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 severalâ€¦ (More)

A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalties of Rogers and HÃ¶lder, Young, andâ€¦ (More)

We study the analogues of the problems of averages and maximal averages over a surface in Rn when the euclidean structure is replaced by that of a vector space over a finite field, and obtain optimalâ€¦ (More)

We consider variants of van der Corputâ€™s lemma in higher dimensions. 1. The very well-known and extremely useful van der Corput lemma is the following: Van der Corputâ€™s lemma. Let I âŠ† R be anâ€¦ (More)

where P(s, t) is a polynomial in s and t with P(0,0)= 0, and âˆ‡P(0,0)= 0. We call H the (local) double Hilbert transform along the surface (s, t,P (s, t)). The operator may be precisely defined for aâ€¦ (More)

The notion of the magnitude of a metric space was introduced by Leinster in [11] and developed in [16], [12], [17], [20] and [13], but the magnitudes of familiar sets in Euclidean space are onlyâ€¦ (More)

for some fixed large N0; we shall call such weights admissible. Rubio de Francia [11] showed that for every w âˆˆ L(R) there is a nonnegative W âˆˆ L(R) such that â€–Wâ€–2 â‰¤ CÎ»â€–wâ€–2, CÎ» < âˆž if Î» > 0, and theâ€¦ (More)

It is known that if q is an even integer then the L(R) norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres â€œsimultaneouslyâ€¦ (More)

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L weighted inequalities. We prove a general inequality of this type for theâ€¦ (More)