Circular average relative to fractal measures

Abstract

<p style='text-indent:20px;'>We prove new <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>–<inline-formula><tex-math id="M2">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> estimates for averages over dilates of the circle with respect to fractal measures, which unify different types of maximal estimates for the circular average. Our results are consequences of <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>–<inline-formula><tex-math id="M4">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.</p>

Remarks on dimension of unions of curves

“…It is well-known that Marstrand's results [20] can be deduced from L p , p = ∞, estimate for the circular maximal function. In [12], some of the authors extended the L p circular maximal estimate to that relative to α-dimensional measures, which recovers the aforementioned Wolff's result in [33]. Moreover, L p maximal bound was extended to space curves (see [28,16,1,17]).…”

Nikodym sets and maximal functions associated with spheres

“…In the paper [HKL22], it was conjectured that (1.8) holds for any p > 4 − α when 1 < α ≤ 2, which was proved for α ≥ 3 − √ 3. This conjecture would imply L p (R 2 )-estimates for the maximal function S T for compact T ⊂ R 2 with finite s-dimensional upper Minkowski content for every 0 ≤ s < 1 and p > 2 + s, which would be sharp except possibly for endpoint.…”

“…Indeed, it is possible to obtain L p -bounds for a smaller range of exponents p by using the local smoothing estimates for the wave equation, which is enough to deduce Corollary 1.12. We learned this observation from the authors of [HKL22]. There is a related result due to Wolff [Wol00] and Oberlin [Obe06] (see also [Mit99]): a Borel set containing a set of spheres of Hausdorff dimension larger than 1 (as a subset of R n × R + ) must have positive Lebesgue measure.…”

“…For fractal measures, the problem has been studied in [Wol99a,Eg04,CHL17,Har19,Har18] for the case p = 2, and sharp results are known for n = 2, 3. For the case p > 2, Ham-Ko-Lee [HKL22] obtained some sharp estimates and used them to prove L p -improving estimates for the circular maximal function relative to a class of fractal measures as a corollary. We use results obtained in the papers [CHL17,HKL22] for the proof of Theorem 1.10, which is summarized in Theorem 4.10.…”

$$L^p-L^q$$ Local Smoothing Estimates for the Wave Equation via k-Broad Fourier Restriction

“…As the exponents pn,k decrease with k, it is natural to explore if higher orders of multilinearity imply further progress on Conjecture 1.1. This line of investigation was considered by Lee [23] for n = 2 using the trilinear reduction of Lee and Vargas [25]; see also the recent work [17]. In this note, we further extend the multilinear approach to any dimension and any level of linearity in the case q > p. We remark that whereas partial results for q = p using this method were discussed in [8], our focus is on sharp results with q > p. Rather than working with k-linear estimates, we will work with their k-broad variants (see §3), which hold in the full range p ≥ pn,k .…”