Abstract:
In the recent paper [5] by P. A. Kulikov, a weak factorization of the Paley–Wiener space $\mathrm{PW}^{1}_{2 a}$ into a sum of products $\mathrm{PW}^{p}_{a} \cdot \mathrm{PW}^{q}_{a}$ was established for $1/p + 1/q = 1$, $1 < p < \infty$, by using a certain operator technique. In the present paper, it is shown that a method from A. L. Volberg's paper [9] allows one to decompose $\mathrm{PW}^{1}_{2 a}$ into a finite sum of products $\mathrm{PW}^{p}_{2 a / p} \cdot \mathrm{PW}^{q}_{2 a / q}$ for suitable rational exponents. Moreover, the same method yields a weak factorization of the Paley–Wiener spaces $\mathrm{PW}^{X^2}_{2 a}$ into a sum of three terms $\mathrm{PW}^{X}_{a} \cdot \mathrm{PW}^{X}_{a}$ for arbitrary rearrangement invariant lattices of measurable functions $X$ having the Fatou property and $r$-convex with some $r > 0$, together with similar results for spaces of polynomials on the circle.
In the case of the Hardy spaces on the bidisk, another weak factorization result of A. L. Volberg from the same paper admits a generalization to Hardy-type spaces for $1$-concave rearrangement invariant lattices. In particular, this partially generalizes a notable weak factorization result of S. Ferguson and M. Lacey for the space $\mathrm{H}_{1} [\mathbb T^2]$.
Key words and phrases:weak factorization, Paley-Wiener spaces, polynomial spaces, rearrangement invariant spaces, Hardy spaces.
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