Averages Along the Primes: Improving and Sparse Bounds

Han, Rui; Krause, Ben; Lacey, Michael T.; Yang, Fan

doi:10.1515/conop-2020-0003

Cited by 11 publications

(10 citation statements)

References 30 publications

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“…This was known to be true for p > d+1 d−1 , it was independently established by Hughes [9] and Kesler-Lacey [11]. These types of l p -improving inequalities are a newer object of study in the discrete setting and the precise decay obtained in them provides information for a variety of applications which are just beginning to be explored; for example, see [6] for information about polynomial sequences and [1], [7] for information about primes. This paper is organized as follows: Section 2 contains the proof of our main result Theorem 1.2, and Section 3 concerns our l p -improving result Theorem 1.4.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

New bounds for discrete lacunary spherical averages

Anderson¹,

Madrid²

2020

Preprint

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We show that the discrete lacunary spherical maximal function is bounded on l p (Z d ) for all p > d+1 d−1 . Our range is new in dimension 4, where it appears that little was previously known for general lacunary radii. Our technique, using the Kloosterman refinement, also allows us to recover the range of l p improving estimates in all dimensions d ≥ 4. Though our range does not improve on the current known l p bounds for dimensions six and higher, our proof allows us to get estimates for the main term of the multiplier decomposition that match the conjectured overall sharp bounds.

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Section: Introduction and Main Resultsmentioning

confidence: 93%

New bounds for discrete lacunary spherical averages

Anderson¹,

Madrid²

2020

Preprint

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“…Our goal is to prove an estimate for a particular type of stopping times, the definition of which can also be found in [HKLY20].…”

Section: Sparse Boundsmentioning

confidence: 99%

“…The proof of Theorem 5.2 follows the idea of [HKLY20]. We will again use the auxiliary high-low construction, slightly modified this time.…”

Section: Sparse Boundsmentioning

confidence: 99%

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Averaging with the Divisor Function: $\ell^p$-improving and Sparse Bounds

Giannitsi¹

2021

Preprint

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We study averages along the integers using the divisor function d(n), and definedWe shall show that these averages satisfy a uniform, scale free ℓ p -improving estimate for p ∈ (1, 2), that isas long as f is supported on [0, N ].We will also show that the associated maximal functionsparse founds for p ∈ (1, 2), which implies that K * is bounded on ℓ p (w) for p ∈ (1, ∞), for all weights w in the Muckenhoupt Ap class.

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“…It is natural to suppose that N is sufficiently large, as a function of y. For the average over all primes, this inequality was established in [7], with study of the endpoint case in [11]. The novelty here is the uniformity in choice of arithmetic progression.…”

Section: Introductionmentioning

confidence: 99%

Improving and Maximal Inequalities for Primes in Progressions

Giannitsi¹,

Lacey²,

Mousavi³

et al. 2021

Preprint

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Assume that y < N are integers, and that (b, y) = 1. Define an average along the primes in a progression of diameter y, given by integer (b, y) = 1.Above, Λ is the von Mangoldt function and φ is the totient function. We establish improving and maximal inequalities for these averages. These bounds are uniform in the choice of progression. For instance, for 1 < r < ∞ there is an integer Ny,r so that supThe implied constant is only a function of r. The uniformity over progressions imposes several novel elements on the proof.

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Averages Along the Primes: Improving and Sparse Bounds

Abstract: Consider averages along the prime integers P given by

Cited by 11 publications

References 30 publications

New bounds for discrete lacunary spherical averages

New bounds for discrete lacunary spherical averages

Averaging with the Divisor Function: $\ell^p$-improving and Sparse Bounds

Improving and Maximal Inequalities for Primes in Progressions

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