On the Sharpness of Mockenhaupt’s Restriction Theorem

Abstract

We prove that the range of exponents in Mockenhaupt's restriction theorem for Salem sets [12], with the endpoint estimate due to Bak and Seeger [1], is optimal.Mathematics Subject Classification: 28A78, 42A32, 42A38, 42A45Date: May 30, 2013 (revised).

Convolution Powers of Salem Measures With Applications

“…Furthermore, he remarked that for suitable examples there is a possibility that even the stronger Stein-Tomas L p → L 2 (µ) bound could hold in this range. Recently Hambrook and Laba [11] gave, for a dense set of α's (and d = 1), examples of Salem sets of dimension α, which show that the p range for the L p → L 2 (µ) bound in [1] cannot be improved in general. Their examples carry randomness and arithmetic structures at different scales.…”

A Class of Random Cantor Measures, with Applications

“…(Note that taking s = σ = d − 1, this recovers the Stein-Tomas estimate in the case of the sphere). Hambrook and Laba [9] (see also [5] for a generalization) constructed, for a dense set of t ∈ [0, 1], measures µ on the real line satisfying (4.6) and (4.7) for s, σ arbitrarily close t, and supported on sets of Hausdorff dimension t, for which the restriction estimate (4.5) does not hold for any p < p t,t,1 . This shows that in general the result of Mockenhaupt, Bak and Seeger is sharp also for fractal measures.…”

Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions

“…The sequence (A j ) ∞ j=0 can be constructed by making trivial modifications to the construction of Chen [3] (cf. [6,7]). Note that A j ⊆ N −1 j [N j ] and |A j | = T j .…”