u and magnetic B converge to zero at the L2-rate (1+t)-134 {L^2} - {rm{rate}},{(1 + t)^{- {{13} over 4}}} , which are faster than ones of themselves, and totally new as compared to Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019].
{"title":"Optimal decay rate for higher–order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model","authors":"Juan Wang, Yinghui Zhang","doi":"10.1515/anona-2021-0219","DOIUrl":"https://doi.org/10.1515/anona-2021-0219","url":null,"abstract":"Abstract We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H5 × H4 × H4 framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 {L^2} - {rm{rate}},{(1 + t)^{- {{11} over 4}}} , which is same as one of the heat equation, and particularly faster than the L2-rate (1+t)-54 {L^2} - {rm{rate}},{(1 + t)^{- {5 over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L2-rate (1+t)-94 {L^2} - {rm{rate}},{(1 + t)^{- {9 over 4}}} , in Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Second, we prove that fifth–order spatial derivative of density ρ converges to zero at the L2-rate (1+t)-134 {L^2} - {rm{rate}},{(1 + t)^{- {{13} over 4}}} , which is same as that of the heat equation, and particularly faster than ones of Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity u and magnetic B converge to zero at the L2-rate (1+t)-134 {L^2} - {rm{rate}},{(1 + t)^{- {{13} over 4}}} , which are faster than ones of themselves, and totally new as compared to Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019].","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"830 - 849"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42280589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let X be the direct product of Xi where Xi is smooth manifold for 1 ≤ i ≤ k. As is known, if every Xi satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one Xi does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties of M when X1 has exponential volume growth and X2 satisfies the doubling condition. Next, we consider the product space of two weighted hyperbolic spaces X1 = (ℍn+1, d, yα−n−1dydx) and X2 = (ℍn+1, d, yβ−n−1dydx) which both have exponential volume growth. The mapping properties of M are discussed for every α,β≠n2 alpha,beta ne {n over 2} . Furthermore, let X = X1 × X2 × … Xk where Xi = (ℍni+1, yαi−ni−1dydx) for 1 ≤ i ≤ k. Under the condition αi>ni2 {alpha_i} > {{{n_i}} over 2} , we also obtained the mapping properties of M.
{"title":"Centered Hardy-Littlewood maximal function on product manifolds","authors":"Shiliang Zhao","doi":"10.1515/anona-2021-0233","DOIUrl":"https://doi.org/10.1515/anona-2021-0233","url":null,"abstract":"Abstract Let X be the direct product of Xi where Xi is smooth manifold for 1 ≤ i ≤ k. As is known, if every Xi satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one Xi does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties of M when X1 has exponential volume growth and X2 satisfies the doubling condition. Next, we consider the product space of two weighted hyperbolic spaces X1 = (ℍn+1, d, yα−n−1dydx) and X2 = (ℍn+1, d, yβ−n−1dydx) which both have exponential volume growth. The mapping properties of M are discussed for every α,β≠n2 alpha,beta ne {n over 2} . Furthermore, let X = X1 × X2 × … Xk where Xi = (ℍni+1, yαi−ni−1dydx) for 1 ≤ i ≤ k. Under the condition αi>ni2 {alpha_i} > {{{n_i}} over 2} , we also obtained the mapping properties of M.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"888 - 906"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45925046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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