arXiv subjects

Christoph Thiele

Publications and source records attributed to Christoph Thiele.

62 records · Page 4Linked to original sources

arXiv2002

A counterexample to a multilinear endpoint question of Christ and Kiselev

Christ and Kiselev have established that the generalized eigenfunctions of one-dimensional Dirac operators with $L^p$ potential $F$ are bounded for almost all energies for $p < 2$. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series $\sum_n T_n(F, ..., F)$ and carefully bounding each term $T_n(F, ..., F)$. It is conjectured that the results of Christ and Kiselev also hold for $L^2$ potentials $F$. However in this note we show that the bilinear term $T_2(F,F)$ and the trilinear term $T_3(F,F,F)$ are badly behaved on $L^2$, which seems to indicate that multilinear expansions are not the right tool for tackling this endpoint case.

math.CA↗

arXiv2002

On the bi-carleson operator I. The Walsh case

We prove $L^p$ estimates for the Walsh model of the maximal bi-Carleson operator (which is a hybrid of the bilinear Hilbert transform and the Carleson maximal operator which appears naturally in the eigenfunction problem for one-dimensional Schrodinger operators). The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.

math.CA↗

arXiv2001

Uniform estimates on multi-linear operators with modulation symmetry

We prove uniform $L^p$ bounds for multilinear operators which are given by multipliers whose symbols are singular on a one dimensional subspace. The novelty is that these bounds are uniform in the choice of the subspace.

math.CA↗

arXiv2001

Uniform estimates on paraproducts

We prove uniform $L^p$ estimates for a family of paraproducts and corresponding maximal operators.

math.CA↗

arXiv2001

$L^p$ estimates for the biest I. The Walsh case

We prove L^p estimates for the Walsh model of the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.

math.CA↗

arXiv2001

$L^p$ estimates for the biest II. The Fourier case

We prove L^p estimates for the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. In a previous paper these estimates were obtained for a simpler Walsh model for this operator, but in the Fourier case additional complications arise due to the inability to perfectly localize in both space and frequency.

math.CA↗

arXiv1999

Multi-linear operators given by singular multipliers

We prove L^p estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer, as well as the bilinear Hilbert transform.

math.CA↗

arXiv1999

On Calderón's conjecture

This paper is a successor of \cite{laceyt}. In that paper we considered bilinear operators of the form H_alpha(f_1,f_2)(x) = p.v. \int f_1(x-t) f_2(x + alpha t)/t dt, which are originally defined for f_1, f_2 in the Schwartz class S(R). The natural question is whether estimates of the form H_alpha(f_1,f_2)|_p <= C_{alpha,p_1,p_2} |f_1|_{p_1} |f_2|_{p_2} with constants C_{alpha,p_1,p_2} depending only on alpha,p_1,p_2 and p = p_1p_2/(p_1+p_2) hold. The purpose of the current paper is to extend the range of exponents p_1 and p_2 for which the estimate is known. In particular, the case p_1=2, p_2=\infty is solved to the affirmative. This was originally considered to be the most natural case and is known as Calderón's conjecture.

math.CA↗