Failure of the H¹× H¹ L1/2,∞ estimate for the lacunary bilinear spherical maximal function and applications

In contrast to Christ's theorem that the lacunary spherical maximal operator is bounded from H¹( Rᵈ) to L1,∞( Rᵈ) (d≥2), we show that the lacunary bilinear spherical maximal operator Blac(f,g)(x) =∈ Z |∫_ S²ᵈ⁻¹f(x-2^ y)\,g(x-2^ z)\,dσ(y,z)| is not bounded from H¹( Rᵈ)× H¹( Rᵈ) to L1/2,∞( Rᵈ) for any d≥1. In particular, it is not bounded from L¹( Rᵈ)× L¹( Rᵈ) to L1/2,∞( Rᵈ). More precisely, for every sufficiently large integer N, we construct two finite linear combinations of smooth, compactly supported, mean-zero H¹( Rᵈ)-atoms for which the ratio of the weak L1/2 quasinorm to the product of the two Hardy norms is at least cdlog(N+1). This lower bound holds already for the supremum over a suitable set of N dyadic radii. For $d=1$ this answers the case $p=q=1$ of a question in Borges's survey. For every k≥3 and d≥1, a modification of the construction proves that the lacunary k-linear spherical maximal operator does not map (H¹( Rᵈ))ᵏ into L1/k,∞( Rᵈ). We also indicate extensions to certain hypersurfaces and to horizontal spherical averages on the Heisenberg group.