Affine Arclength Measure Research Articles

On the Oberlin affine curvature condition

We generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension d in Rn, 1≤d≤n−1. We show that a canonical equiaffine-invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of Oberlin with an exponent which is best possible. The proof combines aspects of geometric invariant theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.

Existence of Extremizers for a Model Convolution Operator

The operator $T$, defined by convolution with the affine arc length measure on the moment curve parametrized by $h(t)=(t,t^{2},...,t^{d})$ is a bounded operator from $L^{p}$ to $L^{q}$ if $(\frac{1}{p}, \frac{1}{q})$ lies on a line segment. In this article we prove that at non-end points there exist functions which extremize the associated inequality and any extremizing sequence is pre compact modulo the action of the symmetry of $T$. We also establish a relation between extremizers for $T$ at the end points and the extremizers of an X-ray transform restricted to directions along the moment curve. Our proof is based on the ideas of Michael Christ on convolution with the surface measure on the paraboloid.

Convolution estimates for measures on some complex curves

We consider the convolution operator for a measure supported on complex curves. The measure which we consider here is an analogue of the affine arclength measure for real curves. By modifying a combinatorial argument called the band structure argument, we prove the (nearly) optimal Lorentz space estimates. This includes the optimal strong type estimates as special cases. The complex curves we consider here are the ones considered for the Fourier restriction estimates for complex curves in \cite{BH}.

Uniform estimates for Fourier restriction to polynomial curves in ℝ d

We prove uniform $L^p \to L^q$ bounds for Fourier restriction to polynomial curves in ${\Bbb R}^d$ with affine arclength measure, in the conjectured range.

Uniform Lxp–Lx,rq improving for dilated averages over polynomial curves

Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator A given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially, A takes weighted averages over translates of the curve. In this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered. The sharp range of estimates for the resulting operator is obtained in all dimensions, except for an endpoint. The techniques used are redolent of those previously applied in the study of A . In particular, the arguments are based upon the refinement method of Christ, although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates.

UniformLp-improving for weighted averages on curves

We define variable parameter analogues of the affine arclength measure on curves and prove near-optimal Lp-improving estimates for associated multilinear generalized Radon transforms. Some of our results are new even in the convolution case.

Uniform Lorentz norm estimates for convolution operators

Uniform endpoint Lorentz norm improving estimates for convolution operators with affine arclength measure supported on simple plane curves are established. The estimates hold for a wide class of simple curves, and the condition is stated in terms of averages of the square of the affine arclength weight, extending previously known results.

Uniform estimates for the local restriction of the Fourier transform to curves

We prove sharp estimates, with respect to the affine arclength measure, for the restriction of the Fourier transform to a class of curves in R that includes curves of finite type. This measure possesses certain invariance and mitigation properties which are important in establishing uniform results.

Restriction of Fourier transforms to curves: An endpoint estimate with affine arclength measure

Abstract. Consider the Fourier restriction operators associated to curves in ℝ d , . 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.

Endpoint [formula omitted] bounds for integration along certain polynomial curves

We establish strong-type endpoint L p ( R d ) → L q ( R d ) bounds for the operator given by convolution with affine arclength measure on polynomial curves for d ⩾ 4 . The bounds established depend only on the dimension d and the degree of the polynomial.

An affine-invariant inequality for rational functions and applications in harmonic analysis

AbstractWe extend an affine-invariant inequality for vector polynomials established by Dendrinos and Wright to general rational functions. As a consequence we obtain sharp universal estimates for various problems in Euclidean harmonic analysis defined with respect to the so-called affine arc-length measure.

Fourier restriction to polynomial curves I: a geometric inequality

We prove a Fourier restriction result for general polynomial curves in ${\Bbb R}^d$. Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established.

Convolution with measures on flat curves in low dimensions

We prove L p → L q convolution estimates for the affine arclength measure on certain flat curves in R d when d ∈ { 2 , 3 , 4 } . For d = 2 , 3 , we also establish certain related Lorentz space estimates.

Endpoint bounds for a generalized Radon transform

We prove that convolution with affine arclength measure on the curve parametrized by $h(t) := (t,t^2,...,t^n)$ is a bounded operator from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds.

Universal [formula omitted] improving for averages along polynomial curves in low dimensions

We prove sharp L p → L q estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree. Our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to M. Christ. Almost sharp Lorentz space estimates are obtained as well.

Fourier restriction, polynomial curves and a geometric inequality

We announce a Fourier restriction result for general polynomial curves in R d . Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal bound for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. There are applications of this geometric inequality to other problems in euclidean harmonic analysis. To cite this article: S. Dendrinos, J. Wright, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The Lp-Lp mapping properties of convolution operators with the affine arclength measure on space curves

AbstractThe Lp-improving properties of convolution operators with measures supported on space curves have been studied by various authors. If the underlying curve is non-degenerate, the convolution with the (Euclidean) arclength measure is a bounded operator from L3/2()3 into L2(3). Drury suggested that in case the underlying curve has degeneracies the appropriate measure to consider should be the affine arclength measure and the obtained a similar result for homogeneous curves t→(t, t2, tk), t >0 for k ≥ 4. This was further generalized by Pan to curves t → (t, tk, tt), t > 0 for l < k < l, k+l ≥ 5. In this article, we will extend Pan's result to (smooth) compact curves of finite type whose tangents never vanish. In addition, we give an example of a flat curve with the same mapping properties.

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in R n

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in R^<i>n</i>

Convolution estimates for some degenerate curves

Let k ∈ R, k ≥ 3. Consider the curve of class C3 in R3 defined byLet σk(t) = t(k−3)/6dt be the affine arclength measure on γk. Define the convolution operator T (= Tk) by

Degenerate curves and harmonic analysis

This article deals with several related questions in harmonic analysis which are well understood for non-degenerate curves in ℝn, but poorly understood in the degenerate case. These questions invariably involve a positive ‘reference’ measure on the curve. In the non-degenerate case the choice of measure is not particularly critical and it is usually taken to be the Euclidean arclength measure. Since the questions considered here are invariant under the group of affine motions (of determinant 1), the correct choice of reference measure is the affine arclength measure. We refer the reader to Guggenheimer [8] for information on affine geometry. When the curve has degeneracies, the choice of measure becomes critical and it is the affine arclength measure which yields the most powerful results. From the Euclidean point of view the affine arclength measure has correspondingly little mass near the degeneracies and thus compensates automatically for the poor behaviour there. This principle should also be valid for general submanifolds of ℝn but alas the affine geometry of submanifolds is itself not well understood in general.