arXiv subjects
Publications and source records attributed to Ioannis Parissis.
Let M denote the maximal function along the polynomial curve p(t)=(t,t^2,...,t^d) in R^d: M(f)=sup_{r>0} (1/2r) \int_{|t| 0 is an absolute constant. The proof depends on the explicit construction of a "parabolic" semi-group of operators which is a mixture of stable semi-groups.
Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On the other hand we use a digit scrambled version of the van der Corput set to show that this bound is tight in the case N=2^n, for some positive integer n. These results unify the corresponding classical results of Roth and Schmidt in a sharp fashion.
Let Pd denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger that for every polynomial P in Pd: |p.v.\int_R {e^{iP(t)} dt/t} | < C(d) for some constant C(d) depending only on d. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is log d. We prove this conjecture.