An answer to a question on the convolution of functions

Akbarbaglu, Ibrahim; Maghsoudi, S.

doi:10.1007/s00013-012-0396-9

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…Implication (i)⇒(ii) is trivial. Implication (ii)⇒(iii) follows from the Young inequality (note that 1/p + 1/q ≤ 2), and implication (iii)⇒(i) follows immediately from [5], Theorem 2.4 (in the same way as this implication in the proof of Theorem 18.41).…”

Section: Dichotomies For Convolution In L P Spacesmentioning

confidence: 79%

“…The main result of the paper [5] of Akbarbaglu and Maghsoudi implies the following result which together with Theorem 18.41 show that the assumptions on p and q in the Minkowski inequality cannot be relaxed.…”

Section: Dichotomies For Convolution In L P Spacesmentioning

confidence: 87%

“…It seems that the proof of [5], Theorem 2.4, can easily be rewritten so that the assertion of Theorem 18.42 holds also for the case p ∈ (1, ∞) and q ∈ (0, 1). Also, the case q = ∞ seems to be easy to handle.…”

Section: Proof the Implication (I)mentioning

confidence: 99%

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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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Section: Dichotomies For Convolution In L P Spacesmentioning

confidence: 79%

Section: Dichotomies For Convolution In L P Spacesmentioning

confidence: 87%

See 1 more Smart Citation

Bilinear mappings – selected properties and problems

Balcerzak¹,

Strobin²,

Wachowicz³

2013

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

View full text Add to dashboard Cite

show abstract

“…Recently, Głąb and Strobin [4], using the notion of porosity, generalised and considerably extended some interesting results on the convolution of functions essentially due to Rickert [7] andŻelazko [10]. See also [1,3,8] for some related results.…”

Section: Introductionmentioning

confidence: 99%