We consider the Lp norms of sums of characteristic functions of affine subspaces of a vector space V over a finite field under certain restrictions on p, dim V and the dimensions of the subspaces involved. We investigate the conditions under which these norms are increased when the affine subspaces are replaced by their parallel translates passing through 0. Applications to extremal configurations for Kakeya maximal-type inequalities are given and open questions are raised.

In this note we intend to discuss the method of A. Córdoba and R. Fefferman of using covering lemmas to control maximal functions, and make some simplifications which allow us to obtain alternative proofs of some of their results.

Weprove optimal radially weighted L2-norm inequalities for the Fourier extension operator associated to the unit sphere in ℝn. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

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.

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on $L^{2}(\mathbb{S}^{n-1})\times \ldots\times L^{2}(\mathbb{S}^{n-1})$.

Suppose that $R$ goes to infinity through a second-order lacunary set. Let $S_R$ denote the $R$th spherical partial inverse Fourier integral on ${\rm I\!R}^d$. Then $S_R f$ converges almost everywhere to $f$, provided that $f$ satisfies\[\int \widehat{f}(\xi)\log\log(8+|\xi|)^2\,d\xi < \infty.\]

We introduce an analogue of Calderón's first commutator along a parabola and establish its L2 boundedness under essentially sharp hypotheses.

Well over two decades have now passed since the publication of the classic books Singular integrals and differentiability properties of functions by E. M. Stein [32] and An introduction to Fourier analysis on Euclidean spaces by E. M. Stein and G. Weiss [37]. These two texts would, I am sure, be universally regarded as defining the ‘common core’ of harmonic analysis in the Calderón–Zygmund tradition in the early 1970s. What has been going on in the subject since then?

Let Σ ⊆ ℝn be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝn (U open patches in ℝn−1). The restriction problem for Σ is the question of the a priori estimate (for f ∈ S(ℝ))

(^denoting the Fourier transform). The Bochner-Riesz problem for Σ is the question of whether the functions

define Lp-bounded Fourier multiplier operators on ℝn in the range

.

Until the comparatively recent past, writers on the British Constitution have been somewhat insular, if not faintly xenophobic. The inference was that all was well with the British Constitution and that it had nothing to learn from outside example. This satisfaction with the state of affairs was given what was perhaps its most dramatic, certainly its best written expression by Professor Chrimes when he said—“This magic formula which better than any other reconciles the apparently irreconcilable fundamentals.”