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Strong variational and jump inequalities in harmonic analysis

We prove variational and jump inequalities for a large class of linear operators arising in harmonic analysis.
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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…
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A variation norm Carleson theorem

By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues…
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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…
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Imaginary powers of Laplace operators

We show that if L is a second-order uniformly elliptic operator in divergenceformonRd, then Ci(1+lol)d/2 < |ILi'I|,1-L,,. <? C2(1+1kaI)d/2. We also prove that the upper bounds remain true for any…
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Universal L-p improving for averages along polynomial curves in low dimensions

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A non-linear generalisation of the Loomis-Whitney inequality and applications

We establish a diffeomorphism–invariant generalisation of the classical Loomis–Whitney inequality in R. As a consequence we obtain a sharp trilinear restriction theorem for the Fourier transform in…
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Distributional and L-q norm inequalities for polynomials over convex bodies in R-n

K |p|q ) 1 q are all equivalent to each other. Recently there has been considerable interest in the behaviour of the constants in these equivalences as q varies when we consider arbitrary unit-volume…
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What is van der Corput’s lemma in higher dimensions?

We consider variants of van der Corput's lemma in higher dimensions. [Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid),…
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Singular maximal functions and Radon transforms near L1

<abstract abstract-type="TeX"><p>We show that some singular maximal functions and singular Radon transforms satisfy a weak type <i>L</i> log log <i>L</i> inequality. Examples include the maximal…
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