Sobolev spaces and symbolic calculus

We use elementary theory of distributions and geometry of Euclidean spaces to obtain a theorem on the symbolic calculus of several variables in spaces of Fourier series with weights.     

Weighted estimates for multilinear pseudodifferential operators

In this paper,we study the weighted estimates for multilinear pseudodifferential operators.We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions.Our result generalizes similar ones from the classical Ap weights to multiple weights.     

Dispersive estimates for principally normal pseudodifferential operators

In this article we construct parametrices and obtain dispersive estimates for a large class of principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove L^q Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough potentials. © 2004 Wiley Periodicals, Inc.     

Heat kernel estimates for pseudodifferential operators,fractional Laplacians and Dirichlet-to-Neumann operators

The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(?t P) associated with a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to ${t \in{\mathbb C}_+}$ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.     

Weighted estimates for bilinear fractional integral operators

Moen (2016) proved weighted estimates for the bilinear fractional integrals where . We improve his results when and consider the case . As a corollary we obtain a bilinear Stein–Weiss inequality where .     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

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.

     

A functional calculus of pseudodifferential operators on unimodular Lie groups

An algebra of pseudodifferential operators on a unimodular Lie group is constructed. It is defined by uniform estimates of the local symbols and by conditions on the decrease of the kernel outside the diagonal, formulated in terms of the weight function, having an intermediate growth between the volume function and a standard exponential. The decrease of the Green function is described, complex powers of elliptic operators of the considered classes are constructed, a meromorphic continuation of the kernels of complex powers is described, from where, in particular, the asymptotic behavior of the spectral function of operators of the considered class is obtained.Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 164–200, 1987.     

Triebel-lizorkin space estimates for multilinear operators of sublinear operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel-Lizorkin space. The operators include Littlewood-Paley operator and Marcinkiewicz operator.     

Sharp Lorentz space estimates for rough operators

We demonstrate the or mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number. Received December 10, 1999 / Published online April 12, 2001     

An FIO calculus for marine seismic imaging, II: Sobolev estimates

We establish sharp L ²-Sobolev estimates for classes of pseudodifferential operators with singular symbols [Guillemin and Uhlmann (Duke Math J 48:251–267, 1981), Melrose and Uhlmann (Commun Pure Appl Math 32:483–519, 1979)] whose non-pseudodifferential (Fourier integral operator) parts exhibit two-sided fold singularities. The operators considered include both singular integral operators along curves in \mathbb R²{\mathbb R^2} with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics [Felea (Comm PDE 30:1717–1740, 2005), Felea and Greenleaf (Comm PDE 33:45–77, 2008), Nolan (SIAM J Appl Math 61:659–672, 2000)].     

Spectral invariance,ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces

The pseudodifferential operators with symbols in the Grushin classes \~S _inf0 ^sup, , 0 < 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (*-algebras) in L(L ²(R ⁿ )) and in L(H ^st ) for weighted Sobolev spaces H _inf ^supst defined via a weight d function . In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and .The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hörmander.     

Modulation space estimates for the fractional integral operators

We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.     

Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces

We obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces. As applications, we improve the known estimates for oscillatory integrals.