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CATEGORIES:Analysis Seminar
SUMMARY:Bochner-Riesz Means on Heisenberg-Type Groups - Ad
am Horwich (University of Birmingham)
DTSTART:20181120T140000Z
DTEND:20181120T150000Z
UID:TALK3443AT
URL:/talk/index/3443
DESCRIPTION:The question of the range of p for which Bochner-R
iesz means of a function f in Lp converge to f has
been well studied on R^n in an attempt to prove t
he Bochner-Riesz conjecture\, which has been prove
n for n=2 but remains open for n>2. For p not smal
ler than 2\, a similar result is expected for maxi
mal Bochner-Riesz means\, and this has led to ques
tions of almost-everywhere convergence\, a weaker
result implied by Lp convergence of the maximal Bo
chner-Riesz means. The almost-everywhere result wa
s proven by Carbery\, Rubio de Francia and Vega (1
988) and has been extended to Heisenberg groups by
Gorges and Muller (2002)\, where Bochner-Riesz me
ans are now defined in terms of the sub-Laplacian
on these groups.\n\nWe prove result regarding conv
ergence of Bochner-Riesz means on Heisenberg-type
(H-type) groups\, a class of 2-step nilpotent Lie
groups that includes the Heisenberg groups. We bro
adly follow the method of Gorges and Muller\, whic
h in term is an adaption of techniques used by Car
bery\, Rubio de Francia and Vega. The implicit res
ults in both papers\, which reduce estimates for t
he maximal Bochner-Riesz operator from Lp to weigh
ted L2 spaces and from the maximal operator to the
non-maximal operator\, have been stated as stand-
alone results\, as well as simplified and extended
to all stratified Lie groups. We also develop for
mulae for integral operators for fractional integr
ation on the dual of H-type groups corresponding t
o pure first and second layer weights on the group
\, which are used to develop `trace lemma' type in
equalities for H-type groups. Obtaining these esti
mates requires an understanding of certain special
functions (in particular Jacobi polynomials).
LOCATION:University House G08
CONTACT:Diogo Oliveira E Silva
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