Algebras of Continuous Fourier Multipliers on Variable Lebesgue Spaces
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Algebras of Continuous Fourier Multipliers on Variable Lebesgue Spaces Alexei Yu. Karlovich Abstract. We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by Mascarenhas, Santos and Seidel. Mathematics Subject Classification. 42B15, 46E30. Keywords. Continuous Fourier multiplier, variable Lebesgue space, Stechkin’s inequality, piecewise continuous function, slowly oscillating function.
1. Introduction Let R˙ and R be the compactifications of the real line R by means of the point ∞ and the two points ±∞, respectively. The space of continuous functions on R that have finite limits at −∞ and +∞ is denoted by C(R), and ˙ := {f ∈ C(R) : f (−∞) = f (+∞)}. C(R) Let C stand for the constant complex-valued functions on R and C0 (R) for the continuous functions on R which vanish at ±∞. Notice that C, C0 (R), ˙ decomposes ˙ and C(R) are closed subalgebras of L∞ (R), and that C(R) C(R), ˙ ˙ into the direct sum C(R) = C+C0 (R). For f ∈ L1 (R), let F f denote the Fourier transform (F f )(x) := f (t)eixt dt, x ∈ R. R
This work was partially supported by the Funda¸c˜ ao para a Ciˆencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matem´ atica e Aplica¸c˜ oes).
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√ If f ∈ L1 (R)∩L2 (R), then F f ∈ L2 (R) and F f L2 (R) = 2πf L2 (R) . Since L1 (R) ∩ L2 (R) is dense in L2 (R), the operator F extends to a bounded linear operator of L2 (R) onto L2 (R), which will also be denoted by F . The inverse of F is given by (F −1 g)(t) = (2π)−1 (F g)(−t) for a.e. t ∈ R. Let p(·) : R → [1, ∞] be a measurable a.e. finite function. By Lp(·) (R) we denote the set of all complex-valued functions f on R such that Ip(·) (f /λ) := |f (x)/λ|p(x) dx < ∞ R
for some λ > 0. This set becomes a Banach function space when equipped with the norm f p(·) := inf λ > 0 : Ip(·) (f /λ) ≤ 1 . It is easy to see that if p is constant, then Lp(·) (R) is nothing but the standard Lebesgue space Lp (R). The space Lp(·) (R) is referred to as a variable Lebesgue space. Let P(R) denote the set of all measurable a.e. finite functions p(·) : R → [1, ∞] such that 1 < p− := ess inf p(x), x∈R
ess sup p(x) =: p+ < ∞. x∈R
If p(·) ∈ P(R), then the space Lp(·) (R) is separable and reflexive, and the set Cc∞ (R) of all infinitely differentiable compactly supported functions is dense Lp(·) (R) (see, e.g., [6, Chap. 2] or [10, Chap. 3]). Let p(·) ∈ P(R). A function a ∈ L∞ (R) is called a Fourier multiplier on the variable Lebesgue space Lp(·) (R) if the operator W 0 (a) := F −1 aF maps the dense set L2 (R) ∩ Lp(·) (R) of Lp(·) (R) into itself and extends to a bounded linear operator on Lp(·) (R). Let Mp(·) st
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