On functions of bounded mean oscillation with bounded negative part (Q6607816)

\textit{J. Bastero} et al. [Proc. Am. Math. Soc. 128, No. 11, 3329--3334 (2000; Zbl 0957.42010)] proved that the commutator \([b,M]\) between multiplication by \(b\) and application of the Hardy-Littlewood maximal function \(Mf(x)=\sup_{x\in Q} m_Q(\vert f\vert)(x)\) (where \((m_Q f)(x)=\frac{1}{\vert Q\vert}\int Q f\)) is bounded on \(L^p\), \(1<p<\infty\), if and only if \(b\in {\mathrm{BMO}}\) with \(b^{-}\in L^\infty\) (\(b^{-}(x)=-\min\{b(x),0\}\)). In contrast, \textit{J. García-Cuerva} et al. [Indiana Univ. Math. J. 40, No. 4, 1397--1420 (1991; Zbl 0765.42012)] previously proved that the maximal commutator \(M_bf(x)=\sup_{x\in Q} m_Q(\vert (b(x)-b(\cdot))f(\cdot)\vert)\) is bounded on \(L^p\), \(1<p<\infty\), if and only if \(b\in {\mathrm{BMO}}\). The present work studies commutators with the bilinear maximal operator \(\mathcal{M}(f,g)(x)=\sup_{x\in Q}m_Q( \vert f(\cdot) g(2x-\cdot)\vert)\). In this case there are two separate commutators, one associated with difference in each of the two arguments of the bilinear operator. Specifically, \[ [b,\mathcal{M}]_1(f,g)(x)= b(x) \mathcal{M}(f,g)(x)-\mathcal{M}(bf,g)\] and \([b,\mathcal{M}]_2(f,g)(x)\) defined similarly with multiplication by \(b\) now applied to \(g\) in the second argument. Corresponding maximal commutators are defined, e.g., \[\mathcal{M}_b^{(1)}(f,g)(x)=\sup_{x\in Q} m_Q(\vert b(x)-b(y)\vert \vert f(y)g(2x-y)\vert)\] with \(\mathcal{M}_b^{(2)}(f,g)(x) =\mathcal{M}_b^{(1)}(g,f)(x)\). Two theorems establish that boundedness of such a commutator on a product space is equivalent with \(b\) belonging to BMO or \(b\in {\mathrm{BMO}}\) and \(b^{-}\in L^\infty\). Two additional theorems provided equivalence of boundedness with membership of \(b\) in a Lipschitz space for suitable exponents.\par In the first two results one chooses \(1<p,p_1,p_2<\infty\) with \(\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}\) and \(b\in L^1_{\mathrm{ loc}}\). Boundedness of \(\mathcal{M}_b^{(1)}\) from \(L^{p_1}\times L^{p_2}\to L^p\) is equivalent to \(b\in {\mathrm{BMO}}\) (Thm.~1,1) while the same boundedness for \([b,\mathcal{M}]_1\) is equivalent to \(b\in {\mathrm{BMO}}\) and \(b^{-}\in L^\infty\) (Thm.~1.2). The remaining two main results consider whether \(b\) belongs to a Lipschitz space. In these results one fixes \(0<\alpha<1\) and \(1<p_1,p_2,q<\infty\) with \(\frac{1}{p_1}+\frac{1}{p_2}-\frac{1}{q}=\frac{\alpha}{n}\). In this setting, boundedness of \(\mathcal{M}_b^{(1)}\) from \(L^{p_1}\times L^{p_2}\to L^q\) is equivalent to \(b\in {\mathrm{ Lip}}_\alpha\) (Thm.~1.3) while the same boundedness for \([b,\mathcal{M}]_1\) is also equivalent to \(b\in {\mathrm{Lip}}_\alpha\) in the event that \(b\geq 0\) (Thm.~1.4). There is not an obvious analog of ``\(b^{-}\in L^\infty\)'' for the Lipschitz case. \par Each of the main results also estblishes that boundedness of the commutator from the product space to the Lebesgue space (\(L^p\) or \(L^q\)) is also equivalent to boundedness from the same product space to the Lorentz space \(L^{p,\infty}\) or \(L^{q,\infty}\) respectively. The latter relies on extension of a result of \textit{J. Bastero} et al., Proc. [Am. Math. Soc. 128, No. 11, 3329--3334 (2000; Zbl 0957.42010)] stating that \(b\in {\mathrm{BMO}}\) and \(b^{-}\in L^\infty\) is equivalent to finiteness of the norm (modulo constants) \(\Vert b\Vert_{{\mathrm{BMO}}_p^-}=\sup_Q m_Q (\vert b-M_Q(b)\vert)\) where \(M_Q(b)=M(\chi_Q b)\). The extension here replaces \(\Vert b\Vert_{{\mathrm{BMO}}_p^-}\) with a suitable weak-type norm. The first results rely on a boundedness property of composition of the sharp maximal function with an \(s\)-maximal commutator \(M_{b,s}f(x)=(\sup_{x\in Q} m_Q(\vert b(x)-b(\cdot)\vert ^s\vert f(\cdot)\vert ^s)^{1/s}\).