We investigate a class of sharp Fourier extension inequalities on the planar curves s " |y| p , p ą 1. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if 1 ă p ă p0, for some p0 ą 4. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schrödinger equation in one spatial dimension. One of our tools is a geometric comparison principle for n-fold convolutions of certain singular measures in R d , developed in the companion paper [32]. We further show that any extremizer exhibits fast L 2 -decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves s " y|y| p´1 , p ą 1.

We prove a quadratic sparse domination result for general non-integral square functions S. That is, we prove an estimate of the formwhere q * 0 is the Hölder conjugate of q 0 /2, M is the underlying doubling space and S is a sparse collection of cubes on M . Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L p (w).2010 Mathematics Subject Classification. 42B20, 42B37.

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We prove a quadratic sparse domination result for general non-integral square functions S. That is, for $$p_0 \in [1,2)$$ p 0 ∈ [ 1 , 2 ) and $$q_0 \in (2,\infty ]$$ q 0 ∈ ( 2 , ∞ ] , we prove an estimate of the form "Equation missing"where $$q_{0}^{*}$$ q 0 ∗ is the Hölder conjugate of $$q_{0}/2$$ q 0 / 2 , M is the underlying doubling space and $${\mathcal {S}}$$ S is a sparse collection of cubes on M. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace–Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space $$L^{p}(w)$$ L p ( w ) .

We prove sufficient conditions for the two-weight boundedness of the Bergman projection on the unit ball. The first condition is in terms of Orlicz averages of the weights, while the second condition is in terms of the mixed B∞-B2 characteristics.