Failure of the $H^1\times H^1⟶L^{1/2,\infty }$ estimate for the lacunary bilinear spherical maximal function and applications

Abstract

In contrast to Christ's theorem that the lacunary spherical maximal operator is bounded from $H^1({\mathbb{R}}^d)$ to $L^{1,\infty }({\mathbb{R}}^d)$ ($d\ge 2$), we show that the lacunary bilinear spherical maximal operator $${\mathcal{B}}_{\mathrm{lac}}(f,g)(x)=\underset{\ell \in \mathbb{Z}}{\sup }|{\int }_{{\mathbb{S}}^{2d-1}}f(x-2^{\ell }y)g(x-2^{\ell }z)d\sigma (y,z)|$$ is not bounded from $H^1({\mathbb{R}}^d)\times H^1({\mathbb{R}}^d)$ to $L^{1/2,\infty }({\mathbb{R}}^d)$ for any $d\ge 1$. In particular, it is not bounded from $L^1({\mathbb{R}}^d)\times L^1({\mathbb{R}}^d)$ to $L^{1/2,\infty }({\mathbb{R}}^d)$. More precisely, for every sufficiently large integer $N$, we construct two finite linear combinations of smooth, compactly supported, mean-zero $H^1({\mathbb{R}}^d)$-atoms for which the ratio of the weak $L^{1/2}$ quasinorm to the product of the two Hardy norms is at least $c_d\log (N+1)$. 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 $k\ge 3$ and $d\ge 1$, a modification of the construction proves that the lacunary $k$-linear spherical maximal operator does not map $(H^1({\mathbb{R}}^d){)}^k$ into $L^{1/k,\infty }({\mathbb{R}}^d)$. We also indicate extensions to certain hypersurfaces and to horizontal spherical averages on the Heisenberg group.