Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, due to a lack of practical estimators. They also include, as special cases, L quantities which are used in many applications. We propose and analyze a family of estimators for Sobolev quantities of unknown prob...

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dyn...

Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $$\mathbb R^n$$ are provided. Namely, of type $$\Vert u-P\Vert _{Y(\mathbb R^n)}\le C\Vert \nabla ^m u\Vert _{X(\mathbb R^n)}$$ , where X and Y either spaces over or Orlicz u is a $$m-$$ times weakly differentiable function whose gradient in X, P polynomial order at most $$m-1$$ depending u, C constant inde...

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal...

We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on generic tori dimension d≥1. study the behavior high Sobolev norms Hs, s≫1, solutions initial conditions in Hs whose Hρ-Sobolev norm, 1≪ρ≪s, is smaller than ε≪1. provide control Hs-norm over time interval order O(ε−2). Due to lack conserved quantities controlling norms, key ingredient proof const...

This paper proposes nonparametric deconvolution density estimation over S 2. Here we would think of the S 2 elements of interest being corrupted by random SO(3) elements (rotations). The resulting density on the observations would be a convolution of the SO(3) density with the true S 2 density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis on SO(3) an...