Riesz transforms and related singular integrals.
Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds
We prove -bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for in a certain interval depending on the Lipschitz character of the domain.
Riesz transforms for Dunkl transform
In this paper we obtain the -boundedness of Riesz transforms for the Dunkl transform for all .
Riesz transforms for the Dunkl Ornstein-Uhlenbeck operator
We propose a definition of Riesz transforms associated to the Ornstein-Uhlenbeck operator based on the Dunkl Laplacian. In the case related to the group ℤ ₂ it is proved that the Riesz transform is bounded on the corresponding spaces, 1 < p < ∞.
Riesz transforms on connected sums
Assume that is a complete Riemannian manifold with Ricci curvature bounded from below and that satisfies a Sobolev inequality of dimension . Let be a complete Riemannian manifold isometric at infinity to and let . The boundedness of the Riesz transform of then implies the boundedness of the Riesz transform of
Riesz-Bochner summability of Fourier integrals and Fourier series
Riordan arrays, orthogonal polynomials as moments, and Hankel transforms.
Rodrigues-type formulae for Hermite and Laguerre polynomials.
Rotation invariant subspaces of Besov and Triebel-Lizorkin space: compactness of embeddings, smoothness and decay of functions.
Let H be a closed subgroup of the group of rotation of Rn. The subspaces of distributions of Besov-Lizorkin-Triebel type invariant with respect to natural action of H are investigated. We give sufficient and necessary conditions for the compactness of the Sobolev-type embeddings. It is also proved that H-invariance of function implies its decay properties at infinity as well as the better local smoothness. This extends the classical Strauss lemma. The main tool in our investigations is an adapted...
Rough Marcinkiewics integral operators.
Rough Marcinkiewicz integral operators on product spaces.
In this paper, we study the Marcinkiewicz integral operators MΩ,h on the product space Rn x Rm. We prove that MΩ,h is bounded on Lp(Rn x Rm) (1< p < ∞) provided that h is a bounded radial function and Ω is a function in certain block space Bq(0,0) (Sn−1 x Sm−1) for some q > 1. We also establish the optimality of our condition in the sense that the space Bq(0,0) (Sn−1 x Sm−1) cannot be replaced by Bq(0,r) (Sn−1 x Sm−1) for any −1 < r < 0. Our results improve some...
Rough maximal functions and rough singular integral operators applied to integrable radial functions.
Let Ω be homogeneous of degree 0 in Rn and integrable on the unit sphere. A rough maximal operator is obtained by inserting a factor Ω in the definition of the ordinary maximal function. Rough singular integral operators are given by principal value kernels Ω(y) / |y|n, provided that the mean value of Ω vanishes. In an earlier paper, the authors showed that a two-dimensional rough maximal operator is of weak type (1,1) when restricted to radial functions. This result is now extended to arbitrary...
Rough Maximal Oscillatory Singular Integral Operators
2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25In this paper, we establish the L^p boundedness of certain maximal oscillatory singular integral operators with rough kernels belonging to certain block spaces. Our L^p boundedness result improves previously known results.
Rough oscillatory singular integrals on ℝⁿ
We establish sharp bounds for oscillatory singular integrals with an arbitrary real polynomial phase P. The kernels are allowed to be rough both on the unit sphere and in the radial direction. We show that the bounds grow no faster than log deg(P), which is optimal and was first obtained by Papadimitrakis and Parissis (2010) for kernels without any radial roughness. Among key ingredients of our methods are an L¹ → L² estimate and extrapolation.
Rough singular integral operators with Hardy space function kernels on a product domain
In this paper we introduce atomic Hardy spaces on the product domain and prove that rough singular integral operators with Hardy space function kernels are bounded on . This is an extension of some well known results.
Rough singular integrals on product spaces.
Roundoff errors in the fast computation of discrete convolutions
The efficient evaluation of a discrete convolution is usually carried out as a repated evaluation of a discrete convolution of a special type with the help of the fast Fourier transform. The paper is concerned with the analysis of the roundoff errors in the fast computation of this convolution. To obtain a comparison, the roundoff errors in the usual (direct) computation of this convolution are also considered. A stochastic model of the propagation of roundoff errors. is employed. The theoretical...