Emanuel Carneiro

Some Extremal Functions in Fourier Analysis, III

We obtain the best approximation in L1(ℝ), by entire functions of exponential type, for a class of even functions that includes e−λ|x|, where λ>0, log |x| and |x|α, where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

A Sharp Inequality for the Strichartz Norm

Let be the solution of the linear Schrodinger equationIn the first part of this paper, we obtain a sharp inequality for the Strichartz norm , where , and , that admits only Gaussian maximizers. As corollaries, we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi [4] and Hundertmark-Zharnitsky [6]) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper, we express Foschis [4] sharp inequalities for the Schrodinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

Gaussian subordination for the Beurling-Selberg extremal problem

We determine extremal entire functions for the problem of ma- jorizing, minorizing, and approximating the Gaussian function e x 2 by en- tire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance (3), (4), (10) and (17)), plus a variety of new interesting functions such asjxj for 1 < ; log (x 2 + 2 )=(x 2 + 2 ) , for 0 < ; log x2 + 2 ; andx2n logx2 , forn2 N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomi- als and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

Convolution Inequalities for the Boltzmann Collision Operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in Lp we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L2 case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some

ON A DISCRETE VERSION OF TANAKA'S THEOREM FOR MAXIMAL FUNCTIONS

{L^{s}_{weak}}

On the regularity of maximal operators

or Ls. The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.

On the endpoint regularity of discrete maximal operators

Let