Neal Bez

A sharp Strichartz estimate for the wave equation with data in the energy space

We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the

Optimal constants and extremisers for some smoothing estimates

L^4_{t,x}(\R^{5+1})

Stability of the Brascamp-Lieb constant and applications

norm of the solution in terms of the energy. We also characterise the maximisers.

Heat-flow monotonicity related to the Hausdorff-Young inequality

AbstractWe establish new results concerning the existence of extremisers for a broad class of Kato-smoothing estimates of the form

boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves

{\left\| {\psi \left( {\left| \nabla \right|} \right)\exp \left( {it\phi \left( {\left| \nabla \right|} \right)f} \right)} \right\|_{{L^2}\left( \omega \right)}} \leqslant C{\left\| d \right\|_{{L^2}}}

New minimal bounds for the derivatives of rational Bézier paths and rational rectangular Bézier surfaces

‖ψ(|∇|)exp(itϕ(|∇|)f)‖L2(ω)≤C‖d‖L2 for solutions of dispersive equations, where the weight ω is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the L2-boundedness of a certain oscillatory integral operator S depending on (ω, ψ, ϕ). Furthermore, when ω is homogeneous, and for certain (ψ, ϕ), we provide an explicit spectral decomposition of S*S and consequently recover an explicit formula for the optimal constant C and a characterisation of extremisers. In certain well-studied cases when ω is inhomogeneous, we obtain new expressions for the optimal constant and the non-existence of extremisers.

Behaviour of the Brascamp–Lieb constant

abstract:We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis.

Some sharp bilinear space-time estimates for the wave equation

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.