On the continuous Cesàro operator in certain function spaces

Albanese, Angela A.; Bonet, José; Ricker, Werner J.

doi:10.1007/s11117-014-0321-5

Cited by 19 publications

(27 citation statements)

References 24 publications

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“…We also show that H n ([0, ∞)) = H n ((0, ∞)), (1.6) where 1 H n ((0, ∞)) := f : (0, ∞) → C f (j) ∈ AC loc ((0, ∞)), j = 0, 1, . .…”

Section: Introductionmentioning

confidence: 82%

“…The case n = 1 holds by (6.31). For n = 2, assume f ∈ AC (1) loc ((0, ∞)) with f, xf , x 2 f ∈ L 2 ((0, ∞)). Then for j = 0, one has again by (6.31), For j = 1, one notes that xf ∈ AC loc ((0, ∞)), see (6.26), and x(xf ) = x 2 f + xf ∈ L 2 ((0, ∞)).…”

Section: )mentioning

confidence: 99%

“…The fact, σ(T 1 ) = C(1; 1), as well as the resolvent formula (6.78) for T 1 are wellknown, we refer, for instance, to [9], and [8] (see also [1], [19], [29], [30], and the references cited therein). What appears to be less well-known is the a.c. nature of the spectrum of T 1 and the spectral representation in terms of the Mellin transform.…”

Section: )mentioning

confidence: 99%

“…, n − 1, will have finite limits f (j) (0 + ) at x = 0. There is an interesting connection with the spaces {H n ([0, ∞))} n∈N , namely that f ∈ H n ([0, ∞)) implies f ∈ H n−1 ([0, ∞)); (1.8) 1 We emphasize from the outset that despite the similarity of notation with Sobolev spaces, neither of the spaces Hn([0, ∞)) nor Hn((0, ∞)) coincides with the standard Sobolev space H n 0 ((0, ∞)). (See, however, Theorem 7.1 in the finite interval context.)…”

Section: Introductionmentioning

confidence: 99%

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On Birman's sequence of Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

Michael

et al. 2018

Journal of Differential Equations

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Dedicated with great pleasure to Eduard Tsekanovskii on the occasion of his 80th birthday.Abstract. In 1961, Birman proved a sequence of inequalities {In}, for n ∈ N, valid for functions in C n 0 ((0, ∞)) ⊂ L 2 ((0, ∞)). In particular, I 1 is the classical (integral) Hardy inequality and I 2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite b > 0, these inequalities hold on the standard Sobolev space H n 0 ((0, b)). Furthermore, in all cases, the Birman constants [(2n − 1)!!] 2 /2 2n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L 2 ((0, ∞)) (resp., L 2 ((0, b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.

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“…We also show that H n ([0, ∞)) = H n ((0, ∞)), (1.6) where 1 H n ((0, ∞)) := f : (0, ∞) → C f (j) ∈ AC loc ((0, ∞)), j = 0, 1, . .…”

Section: Introductionmentioning

confidence: 82%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Birman's sequence of Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

Michael

et al. 2018

Journal of Differential Equations

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“…C, the Cesàro operator Cf is de…ned by In classical analysis, the Cesàro operator was investigated from various aspects and a large number of results have appeared recently [1][2][3][4][5]. Titchmarsh [6] also used the operator as a convergence method for divergent integrals and introduced the Cesàro summability of integrals [7, p.11].…”

Section: Introductionmentioning

confidence: 99%

Cesàro Summability of Integrals of Fuzzy-Number-Valued Functions

Yavuz

Talo²,

Çoşkun³

2018

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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The impact of Lévy noise on the threshold dynamics of a stochastic susceptible‐vaccinated‐infected‐recovered epidemic model with general incidence functions

Kiouach,

Azami El‐idrissi,

Sabbar

2023

Math Methods in App Sciences

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Compartmental systems like the well‐famed SIR, SEIR, SQIR, SVIR, and their variants are efficient tools for the mathematical modeling of infectious illnesses, and they permit us to get a clear picture of how they proliferate. In actuality, the aleatory fluctuations factors present in the natural environment like storm surges, weather changes, and seismic tremors make the dissemination of epidemics susceptible to some randomness. This calls for a stronger mathematical formulation that takes into consideration this stochasticity effect. From this perspective, and in order to highlight in the same time the effect of the vaccination strategy, we survey in this paper an SVIR model with general incidence rates that is disturbed by both Brownian motions and Lévy jumps. Initially, we establish its well‐posedness in the sense that it has a unique positive and global‐in‐time solution. Then, we rely on some assumptions and nonstandard analytic techniques, to derive sufficient and almost necessary conditions for extinction, persistence in the mean, and also weak persistence. More explicitly, we identify firstly under some hypotheses a threshold between extinction and persistence in the mean. In other phrases, if , the infected population dies out while it persists in the mean when . Then, and by modifying the hypothetical framework in order to cover more incidence rates, we prove that can act also as a threshold between extinction and weak persistence. At last, we provide some numerical simulations to corroborate our findings and cover some particular cases of response functions.

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On the continuous Cesàro operator in certain function spaces

Cited by 19 publications

References 24 publications

On Birman's sequence of Hardy–Rellich-type inequalities

On Birman's sequence of Hardy–Rellich-type inequalities

Cesàro Summability of Integrals of Fuzzy-Number-Valued Functions

The impact of Lévy noise on the threshold dynamics of a stochastic susceptible‐vaccinated‐infected‐recovered epidemic model with general incidence functions

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