Optimal Extension of Fourier Multiplier Operators in L p (G)

Mockenhaupt, Gerd; Okada, Susumu; Ricker, Werner J.

doi:10.1007/s00020-010-1829-0

Cited by 9 publications

(6 citation statements)

References 39 publications

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“…Its optimal extension has been investigated in [5]. For convolution (and more general Fourier multipliers) operators in L p (G), 1 ≤ p < ∞, with G a compact abelian group, see [24], [27,Ch.7] and the references therein. The optimal extension of the classical Hardy operator in L p (R), 1 < p < ∞, with kernel K (t, s) := (1/t)χ [0,t] (s) has been investigated in [11].…”

Section: Introductionmentioning

confidence: 99%

Inversion and extension of the finite Hilbert transform on $$\mathbf {(-1,1)}$$

Curbera

Okada

Ricker

2019

Annali di Matematica

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The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible such domain (i.e., its optimal domain). Some classical operators are already optimally defined (e.g., the Hilbert transform in L p (R), 1 < p < ∞), and others are not (e.g., the Hausdorff-Young inequality in L p (T), 1 < p < 2, or the Sobolev inequality in various spaces). In this paper, a detailed investigation is undertaken of the finite Hilbert transform T acting on rearrangement invariant spaces X on (−1, 1), an operator whose singular kernel is neither positive nor does it possess any monotonicity properties. For a large class of such spaces X , it is shown that T is already optimally defined on X (this is known for L p (−1, 1) for all 1 < p < ∞, except p = 2). The case p = 2 is significantly different because the range of T is a proper dense subspace of L 2 (−1, 1). Nevertheless, by a completely different approach, it is established that T is also optimally defined on L 2 (−1, 1). Our methods are also used to show that the solution of the airfoil equation, which is well known for the spaces L p (−1, 1) whenever p = 2 (due to certain properties of T), can also be extended to the class of r.i. spaces X considered in this paper.

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Section: Introductionmentioning

confidence: 99%

Inversion and extension of the finite Hilbert transform on $$\mathbf {(-1,1)}$$

Curbera

Okada

Ricker

2019

Annali di Matematica

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“…given by (2.10) on p. 576 of [21]. Since J (2) is injective, m ϕ and m (2) ϕ have the same null sets.…”

Section: The Optimal Domain and Integral Extension Of T ϕmentioning

confidence: 88%

Integral extension of multiplier operators in A(G)

Okada¹,

Ricker

2011

Rev Mat Complut

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We investigate operator ideal properties of multiplier operators T ϕ (via ϕ ∈ ∞ ( ) with the dual group) acting in the space A(G) of all integrable functions on an infinite, compact abelian group G whose Fourier coefficients are absolutely summable. Of interest is when T ϕ is 1-summing, as this corresponds to ϕ = f with f square integrable relative to Haar measure. Precisely then there is an optimal Banach function space L 1 (m ϕ ) available which contains A(G) densely and continuously and such that T ϕ has a continuous A(G)-valued linear extension I m ϕ to L 1 (m ϕ ). Relevant for studying L 1 (m ϕ ) and I m ϕ is the space S p , 1 ≤ p ≤ ∞, of all functions in L p (G) whose Fourier series is unconditionally convergent in L p (G). Amongst other things, it is shown that I m ϕ is 1-summing iff T ϕ is nuclear iff the vector measure m ϕ (E) := T ϕ (χ E ) has finite variation. Moreover, L 1 (m ϕ ) is a homogeneous Banach space. Hence, it is also a Banach algebra and an L 1 (G)-module under convolution.

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“…[17], [18], [22], [33]), the Hardy operator, [19], and the Hausdorff-Young inequality, [47]. For convolutions with measures in L p -spaces see [53], [54], [58,Ch.7] and for more general Fourier p-multiplier operators we refer to [46].…”

Section: Integral Representation Of the Finite Hilbert Transformmentioning

confidence: 99%

Non-extendability of the finite Hilbert transform

2021

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We present a detailed survey of recent developments in the study of the finite Hilbert transform and its corresponding inversion problem in rearrangement invariant spaces on (−1, 1). Contents 1. Introduction 1 2. Preliminaries 3 3. The airfoil equation: L p -theory 5 4. The finite Hilbert transform in rearrangement invariant spaces 9 5. Inversion of the finite Hilbert transform and the airfoil equation 11 6. Extension of the finite Hilbert transform 13 7. The finite Hilbert transform acting on LlogL 15 8. The fine spectrum of the finite Hilbert transform 19 9. Integral representation of the finite Hilbert transform 25 References 30

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Optimal Extension of Fourier Multiplier Operators in L p (G)

Abstract: Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ψ : Γ → C (with Γ the dual group), we study the optimal domain of the multiplier operator T

Cited by 9 publications

References 39 publications

Inversion and extension of the finite Hilbert transform on $$\mathbf {(-1,1)}$$

Inversion and extension of the finite Hilbert transform on $$\mathbf {(-1,1)}$$

Integral extension of multiplier operators in A(G)

Non-extendability of the finite Hilbert transform

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