Continuity of composition operators in Sobolev spaces

Moussai, Madani

doi:10.1016/j.anihpc.2019.07.002

Cited by 6 publications

(3 citation statements)

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“…• In the general case by Moussai and the author [8], who proved also this "automatic" continuity on the so-called Adams-Frazier spaces W m p ∩ Ẇ 1 mp (R n ), where Ẇ denotes the homogeneous Sobolev space, and on the spaces Ẇ m p ∩ Ẇ 1 mp (R n ), conveniently realized.…”

Section: Continuity Of Composition On Sobolev Spacesmentioning

confidence: 93%

An introduction to composition operators in Sobolev spaces

Bourdaud

2022

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This is an updated version of notes for a series of lectures, given in Padova University in January 2018. We propose a survey on composition operators in classical Sobolev spaces. We mention results obtained in 2019, on the continuity of such operators. NotationN denotes the set of all positive integers, including 0. Z denotes the set of all integers. For x ∈ R n , |x| denotes its euclidean norm.If E, F are topological spaces, then E ֒→ F means that E ⊆ F , as sets, and the natural mapping E → F is continuous. supported functions on Ω, endowed with its natural topology, see [1, 1.56Let E be a subset of L 1,loc (R n ). We say that a function f ∈ L 1,loc (R n ) belongs locally to E if ϕf ∈ E for all ϕ ∈ D(R n ) ; in case E is endowed with a norm, we say that a functionIn all the paper, "ball" means "ball with non zero radius" (we exclude balls reduced to one point).

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Section: Continuity Of Composition On Sobolev Spacesmentioning

confidence: 93%

An introduction to composition operators in Sobolev spaces

Bourdaud

2022

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“…Further results concerning the composition operators in Besov and Triebel-Lizorkin spaces are given [5], [9], [10], [12], [14] and [47]. Recently, Bourdaud and Moussai [13] proved the continuity of the composition operator in W m p (R n ) ∩ Ẇ 1 mp (R n ) to itself, for every integer m 2 and any 1 p < ∞ and in Sobolev spaces W m p (R n ), with m 2 and 1 p < ∞. The author in [24] and [25] gave the necessary and sufficient conditions on G such that…”

Section: > 1 and The Solution Belongsmentioning

confidence: 96%

Composition operators on Herz-type Triebel-Lizorkin spaces with application to semilinear parabolic equations

Drihem¹

2022

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Let G : R → R be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such thatHere Kα p,q F s β are Herz-type Triebel-Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel-Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabolic equationswith initial data in Herz-type Triebel-Lizorkin spaces. Our results cover the results obtained with initial data in some know function spaces such us fractional Sobolev spaces. Some limit cases are given. MSC classification (2010): 46E35, 47H30, 35K45, 35K55.

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“…Theory concerning composition operators or even more general Nemytzkij operators can be found in chapter 5 in [26] and in [1]. One may also consult [4], [6], [7] and [8]. In the theory of nonlinear partial differential equations the operator T + often is called a truncation operator.…”

Section: Notationmentioning

confidence: 99%