Emanuel Carneiro

Emanuel Carneiro; Jeffrey D. Vaaler

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.

Emanuel Carneiro

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.

Emanuel Carneiro; Friedrich Littmann; Jeffrey D. Vaaler

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.

Ricardo J. Alonso; Emanuel Carneiro; Irene M. Gamba

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

Emanuel Carneiro; Vorrapan Chandee; Micah B. Milinovich

Jonathan W. Bober; Emanuel Carneiro; Kevin Hughes; Lillian B. Pierce

{L^{s}_{weak}}

Emanuel Carneiro; Vorrapan Chandee

Emanuel Carneiro; Diego R. Moreira

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.

Emanuel Carneiro; Kevin Hughes

Let

Emanuel Carneiro; Jeffrey D. Vaaler