Bilinear decompositions and commutators of singular integral operators (Q2838111)

Let \(b\in\mathrm{BMO}(\mathbb{R}^n)\), namely, NEWLINE\[NEWLINE \|b\|_{\mathrm{BMO}(\mathbb{R}^n)}:=\sup_{B\subset\mathbb{R}^n} \frac{1}{|B|}\int_B|f(x)-f_B|\,dx<\infty, NEWLINE\]NEWLINE where the supremum is taken over all balls \(B\subset\mathbb{R}^n\) and \(f_B:=\frac{1}{|B|}\int_B f(y)\,dy\). Assume that \(T\) is a Calderón-Zygmund operator. The linear commutator \([b,T]\) is defined for \(f\in C^{\infty}_c(\mathbb{R}^n)\) by NEWLINE\[NEWLINE [b,T](f):=bT(f)-T(bf). NEWLINE\]NEWLINE It is well known that, in general, \([b,T]\) is not bounded from the Hardy space \(H^1(\mathbb{R}^n)\) to the Lebesgue space \(L^1(\mathbb{R}^n)\). However, C. Pérez proved that if \(H^1(\mathbb{R}^n)\) is replaced by a suitable atomic subspace \(\mathcal{H}^1_b(\mathbb{R}^n)\), then \([b,T]\) is bounded from \(\mathcal{H}^1_b(\mathbb{R}^n)\) to \(L^1(\mathbb{R}^n)\).NEWLINENEWLINEIn this article, the author finds the largest subspace \(H^1_b(\mathbb{R}^n)\) of \(H^1(\mathbb{R}^n)\) such that all commutators of Calderón-Zygmund operators are bounded from \(H^1_b(\mathbb{R}^n)\) to \(L^1(\mathbb{R}^n)\). Moreover, several equivalent characterizations of \(H^1_b(\mathbb{R}^n)\) are also presented.NEWLINENEWLINELet \(b\in\mathrm{BMO}(\mathbb{R}^n)\). The author also studies the commutator \([b,T]\), if the operator \(T\) belongs to a class \(\mathcal{K}\) of sublinear operators containing almost all important operators in harmonic analysis. When \(T\) is linear, the author proves that there exists a bilinear operator \(\Re:=\Re_T\), which is bounded from \(H^1(\mathbb{R}^n)\times\mathrm{BMO}(\mathbb{R}^n)\) to \(L^1(\mathbb{R}^n)\), such that, for all \((f,b)\in H^1(\mathbb{R}^n)\times\mathrm{BMO}(\mathbb{R}^n)\), NEWLINE\[NEWLINE [b,T](f)=\Re(f,b)+T(\Im(f,b)),\leqno{(1)} NEWLINE\]NEWLINE where \(\Im\) is a bounded bilinear operator from \(H^1(\mathbb{R}^n)\times\mathrm{BMO}(\mathbb{R}^n)\) to \(L^1(\mathbb{R}^n)\), which is independent of \(T\). In particular, if \(T\) is a Calderón-Zygmund operator satisfying \(T1=T^\ast1=0\), where \(T^\ast\) denotes the adjoint operator of \(T\), and \(b\in\mathrm{BMO}^{\mathrm{log}}(\mathbb{R}^n)\), the generalized \(\mathrm{BMO}\)-type space which has been introduced by E. Nakai and K. Yabuta to characterize pointwise multiples of \(\mathrm{BMO}(\mathbb{R}^n)\), the author further proves that \([b,T]\) is bounded from \(H^1(\mathbb{R}^n)\) to the local Hardy space \(h^1(\mathbb{R}^n)\). Furthermore, if \(b\in\mathrm{BMO}(\mathbb{R}^n)\) and \(T^\ast1=T^\ast b=0\), then \([b,T]\) is bounded from \(H^1_b(\mathbb{R}^n)\) to \(H^1(\mathbb{R}^n)\).NEWLINENEWLINEWhen \(T\) is sublinear, the author proves that there exists a bounded subbilinear operator \(\Re:=\Re_T:\;H^1(\mathbb{R}^n)\times\mathrm{BMO}(\mathbb{R}^n)\rightarrow L^1(\mathbb{R}^n)\) such that, for all \((f,b)\in H^1(\mathbb{R}^n)\times\mathrm{BMO}(\mathbb{R}^n)\), NEWLINE\[NEWLINE |T(\Im(f,b))|-\Re(f,b)\leq|[b,T](f)|\leq \Re(f,b)+|T(\Im(f,b))|,\leqno{(2)} NEWLINE\]NEWLINE where \(\Im\) is as in (1).NEWLINENEWLINEMoreover, it is worth pointing out that the bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong \(L^1(\mathbb{R}^n)\)-estimates for commutators.