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The norm of the Riemann‐Liouville operator on L p [0,1]: a probabilistic approach
Author(s) -
Adell José A.,
GallardoGutiérrez Eva A.
Publication year - 2007
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/bdm041
Subject(s) - mathematics , infimum and supremum , smoothness , probabilistic logic , operator (biology) , uniform norm , upper and lower bounds , mathematical proof , riemann hypothesis , norm (philosophy) , operator norm , pure mathematics , discrete mathematics , mathematical analysis , operator theory , statistics , geometry , repressor , biochemistry , chemistry , political science , transcription factor , law , gene
We obtain explicit lower and upper bounds for the norm of the Riemann–Liouville operator V s on L p [0, 1] which are asymptotically sharp, thus completing previous results by Eveson. Similar statements are shown with respect to the norms || V s f || p , whenever f satisfies certain smoothness properties. It turns out that the correct rate of convergence of || V s f || p as s → ∞ depends both on the infimum of the support of f and on the degree of smoothness of f . We use a probabilistic approach which allows us to give unified proofs.