Andreas Seeger

Local smoothing of Fourier integral operators and Carleson-Sjölin estimates

The purpose of this paper is twofold. First, if Y and Z are smooth paracompact manifolds of dimensions n ~ 2 and n + 1 , respectively, we shall prove local regularity theorems for a certain class of Fourier integral operators !T E /f1.(Z , Y; ~) which naturally arise in the study of wave equations. Secondly, we want to apply these estimates to prove versions of the Carleson-Sjolin theorem on compact two-dimensional manifolds with periodic geodesic flow. The operators we shall study satisfy the curvature condition introduced in [32]. The main example occurs when Z = Y x JR and !T is the solution operator for the Cauchy problem for the wave equation

A variation norm Carleson theorem

By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by Hunt [9] for 1 < p < ∞; see also [7],[15], and [8]. The main purpose of this paper is to sharpen Theorem 1.1 towards control of the variation norm in the parameter ξ. Thus we consider mixed L and V r norms of the type:

Strong variational and jump inequalities in harmonic analysis

We prove variational and jump inequalities for a large class of linear operators arising in harmonic analysis.

Singular integral operators with rough convolution kernels

in all dimensions, again under the assumption Q e L log L. It is conceivable that a variant of the arguments in [3] for the maximal operator could also work for the singular integral operator; in fact, in unpublished work, the authors of [3] obtained a weak type (1, 1) inequality in dimension d < 7. However their arguments if applied to the singular integral operator lead to substantial technical difficulties and no

Wave front sets, local smoothing and Bourgain's circular maximal theorem

The purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of Bourgains circular maximal theorem [1]. We use easy wave front analysis along with techniques previously used in proofs of the Carleson-Sj6lin theorem (see [3],[5],[7]) and in the proof of sharp regularity properties of Fourier integral operators [13]. The circular means operators are defined by

Radon transforms and finite type conditions

The purpose of this paper is to study averaging operators of Radon transform type. We shall formulate suitable finite type conditions and prove LP-Sobolev and LPLq estimates. The results will be essentially sharp for operators associated with families of curves in the plane. Let XC and T be smooth manifolds, dimX3 = nL, dim2 = nR, and let M be a submanifold in XC x T with conormal bundle N*M; we denote by t the codimension of M. We shall always assume that the projections lry M -* X and 7rT M 42 are surjective with rank DirX = nL, rank Dr9 = nR. This in particular implies that N*M C T*X\O x T* \O where 0 refers to the zero sections in T*X and T*2, respectively. This is the usual assumption for the canonical relation associated with Lagrangian distributions arising as kernels of Fourier integral operators. The assumptions on Dirx, D7rT imply that for fixed x E X, y E 2 the sets AMX = {y E T; (X, y) E MI, M = x{yE2X;(x,y) EM}I

Classes of singular integral operators along variable lines

We prove estimates for classes of singular integral operators along variable lines in the plane, for which the usual assumption of nondegenerate rotational curvature may not be satisfied. The main Lp estimates are proved by interpolating L2 bounds with suitable bounds in Hardy spaces on product domains. The L2 bounds are derived by almost-orthogonality arguments. In an appendix we derive an estimate for the Hilbert transform along the radial vector field and prove an interpolation lemma related to restricted weak type inequalities.

Sharp Lorentz space estimates for rough operators

Abstract. We demonstrate the

Some inequalities for singular convolution operators in ^{}-spaces

(H^1,L^{1,2})

On X-ray transforms for rigid line complexes and integrals over curves in

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