We extend Stein's maximal theorem to the bilinear setting. Let $M$ be a homogeneous space with a transitive action of a compact abelian group, and let $1 \le p,q \le 2$ and $1/2 \le r \le 1$ satisfy $1/p + 1/q = 1/r$. For a family of translation-invariant bilinear operators \[ T_m : L^p(M) \times L^q(M) \to L^r(M), \qquad m \in \mathbb{N}, \] that converge almost everywhere, we prove that the associated maximal operator \[ T^*(f,g) = \sup_m |T_m(f,g)| \] is of weak type $L^p(M) \times L^q(M) \to L^{r,\infty}(M)$. The proof relies on probabilistic methods and a bilinear extension of Stein's lemma for double Rademacher series. We also establish a bilinear analogue of Sawyer's extension of Stein's theorem for positive bilinear operators commuting with a mixing family of measure-preserving transformations. Applications include strong-type boundedness of maximal bilinear tail operators associated with ergodic transformations in the natural exponent range $r = (1/p + 1/q)^{-1}$ for $p,q > 1$, as well as almost everywhere convergence results for bilinear Bochner--Riesz means and other bilinear ergodic averages on the torus.
A well-known classical fact is that the L p boundedness of a maximal family of linear operators together with pointwise convergence for a dense subspace of the domain, implies a.e. convergence for all functions on the domain. In contrast to this classical direction, our results show that a.e. convergence of a bilinear sequence forces weak-type bounds for the associated maximal operator. Motivated by the work of Stein [31] and Sawyer [30], we obtain bilinear analogues of their results and we also study some of their consequences in terms of applications. The main results of this paper are Theorems 1 and 3, while our main application is Theorem 5.
In Stein’s celebrated work [31], it is shown that if a sequence of translation-invariant operators tT m u on a compact group acts boundedly on L p pM q, 1 ď p ď 2, and if T m f Ñ T f pointwise for every f P L p pM q, then the maximal operator
satisfies a weak-type pp, pq inequality. The argument is based on probabilistic techniques with Rademacher functions and provides a powerful connection between a.e. convergence
The authors acknowledge the support of the Simons Foundation and the University of Missouri Curators Fund. and weak-type pp, pq inequalities. In this work we develop an analogue for sequences of bilinear operators T m : L p pM q ˆLq pM q Ñ L r pM q with 1{p `1{q " 1{r and 1 ď p, q ď 2. It should be noted that the opposite direction also holds: if a sequence of multilinear operators satisfies a maximal weak-type estimate L p ˆLq Ñ L r,8 , then it converges a.e. for all L p ˆLq functions, assuming it does so for a dense subclass; for a proof of this fact see for instance [10, Proposition 2].
Interest in bilinear operators originated in the pioneering work of Coifman and Meyer [5] [6] in the seventies and by the celebrated work of Lacey and Thiele [23,24] on the bilinear Hilbert transform in the nineties. These works have spurred a resurgence of activity on adaptations of many classical linear results to multilinear analogues, such as the Calderón-Zygmund theory [16] and many other topics. In relation to a.e. convergence of multilinear singular operators and boundedness of the associated maximal operators (which naturally implies a.e. convergence) we refer the reader to [22] [27], [17] [28], [25], [8], [11], [26], [12], [9], [21]. This list is by no means exhaustive, but is representative of the work in this area.
Another contribution of this work is the bilinear analogue of Sawyer’s extension of Stein’s theorem to the case r ą 2 for positive operators under certain mild mixing conditions. The main application of our results concerns improved weak-type bounds for a bilinear tail operator associated with an ergodic measure-preserving transformation on a finite measure space.
We begin by introducing some preliminaries and then develop the probabilistic and measuretheoretic framework necessary for establishing Theorem 3.
Let G be a topological group and let M be a topological space. We say that G acts continuously on M if there is a continuous mapping:
We call M a homogeneous space of G if G acts continuously on M and transitively; the latter means for every pair of points x, y P M there exists an element g P G such that gpxq " y.
We also define the translation operator τ g acting on functions f on M by pτ g f qpxq " f pg ´1xq, g P G.
Now suppose G is a compact group and M is a homogeneous space of G. Then the homogeneous space M inherits a unique normalized G-invariant measure dµ from G that satisfies ż M f pgpxqq dµpxq " ż M f pxq dµpxq, for all f P L 1 pM q and g P G.
If we normalize the measure dµ on M such that
then G also has a finite Haar measure dω G , that satisfies
It is useful to recall the relation between the measures dµ on M and dω G on G. For a fixed point x 0 P M and a Borel subset E of M we define the fiber of E over the point x 0 under the action of G as follows: p E " tg P G : gpx 0 q P Eu.
Then, by translation invariance, for any x 0 P M we have (1) µpEq " ω G p p Eq.
Some classical examples that the reader may keep in mind are the following. First, when both the group and the space coincide with the k-torus, G " M " T k , the action is simply translation modulo 1:
g ¨x " g `x pmod 1q, g, x P T k , that is, addition is taken componentwise on the torus.
A second, closely related example arises when G " M is the dyadic group, in which case the action is given by componentwise dyadic addition: g ¨x " g `x pmod 2q.
Finally, one may think of the rotational setting: let G " SOpnq, the group of rotations in R n , and M " S n´1 , the unit sphere. Here the natural action is the usual rotation of vectors, g ¨x " gpxq, g P SOpnq, x P S n´1 .
These three cases serve as canonical examples of compact group actions and motivate the general framework discussed below. We now discuss the setup in the bilinear setting. Let M and G be as above.
Definition 1. For each m " 1, 2, . . . , let T m be a bilinear operator defined on L p pM, dµq
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