Bounds for lacunary bilinear spherical and triangle maximal functions (Q6632431)

For two functions \(f \in L^p(\mathbb{R}^d)\), \(g \in L^q(\mathbb{R}^d)\) their bilinear spherical average is \N\[\N\mathcal{A}_t(f,g)(x) = \int_{S^{2d-1}} f(x- ty)g(x-tz)d\sigma_{2d-1}(y,z), \N\]\Nhere \(\sigma_{2d-1}\) is the normalized surface measure on the unit sphere \(S^{2d-1}\subset \mathbb{R}^{2d}\).\N\NThe authors study the properties of the lacunar maximal function \N\[\N\mathcal{M}_{lac}(f,g)(x) = \sup_{l\in \mathbb{Z}} | \mathcal{A}_{2^{-l}}(f,g)(x)|, \N\]\Nand compare it with the standard spherical maximal function \N\[\N\mathcal{M}(f,g)(x) = \sup_{t>0} | \mathcal{A}_{t}(f,g)(x)|. \N\]\N\NThey establish the boundedness of the operator \(\mathcal{M}_{lac}(f,g) \to L^r(\mathbb{R}^d)\), proving in particular that, if \(d\geq 2\) \N\[\N\| \mathcal{M}_{lac}(f,g)\|_r = \| \sup | \mathcal{A}_{2^{-l}}(f,g)|\|_r \prec \|f\|_p\|g\|_q \N\]\Nfor \(p,q\in (1,\infty]\) and \( r^{-1}= p^{-1} + q^{-1}\).\N\NThe authors also consider a more general operator of this type where \(\sigma_{2d-1}\) is replaced by a more general measure and also with another kind of averaging, namely triangle averaging.