On the Lp-conjecture for locally compact groups

Abtahi, Fatemeh; Nasr-Isfahani, Rasoul; Rejali, Ali

doi:10.1007/s00013-007-1993-x

Cited by 12 publications

(5 citation statements)

References 18 publications

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“…Saeki [25] proved that for a locally compact group G and 1 < p < ∞, L p (G) is a Banach algebra under convolution if and only if G is compact. We have also considered only this property, that f * g exists for all f, g ∈ L p (G), and proved that, for 2 < p < ∞, the compactness of G is equivalent to the existence of f * g for all f, g ∈ L p (G); see [1]. Furthermore, see [2,3] for the more general case of weighted L p -spaces.…”

Section: Some Aspects Of Il J (G) As a Banach Algebramentioning

confidence: 99%

“…We first note that with a proof similar to that in [25], we can easily prove that G is unimodular. If G is not compact, with an argument similar to [1] for a fixed compact symmetric neighbourhood B of the identity element of G and a real number α with α ≤ 1/2, αM J > 1 and αm J > 1, we may find a sequence (a n ) in G such that…”

Section: Some Aspects Of Il J (G) As a Banach Algebramentioning

confidence: 99%

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AN ARBITRARY INTERSECTION OFL_p-SPACES

Abtahi

Amini

Lotfi

et al. 2012

Bull. Aust. Math. Soc.

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Section: Some Aspects Of Il J (G) As a Banach Algebramentioning

confidence: 99%

Section: Some Aspects Of Il J (G) As a Banach Algebramentioning

confidence: 99%

AN ARBITRARY INTERSECTION OFL_p-SPACES

Abtahi

Amini

Lotfi

et al. 2012

Bull. Aust. Math. Soc.

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“…Note that the presented results generalize some earlier ones (cf. [1], [36]). The first result is connected with the famous L p -conjecture.…”

Section: Dichotomies For Convolution In L P Spacesmentioning

confidence: 99%

Bilinear mappings – selected properties and problems

Balcerzak¹,

Strobin²,

Wachowicz³

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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This work is a survey of results and problems connected with selected properties of bilinear mappings between function spaces. Section 1 is devoted to the lack of the counterpart of the Banach openness principle for bilinear surjections between Banach spaces. Some counterexamples and positive results are discussed. Then we deal with variants of openness for multiplication in spaces C(K), for a compact K, and in other Banach algebras and function spaces. In Section 2, several phenomena of dichotomies for operators of multiplication and convolution in spaces of integrable or continuous functions are presented. Some natural properties of these operators hold either always or the sets of objects having the given property are small. The smallness is described by meagerness and porosity. Let us introduce some basic notation. The ball with center x and radius r in a given metric space X is denoted by B X (x, r) (if X is fixed, index X will be omitted). The interior of a set A ⊂ X will be written as int(A).

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“…Motivated by the L p -conjecture, in [1], we have considered only the property that f * g exists for all f, g ∈ L p (G) and proved the following result which was indeed our purpose of that work.…”

Section: Introductionmentioning

confidence: 98%

On the weighted $\ell^p$-space of a discrete group

Abtahi¹,

Nasr-Isfahani²,

Rejali³

2009

Publ. Math. Debrecen

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Let G be a locally compact group and 1 < p < ∞. The L p -conjecture asserts that L p (G) is closed under the convolution if and only if G is compact. For 2 < p < ∞, we have recently shown that f * g exists and belongs to L ∞ (G) for all f, g ∈ L p (G) if and only if G is compact. Here, we consider the weighted case of this result for a discrete group G and a weight function ω on G; we prove that f * g exists and belongs to ℓ ∞ (G, 1/ ω) for all f, g ∈ ℓ p (G, ω) if and only if ℓ p (G, ω) ⊆ ℓ q (G, 1/ ω), the dual of ℓ p (G, ω).

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On the Lp-conjecture for locally compact groups

Cited by 12 publications

References 18 publications

AN ARBITRARY INTERSECTION OFL_p-SPACES

AN ARBITRARY INTERSECTION OFL_p-SPACES

Bilinear mappings – selected properties and problems

On the weighted $\ell^p$-space of a discrete group

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On the Lp-conjecture for locally compact groups

Cited by 12 publications

References 18 publications

AN ARBITRARY INTERSECTION OFLp-SPACES

AN ARBITRARY INTERSECTION OFLp-SPACES

Bilinear mappings – selected properties and problems

On the weighted $\ell^p$-space of a discrete group

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AN ARBITRARY INTERSECTION OFL_p-SPACES

AN ARBITRARY INTERSECTION OFL_p-SPACES