Multilinear paraproducts revisited

Abstract

We prove that multilinear paraproducts are bounded from products of Lebesgue spaces L p 1 ×· · ·×L p m+1 to L p,∞ , when 1 ≤ p1, . . . , pm, pm+1 < ∞, 1/p1 + · · · + 1/pm+1 = 1/p. We focus on the endpoint case when some indices pj are equal to 1, in particular we obtain a new proof of the estimate L 1 × · · · × L 1 → L 1/(m+1),∞ .Mathematics Subject Classification (2010): Primary 42B20; Secondary 42E30.

Variable Hardy–Lorentz spaces

“…In a recent paper, Grafakos and He [29] provided a careful treatment of the weak Hardy space ,∞ (ℝ ) for all indices 0 < < ∞, starting from the classical definition of the Poisson maximal function. He [31] obtained a new square function characterization for the space ,∞ (ℝ ), which was used to obtain weak type endpoint estimates for multilinear paraproducts in [30]. It is noticed that Kempka and Vybíral [34] recently introduced the variable Lorentz space (⋅), (⋅) (ℝ ), and showed that these spaces arise through real interpolation between variable Lebesgue spaces and ∞ (ℝ ) when (⋅) ≡ is a constant.…”

Square function characterization of weak Hardy spaces

“…The preceding corollary has applications in the theory of paraproducts. See [14]. Moreover, the following corollary can be proved similarly to the previous corollary.…”