Optimal constants and extremisers for some smoothing estimates

2012/06/22 by Neal Bez, Mitsuru Sugimoto, Bez, Neal +1 · 1 citation
Mathematics · #35B45 #35B65 #35P10 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.AP #math.CA #msc:35B45 #msc:35B65 #msc:35P10

paper · pdf · doi:10.48550/arxiv.1206.5110

21 pages, statement of Theorem 1.6 fixed, further minor modifications, references added

arxiv created 2012/11/11 · arxiv updated 2012/11/13

Abstract

We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form ‖ψ(|∇|) exp(itϕ(|∇|)f ‖_L²(w) ≤ C‖f‖_L², where the weight w 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 (w,ψ,ϕ). Furthermore, when w 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 w is inhomogeneous, we obtain new expressions for the optimal constant.

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