Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.     

Directional derivative estimates for Berezin's operator calculus

Directional derivative estimates for Berezin symbols of bounded operators on Bergman spaces of arbitrary bounded domains in are obtained. These estimates also hold in the setting of the Segal-Bargmann space on . It is also shown that our estimates are sharp at every point of by exhibiting the optimizers explicitly.

     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

Sobolev estimates for operators given by averages over cones

We prove a result related to work by A. Greenleaf and G. Uhlmann concerning Sobolev estimates for operators given by averages over cones. This is done using the almost orthogonality lemma of Cotlar and Stein, and the van der Corput lemma on oscillatory integrals.

     

Sobolev estimates for the Radon transform of Melrose and Taylor

We derive the regularity properties of the Radon transform of Melrose and Taylor for the scattering on a compact, convex obstacle with a smooth boundary. The result is formulated in terms of the highest order of contact of tangent lines with the boundary of an obstacle. The main ingredients of the proof are the estimates for degenerate oscillatory integral operators and almost orthogonal decompositions. © 1998 John Wiley & Sons, Inc.     

Randomized approximation of Sobolev embeddings, II

We study the approximation of Sobolev embeddings by linear randomized algorithms based on function values. Both the source and the target space are Sobolev spaces of non-negative smoothness order, defined on a bounded Lipschitz domain. The optimal order of convergence is determined. We also study the deterministic setting. Using interpolation, we extend the results to other classes of function spaces. In this context a problem posed by Novak and Woźniakowski is solved. Finally, we present an application to the complexity of general elliptic PDE.     

-error estimates for ``shifted' surface spline interpolation on Sobolev space

The accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant is reasonably smooth. However, for functions which have smoothness below a certain order associated with the basis function , no approximation power has yet been established. Hence, the purpose of this study is to discuss the -approximation order ( ) of interpolation to functions in the Sobolev space with \max(0,d/2-d/p)$">. We are particularly interested in using the ``shifted' surface spline, which actually includes the cases of the multiquadric and the surface spline. Moreover, we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met.

     

Sobolev estimates for constructive uniform-grid FFT interpolatory approximations of spherical functions

The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications. In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations \(\left (\frac {j\pi }{N}, \frac {k \pi }{N}\right )\), for j=1,…,N?1, k=0,…,2N?1. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than N. Using the \({\mathcal {O}}(N^{2})\) data, the spherical interpolatory approximation is efficiently constructed by applying the FFT techniques (in both azimuthal and latitudinal variables) with only \({\mathcal {O}}(N^{2} \log N)\) complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework that are well suited for quantification of various computer models. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.     

Sobolev spaces with only trivial isometries, II

In this article, we obtain a canonical form for surjective linear isometries provided U is an open, bounded, connected, domain with Lipschitz boundary, and . We will show there exists |c| = 1 and mapping τ that is a composition of a translation and a sign-changing permutation of coordinates such that Tf = cf(τ). As a corollary, if , all surjective isometries have this trivial form by the Sobolev Imbedding Theorem.      

Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in ${{\mathbb R}^n}$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L² estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.     

A counterexample to weighted estimates for multilinear Fourier multipliers with Sobolev regularity

The problem whether the weighted estimates for multilinear Fourier multipliers with Sobolev regularity hold under weak condition on weights is considered.     

Error estimates in Sobolev spaces for interpolating thin plate splines under tension

This paper discusses _Lp$L_p$-error estimates for interpolation by thin plate spline under tension of a function in the classical Sobolev space on an open bounded set with a Lipschitz-continuous boundary. A property of convergence is also given when the set of interpolating points becomes more and more dense.     

Distributed-order calculus: An operator-theoretic interpretation

Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 478–486, April, 2008.     

Sharp estimates for Jacobi matrices and chain sequences, II

Chain sequences are positive sequences {c_n} of the form c_n=g_n(1−g_n−1) for a nonnegative sequence {g_n}. They are very useful in estimating the norms of Jacobi matrices and for localizing the interval of orthogonality for orthogonal polynomials. We give optimal estimates for the chain sequences which are more precise than the ones obtained in the paper (Constructive Approx. 6 (1990) 363) and in our earlier paper (J. Approx. Theory 118 (2002) 94).     

Sobolev estimates for the complex green operator on a class of weakly pseudoconvex boundaries

On the boundary of a smooth bounded convex domain in C_n (more generally, a domain that admits a defining funtion that is plurisubharmonic on the boundary), the canonicl (narm-minimizing) solution operator of the -equation, the inverse of □_b (the complex Gereen Operator) and the Szegö projection are continuous is Sobolev norms.