On spectral gaps of Markov maps

Conde-Alonso, José M.; Parcet, Javier; Ricard, Éric

doi:10.1007/s11856-018-1693-1

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…Observe that here the restriction on the dimension arises for the case of the full spherical maximal operator due to the estimate (9), where the L 2 boundedness of M full is required (following from Theorem 5.3). Indeed, M full is not bounded for p ≤ 2 in dimension n = 2.…”

Section: 3mentioning

confidence: 99%

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Bilinear spherical maximal functions of product type

Roncal¹,

Shrivastava²,

Shuin³

2020

Preprint

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In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón-Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. in [15]. We deal with lacunary and full versions of this operator, and we prove weighted estimates with respect to bilinear weights. Our approach involves sparse forms following ideas by Lacey [20], but we also use other techniques to handle the particular triplet (2, 2, 1).

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Section: 3mentioning

confidence: 99%

“…Then, M lac (f 1 , f 2 ), h can be dominated by the sum of finitely many sparse forms. Finally, one can find a universal sparse form (see [9,Proposition 2.1]) in the sparse domination. We proceed to prove the sparse domination result for the operator M D∩Q 0 .…”

Section: Sparse Domination: Proof Of Theorem 22mentioning

confidence: 99%

Bilinear spherical maximal functions of product type

Roncal¹,

Shrivastava²,

Shuin³

2020

Preprint

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“…This is clearly equivalent to δ P (T ) < 1. When X is taken as a non-commutative L p -spaces, the spectral gap of Markov operator has been recently studied in [5]. In the classical setting, this gap has been extensively investigated by many authors (see for example, [17]).…”

Section: Preliminariesmentioning

confidence: 99%

Spectral conditions for uniform $P$-ergodicities of Markov operators on abstract states spaces

Erkurşun-Özcan¹,

Mukhamedov²

2020

Preprint

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In the present paper deals with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P -ergodicity, i.e. T n − P → 0, here P is a projection. We have showed that T is uniformly P -ergodic if and only if T n − P ≤ Cβ n , 0 < β < 1. In this paper, we prove that such a β is characterized by the spectral radius of T − P . Moreover, we give Deoblin's kind of conditions for the uniform P -ergodicity of Markov operators.

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“…This is clearly equivalent to δ P (T ) < 1. When X is taken as a noncommutative L p -spaces, the spectral gap of Markov operator has been recently studied in [8]. In the classical setting, this gap has been extensively investigated by many authors (see for example, [25]).…”

Section: Thenmentioning

confidence: 99%

Generalized Dobrushin ergodicity coefficient and uniform ergodicities of Markov operators

Mukhamedov

Al-Rawashdeh

2019

Positivity

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In our earlier paper, a generalized Dobrushin ergodicity coefficient of Markov operators (acting on abstract state spaces) with respect to a projection P , has been introduced and studied. It turned out that the introduced coefficient was more effective than the usual ergodicity coefficient. In the present work, by means of a left consistent Markov projections and the generalized Dobrushin's ergodicity coefficient, we investigate uniform and weak P -ergodicities of non-homogeneous discrete Markov chains (NDMC) on abstract state spaces. It is easy to show that uniform P -ergodicity implies a weak one, but in general the reverse is not true. Therefore, some conditions are provided together with weak P -ergodicity of NDMC which imply its uniform P -ergodicity. Furthermore, necessary and sufficient conditions are found by means of the Doeblin's condition for the weak P -ergodicity of NDMC. The weak P -ergodicity is also investigated in terms of perturbations. Several perturbative results are obtained which allow us to produce nontrivial examples of uniform and weak P -ergodic NDMC. Moreover, some category results are also obtained. We stress that all obtained results have potential applications in the classical and non-commutative probabilities.MSC : 47A35; 60J10, 28D05

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On spectral gaps of Markov maps

Cited by 10 publications

References 11 publications

Bilinear spherical maximal functions of product type

Bilinear spherical maximal functions of product type

Spectral conditions for uniform $P$-ergodicities of Markov operators on abstract states spaces

Generalized Dobrushin ergodicity coefficient and uniform ergodicities of Markov operators

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