P. K. Ratnakumar

On bilinear Littlewood-Paley square functions

In this paper, we study bilinear Liitlewood-Paley square function introduced by M. Lacey. We give an easy proof of it’s boundedness from Lp(Rd) × Lq(Rd) into Lr(Rd), d ≥ 1, for all possible values of exponents p, q, r, i.e. for 2 ≤ p, q ≤ ∞, 1 ≤ r ≤ ∞ satisfying 1 p + 1 q = 1 r . We also prove analogous results for bilinear square functions on Torus group Td.

Gelfand pairs,K-spherical means and injectivity on the Heisenberg group

We study the injectivity properties of the spherical mean value operators associated to the Gelfand pairs (Hn,K), whereK is a compact subgroup ofU(n). We show that these spherical mean value operators are injective onLp Hn) for 1≤p<∞. Forp=∞, these operators are not injective. Nevertheless, if the spherical meansf*μi overK-orbits of sufficiently many points (zi,ti) ∈Hn vanish, we identify a necessary and sufficient condition on the points (zi,ti) which guaranteesf=0. ForK=U(n), this is equivalent to the condition for the two-radius theorem.

A Hardy-Sobolev inequality for the twisted Laplacian

We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on C n . The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.