Mathematics Seminar next week

[math]

MATHEMATICS SEMINARS - 2011


Tuesday, 25 October, at 15:30 hrs

Dr. Isroil A. Ikromov
(Uzbek Academy of Sciences, Samarkand, Uzbekistan)

"Boundedness Problem for Maximal Operators associated with
hypersurfaces and related oscillatory integrals.

(Joint research with D. M\"uller and M. Kempe)

Abstract.
It is well-known results proved by E.M. Stein (1976 for the case
$n\ge3$) and J. Bourgain (1985 for the case $n=2$) that the spherical 
maximal operator is bounded on $L^p({\Bbb R}^n)$ if and only if 
$p>n/(n-1)$. Further, the bounded-ness problem had been considered by 
many authors for more general hypersurfaces. The problem is
connected to the geometric properties of the surface.
There is still the open conjecture of E.M. Stein about connection 
between the decay rate of the associated oscillatory integrals and the
exponent $p$ such that the maximal operator is bounded on $L^p({\Bbb 
R}^n)$. Following A.N. Varchenko we introduce notion of "height" of 
hyper-surface at a point. This notion allows us to obtain the sharp
exponent of boundedness for the maximal operator defined by any smooth
hypersurface in $L^p({\Bbb R}3)$.
In particular, we get a confirmation of the E.M. Stein conjecture
for arbitrary two-dimensional smooth hypersurfaces. In the talk we also
give some estimates for related oscillatory integrals with smooth phase
functions.


Venue: Luigi Stasi Seminar Room, ICTP Leonardo Building, first floor.