LITTLEWOOD – PALEY INEQUALITY FOR ARBITRARY RECTANGLES IN R 2 FOR 0 < p ≤ 2

The one-sided Littlewood–Paley inequality for pairwise disjoint rectangles in R2 is proved for the Lp-metric, 0 < p ≤ 2. This result can be treated as an extension of Kislyakov and Parilov’s result (they considered the one-dimensional situation) or as an extension of Journé’s result (he considered disjoint parallelepipeds in Rn but his approach is only suitable for p ∈ (1, 2]). We combine Kislyakov and Parilov’s methods with methods “dual” to Journé’s arguments. §1. Statement of the theorem and the history of the problem Let Δm be pairwise disjoint intervals on R. In 1983, Rubio de Francia (see [1]) proved that (1) ∥ ∥ ∥ (∑ m |MΔmf | )1/2∥ ∥ ∥ Lp(R) ≤ Cp‖f‖Lp(R) for 2 ≤ p < ∞, where MΔf = (f̂χΔ) is the Fourier multiplier corresponding to a set Δ, and Cp depends on p only. Shortly after that, Journé (see [2]) extended this result to R , proving an estimate similar to (1) in the case where f is defined on R and Δm are pairwise disjoint parallelepipeds with sides parallel to coordinate axes. Note that the ndimensional version of (1) cannot be proved by n-fold application of the one-dimensional estimate. For this purpose Journé in [2] described and then used the theory of singular integral operators on product domains. Later, Fernando Soria offered (see [3]) a simpler proof in the case of R. Now we note that the dual version of estimate (1) can be written as (2) ∥ ∥ ∥ ∑ m fm ∥ ∥ ∥ Lp(R) ≤ Cp ∥ ∥ { fm }∥ ∥ Lp(R, 2) , 1 < p ≤ 2, where {fm} is a collection of functions with supp f̂m ⊂ Δm. In 1984, Bourgain (see [4]) proved that (2) remains true when p = 1. He used a more complicated method than that employed by Rubio de Francia. In 2005, Kislyakov and Parilov, acting in the spirit of Rubio de Francia’s methods, established (see [5]) that (2) is true for all 0 < p ≤ 2 (they considered the unit circle T rather than R, but this does not play a significant role). In this paper we extend their result (namely, inequality (2) for 0 < p ≤ 2) to R, proving the following theorem. Theorem. Let {fm} be a sequence of functions such that fm ∈ L(R) and supp f̂m ⊂ Δm, where the Δm are disjoint rectangles in R 2 with sides parallel to coordinate axes. 2010 Mathematics Subject Classification. Primary 42B25, 42B15.