Anthony Carbery

Multidimensional van der Corput and sublevel set estimates

If a function has a large derivative, then it changes rapidly, and so spends little time near any particular value. This paper is devoted to quantifying that principle for functions of several variables, particularly as it pertains to two problems in harmonic analysis. Along the way we shall encounter diverse problems and techniques, and shall be led to issues distinctly combinatorial in nature. We begin by reviewing two well-known questions in one-dimensional analysis. Suppose that u is a (smooth) real valued function on the real line R such that for some k ∈ N, u(t) ≥ 1 for all t ∈ R. Here, and in what follows, u denotes the k’th derivative of u. a) How small are the sublevel sets {t ∈ R : |u(t)| ≤ α} for small α? In particular, at what rate does the Lebesgue measure |{t ∈ R : |u(t)| ≤ α}| tend to zero as α→ 0? b) How quickly does the oscillatory integral I(λ) = ∫ b a edt tend to zero as λ→∞? Answers are given by the following two results. The first is known as van der Corput’s lemma; see, for example [S].

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.

Variants of the Calderón-Zygmund theory for Lp-spaces.

The purposes of this paper may be described as follows: (i) to provide a useful substitute for the Cotlar-Stein lemma for Lp-spaces (the orthogonality conditions are replaced by certain fairly weak smoothness asumptions); (ii) to investigate the gap between the Hormander multiplier theorem and the Littman-McCarthy-Riviere example - just how little regularity is really needed? (iii) to simplify and extend the work of Duoandikoetxea and Rubio de Francia and Christ and Stein, which sometimes has unnecessarily strong assumptions, and to introduce a sensitivity to different Lp-spaces which does not appear in their work.

Sets of divergence for the localization problem for Fourier integrals

Abstract A set E ⊆e 1283-1 ¦ ¦x¦ 2 ( 1283-2 ), with ƒ 0 on B, such that S R ƒ(x) diverges on E as R → ∞, where S R ƒ is the partial Fourier integral of ƒ over the ball ξ ¦ ¦ξ¦ ≤ R). It is asserted that every E of dimension less than n - 1 is an SDLP. When E is radial, E is an SDLP if E is compact and dim E n - 1/2. A precise formulation of this result in terms of capacity is given.

Strichartz inequalities with weights in Morrey-Campanato classes

We prove some weighted refinements of the classical Strichartz inequalities for initial data in the Sobolev spaces Ḣs(ℝn). We control the weightedL2-norm of the solution of the free Schrödinger equation whenever the weight is in a Morrey-Campanato type space adapted to that equation. Our partial positive results are complemented by some necessary conditions based on estimates for certain particular solutions of the free Schrödinger equation.

Heat-flow monotonicity related to the Hausdorff-Young inequality

It is known that if q is an even integer, then the L q (ℝ d ) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres ‘simultaneously slide’ to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if q > 2 is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff–Young inequality.

Averages in vector spaces over finite fields

We study the analogues of the problems of averages and maximal averages over a surface in when the euclidean structure is replaced by that of a vector space over a finite field, and obtain optimal results in a number of model cases

A multilinear generalisation of the Cauchy-Schwarz inequality

We prove a multilinear inequality which in the bilinear case reduces to the Cauchy-Schwarz inequality. The inequality is combinatorial in nature and is closely related to one established by Katz and Tao in their work on dimensions of Kakeya sets. Although the inequality is elementary in essence, the proof given is genuinely analytical insofar as limiting procedures are employed. Extensive remarks are made to place the inequality in context.

A NOTE ON LOCALISED WEIGHTED INEQUALITIES FOR THE EXTENSION OPERATOR

We prove optimal radially-weighted L 2 -norm inequalities for the Fourier extension operator associated to the unit sphere in R n . Such inequal- ities valid at all scales are well-understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

A Multilinear Extension Inequality In Rn

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on L2 (Sn−1) × … × L2 (Sn−1). 2000 Mathematics Subject Classification 42B10.