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On discrete fractional integral operators and mean values of Weyl sums
Author(s) -
Pierce Lillian B.
Publication year - 2011
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdq127
Subject(s) - mathematics , multiplier (economics) , operator (biology) , fourier transform , integer (computer science) , perspective (graphical) , pure mathematics , algebra over a field , discrete mathematics , mathematical analysis , biochemistry , chemistry , geometry , macroeconomics , repressor , computer science , transcription factor , economics , gene , programming language
In this paper, we prove new ℓ p →ℓ q bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number r s , k ( l ) of representations of a positive integer l as a sum of s positive k th powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.