Lower bounds for uncentered maximal functions in any dimension (joint work with B. Jaye and F. Nazarov)

Tuesday, September 20, 2016 - 2:00pm
Math Sci 111
Paata Ivanishvili (Kent State Univ)

Does the centered maximal function operator  have a nontrivial  fixed point in Lp over the n-dimensional Euclidean space? The answer is Yes if p>n/(n-2) with  n>2, and No otherwise. However, if one considers other maximal functions, for example,  uncentered (or ``almost centered'') maximal function operators defined over the family of shifts and dilates of a centrally symmetric convex body then the situation is much worse:   we will show that for  any n>0 and any p>1 there exists a constant  A=A(p,n) strictly greater than one  such that  | Mf|_p >  A |f|_p for any nonnegative f from Lp. In particular, this answers the question raised to the authors by Andrei Lerner. 

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