Uniformly continuous composition operators in the space of functions of -variation with weight in the sense of Riesz

Aziz, Wadie; Azócar, Antonio; Guerrero, José Atilio; Merentes, Nelson

doi:10.1016/j.na.2010.09.010

Cited by 3 publications

(3 citation statements)

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“…Clearly, the continuity of γ at 0 and γ(0) = 0, imply that the uniformly bounded operator H is uniformly continuous. It follows that Theorem 3.2 improves the result of [3] where H is assumed to be uniformly continuous.…”

Section: Resultsmentioning

confidence: 52%

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Uniformly Bounded Composition Operators

Głazowska¹,

Matkowski²

2015

Bull. Aust. Math. Soc.

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We prove that if a uniformly bounded (or equidistantly uniformly bounded) Nemytskij operator maps the space of functions of bounded ${\it\varphi}$-variation with weight function in the sense of Riesz into another space of that type (with the same weight function) and its generator is continuous with respect to the second variable, then this generator is affine in the function variable (traditionally, in the second variable).

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Section: Resultsmentioning

confidence: 52%

“…In [3] it has been proved that if H maps the space GRV ϕ,λ (I, C) of functions of bounded ϕ-variation with weight λ in the sense of Riesz into the space GRV ψ,λ (I, Y) and is uniformly continuous, then h, the generator function of the operator H, is affine in the second variable.…”

Section: Introductionmentioning

confidence: 99%

Uniformly Bounded Composition Operators

Głazowska¹,

Matkowski²

2015

Bull. Aust. Math. Soc.

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“…The uniformly continuous composition operators were firstly considered in [12] for the space of differentiable functions and absolutely continuous functions, later in [13] for the space of Hölder function, and in [14] for the space of bounded variation functions. Later, these were used in the main result of the papers [1,2,3,5].…”

Section: Introductionmentioning

confidence: 99%

SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV²_α(I)

Aziz¹,

Guerrero²,

Merentes³

2015

Bulletin of the Korean Mathematical Society

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Abstract. The space BV 2 α (I) of all the real functions defined on interval I = [a, b] ⊂ R, which are of bounded second α-variation (in the sense De la Vallé Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function h : I × R −→ R, and prove, in particular, that if H maps BV 2 α (I) into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.

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