arXiv subjects
Publications and source records attributed to Andreas Seeger.
The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation between classes of endpoint Mikhlin-Hörmander type Fourier multipliers, and one on the constant in the triangle inequality for the spaces $L^{p,r} $ when $p<1$.
Consider the surface measure $μ$ on a sphere in a nonvertical hyperplane on the Heisenberg group $\mathbb{H}^n$, $n\ge 2$, and the convolution $f*μ$. Form the associated maximal function $Mf=\sup_{t>0}|f*μ_t|$ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain-Demeter to prove $L^p$-boundedness of $M$ in an optimal range.
We show that, for suitable enumerations, the multivariate Haar system is a Schauder basis in the classical Sobolev spaces on $\mathbb R^d$ with integrability $1<p<\infty$ and smoothness $1/p-1<s<1/p$. This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a {conditional} Schauder basis for a nonempty open subset of the $(1/p,s)$-diagram. The results extend to (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type in the range of parameters $\frac{d}{d+1}<p<\infty$ and $\max\{d(1/p-1),1/p-1\}<s<\min\{1,1/p\}$, which is optimal except perhaps at the end-points.
We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy-Sobolev spaces $\dot H^1_α$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.
We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem.
We give an alternative proof of recent results by the authors on uniform boundedness of dyadic averaging operators in (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type. This result served as the main tool to establish Schauder basis properties of suitable enumerations of the univariate Haar system in the mentioned spaces. The rather elementary proof here is based on characterizations of the respective spaces in terms of orthogonal compactly supported Daubechies wavelets.
In May 2016, we organized a conference in harmonic analysis in honor of Professor Michael Christ, on the campus of the University of Wisconsin in Madison. We are happy to present sixteen open problems, almost all of which were contributed by participants of a problem session held in the afternoon of May 19, 2016.
We prove $L^{p_1}(\mathbb R^d)\times \dots \times L^{p_{n+2}}(\mathbb R^{d})$ polynomial growth estimates for the Christ-Journé multilinear singular integral forms and suitable generalizations.
In a previous paper by the authors the existence of Haar projections with growing norms in Sobolev-Triebel-Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel-Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.
We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $α$ of the form ${d}/{n}, n=2,3,\cdots$ there exist $α$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-α}$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $α$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb N$.
For $1<p<\infty$ we determine the precise range of $L_p$ Sobolev spaces for which the Haar system is an unconditional basis. We also consider the natural extensions to Triebel-Lizorkin spaces and prove upper and lower bounds for norms of projection operators depending on properties of the Haar frequency set.
Consider the wave equation associated with the Kohn Laplacian on groups of Heisenberg type. We construct parametrices using oscillatory integral representations and use them to prove sharp $L^p$ and Hardy space regularity results.
We study a.e. convergence on $L^p$, and Lorentz spaces $L^{p,q}$, $p>\tfrac{2d}{d-1}$, for variants of Riesz means at the critical index $d(\tfrac 12-\tfrac 1p)-\tfrac12$. We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on $L^2$ spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted $L^2$ spaces, and a sharp endpoint bound for Stein's square-function associated with the Riesz means.
A weak type $(1,1)$ estimate is established for the first order $d$-commutator introduced by Christ and Journé, in dimension $d\ge 2$.
We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radial Fourier multipliers.
We prove mixed norm space-time estimates for solutions of the Schroedinger equation, with initial data in $L^p$ Sobolev or Besov spaces, and clarify the relation with adjoint restriction.
Consider the Fourier restriction operator associated to a curve in $R^d$, $d\ge 3$. We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.
We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some $p>1$. Secondly, we obtain a necessary and sufficient condition for $L^2$ boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$ regularity result for such averages. We formulate various open problems.