arXiv subjects
Publications and source records attributed to Andreas Seeger.
We show that maximal operators formed by dilations of Mikhlin-H"ormander multipliers are typically not bounded on $L^p(R^d)$. We also give rather weak conditions in terms of the decay of such multipliers under which $L^p$ boundedness of the maximal operators holds.
Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $μ_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*μ_t|$. We prove for $n\ge 2$ that $M$ defines an operator bounded on $L^p(H^n)$ provided that $p>2n/(2n-1)$. This improves an earlier result by Nevo and Thangavelu, and the range for $L^p$ boundedness is optimal. We also extend the result to a more general setting of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.
A locally compact group $G$ is said to be weakly amenable if the Fourier algebra $A(G)$ admits completely bounded approximative units. Consider the family of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the $2n+1$ dimensional Heisenberg group and $SL(2,\Bbb R)$ acts via the irreducible representation of dimension $2n$ fixing the center of $H_n$. We show that these groups fail to be weakly amenable. Following an idea of Haagerup for the case $n=1$ one can reduce matters to the problem of obtaining nontrivial uniform bounds for a family of singular oscillatory integral operators with product type singularities and polynomial phases. The result on the family $G_n$ and various other previously known results are used to settle the question of weak amenability for a large class of Lie groups, including the algebraic groups; we assume that the Levi-part has finite center.
For a function $f\in L^p(\Bbb R^d)$, $d\ge 2$, let $A_t f(x)$ be the mean of $f$ over the sphere of radius $t$ centered at $x$. Given a set $E\subset (0,\infty)$ of dilations we prove endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E} |A_t f(x)|$.
We show that if $f$ is locally in $L\log\log L$ then the lacunary spherical means converge almost everywhere. The argument given here is a model case for more general results on singular maximal functions and Radon transforms (see ref. 6).
We show that some singular maximal functions and singular Radon transforms satisfy a weak type $L\log\log L$ inequality. Examples include the maximal function and Hilbert transform associated to averages along a parabola. The weak type inequality yields pointwise convergence results for functions which are locally in $L\log\log L$.
We prove various estimates for the mean square lattice point discrepancy for dilates of a convex body.
We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors.
We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption.
We prove variable coefficient versions of L^p boundedness results on Hilbert transforms and maximal functions along convex curves in the plane.
Sharp L^2 estimates for oscillatory integral operators and Fourier integral operators associated with canonical relations having two-sided cusp or one-sided swallowtail singularities are obtained.
We demonstrate the $(H^1,L^{1,2})$ or $(L^p,L^{p,2})$ 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.
Let $A_tf(x)=\int f(x+ty)dσ(y)$ denote the spherical means in $\Bbb R^d$ ($dσ$ is surface measure on $S^{d-1}$, normalized to $1$). We prove sharp estimates for the maximal function $M_E f(x)=\sup_{t\in E}|A_tf(x)|$ where $E$ is a fixed set in $\Bbb R^+$ and $f$ is a {\it radial} function $\in L^p(\Bbb R^d)$. Let $p_d=d/(d-1)$ (the critical exponent of Stein's maximal function). For the cases (i) $p<p_d$, $d\ge 2$ and (ii) $p=p_d$, $d\ge 3$, and for $p\le q\le\infty$ we prove necessary and sufficient conditions for $L^p\to L^{p,q}$ boundedness of the operator $M_E$.