On discrete fractional integral operators and related Diophantine equations

Jongchon Kim

doi:10.4310/MRL.2015.v22.n3.a11

On discrete fractional integral operators and related Diophantine equations

Jongchon Kim^*

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

2 Citations (Scopus)

Abstract

We study discrete versions of fractional integral operators along curves and surfaces. l ^p→l ^q estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve γ (m)=(m,m²,…,m^k) to Vinogradov’s mean value theorem. Sharp l ^p→l ^q estimates of the discrete fractional integral along the hyperbolic paraboloid in Z³ are also obtained except for endpoints.

Original language	English
Pages (from-to)	841-857
Number of pages	17
Journal	Mathematical Research Letters
Volume	22
Issue number	3
Online published	20 May 2015
DOIs	https://doi.org/10.4310/MRL.2015.v22.n3.a11
Publication status	Published - 2015
Externally published	Yes

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© 2015, International Press of Boston, Inc. All rights reserved.

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abstract = "We study discrete versions of fractional integral operators along curves and surfaces. l p→l q estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve γ (m)=(m,m2,…,mk) to Vinogradov{\textquoteright}s mean value theorem. Sharp l p→l q estimates of the discrete fractional integral along the hyperbolic paraboloid in Z3 are also obtained except for endpoints.",

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AU - Kim, Jongchon

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AB - We study discrete versions of fractional integral operators along curves and surfaces. l p→l q estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve γ (m)=(m,m2,…,mk) to Vinogradov’s mean value theorem. Sharp l p→l q estimates of the discrete fractional integral along the hyperbolic paraboloid in Z3 are also obtained except for endpoints.

UR - https://www.scopus.com/pages/publications/84929885637

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84929885637&origin=recordpage

U2 - 10.4310/MRL.2015.v22.n3.a11

DO - 10.4310/MRL.2015.v22.n3.a11

M3 - RGC 21 - Publication in refereed journal

AN - SCOPUS:84929885637

SN - 1073-2780

VL - 22

SP - 841

EP - 857

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 3

ER -

On discrete fractional integral operators and related Diophantine equations

Abstract

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