We study the problem of dominating the dyadic strong maximal function by (1, 1)-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance.
Let H be a .d 1/-dimensional hyperbolic paraboloid in R d and let Ef be the Fourier extension operator associated to H, with f supported in B d 1 .0; 2/. We prove that kEf k L p .B.0;R// Ä C " R " kf k L p for all p 2.d C 2/=d whenever d=2 m C 1, where m is the minimum between the number of positive and negative principal curvatures of H. Bilinear restriction estimates for H proved by S. Lee and Vargas play an important role in our argument.
We show that $ \bigg \|\sup _{0 < t < 1} \big |\sum _{n=1}^{N} e^{2\pi i (n(\cdot ) + n^2 t)}\big | \bigg \|_{L^{4}([0,1])} \leq C_{\epsilon } N^{3/4 + \epsilon } $ and discuss some applications to the theory of large values of Weyl sums. This estimate is sharp for quadratic Weyl sums, up to the loss of $N^{\epsilon }$.
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