There are presented two extrapolation methods for multi-sublinear operators that allow to derive estimates for functions from the given estimates on characteristic functions. The results are stated only for bi-sublinear operators, but they stay valid also for \(m\)-sublinear operators for any \(m\in\mathbb{N}\). The paper consists of four sections. The first one explains the concepts and provides notations and definitions. The second one is devoted to a general method that allows to extend the estimation imposed only on characteristic functions to specific Lorentz spaces. The authors start this section with sublinear operators, and then generalize to bi-sublinear operators. We state here the main result only for sublinear operators. The bi-sublinear case of operators is more complicated, but the main ideas are similar.
Given an increasing function \(D\) such that \(D(0)=0\) and \(0<q<\infty\), the Lorentz space \(\Lambda^q(dD)\) contains all measurable functions such that \(\|f\|_{\Lambda^q(dD)}= (\int_0^\infty f^{*q}\,dD)^{1/q} \approx (\int_0^\infty \lambda^qD(\mu_f(\lambda))\frac{d\lambda}{\lambda})<\infty\), where \(f^*\) is a decreasing rearrangement of \(f\) and \(\mu_f\) is a distribution function of \(f\). It is a quasi-Banach space under the assumption of the \(\Delta_2\) condition for \(D\), i.e., \(D(2t) \leq CD(t)\) for some \(C>0\) and all \(t>0\). By Galb\((X)\), we denote a sequence space called galb of a quasi-Banach space \(X\), the notion introduced in [P.Turpin, “Convexités dans les espaces vectoriels topologiques généraux” (Dissertationes Math., Warszawa 131) (1976; Zbl 0331.46001)].
Theorem. Let \(T\) be a sublinear operator, let \(X\) be a quasi-Banach lattice space and let \(D\) be an increasing function such that \(D(0)=0\). Assume that, for any measurable set \(E\) with \(|E|<\infty\), we have
\[ \|T(\chi_E)\|_X \leq CD(|E|). \]
Then the following are statement valid:

(a)

If Galb\((X) = \ell^1\), then
\[ T: \Lambda^1(dD) \to X. \]

(b)

If Galb\((X) = \ell^p\) with \(0<p<1\), then
\[ T: \Lambda^p(dD^p) \to X. \]

(c)

If Galb\((X) = \ell(\log \ell)^\alpha\) with \(\alpha > 0\), then
\[ T: \Lambda_\alpha^*(dD) \to X, \]
where \(\Lambda_\alpha^*(dD)\) is the subspace of \(\Lambda^1(dD)\) defined by the functional
\[ \begin{split} \|f\|_{\Lambda_\alpha^*(dD)} = \int_0^\infty \lambda D(\mu_f(\lambda))\left(1 + \log^+ \frac{\|f\|_{\Lambda^1(dD)}}{\lambda D(\mu_f(\lambda))}\right)^\alpha \frac{d\lambda}{\lambda}\\ = \|f\|_{\Lambda^1(dD)} \int_0^\infty \varphi_\alpha \left( \frac{\lambda D(\mu_f(\lambda))}{\|f\|_{\Lambda^1(dD)}}\right)\, \frac{d\lambda}{\lambda} \end{split} \]
with \(\varphi_\alpha (t) = t\left(1+ \log^+ 1/t\right)^\alpha\).

In Section 3, the authors consider bi-sublinear operators which, for given \(\varepsilon, \delta > 0\), are \((\varepsilon, \delta)\)-atomic, \((\varepsilon, \delta)\)-atomic approximable, or iterative \((\varepsilon, \delta)\)-atomic (approximable). These notions are modifications of the parallel ones considered in [J. Lond.Math.Soc., II.Ser.70, No.3, 750–762 (2004; Zbl 1071.47012)] in the case of sublinear operators. The main result goes as follows.
Theorem. Let \(T\) be a bi-sublinear operator that is \((\varepsilon,\delta)\)-atomic approximable or iterative \((\varepsilon,\delta)\)-atomic approximable.

(i)

Assume that, for all measurable sets \(E_1\), \(E_2\), \((T(\chi_{E_1},\chi_{E_2}))^*(t) \leq h(t; |E_1|, |E_2|)\), where, for all \(s_1,s_2 > 0\), \(h(t; s_1,s_2)\) is continuous as a function of \(t>0\). Then, for all \(f_1, f_2 \in L^1\) such that \(\|f_1\|_\infty, \|f_2\|_\infty \leq 1\), we have that
\[ (T(f_1,f_2))^*(t) \leq h(t; \|f_1\|_1, \|f_2\|_1). \]

(ii)

Let \(X\) be a quasi-Banach r.i.space and assume that, for all measurable sets \(E_1, E_2\), we have \(\|T(\chi_{E_1}, \chi_{E_2})\|_X \leq D(|E_1|, |E_2|)\), where \(D\) is increasing in each variable and \(D(0,\cdot) = D(\cdot,0) =0\). Then, for all \(f_1, f_2\in L^1\) such that \(\|f_1\|_\infty, \|f_2\|_\infty \leq 1\), we have
\[ \|T(f_1,f_2))\|_X \lesssim D(\|f_1\|_1, \|f_2\|_1). \]

Section 4 is devoted to similar studies for the bilinear Hilbert transform in Lorentz and Orlicz spaces. The paper contains several examples illustrating the main results.