Compact bilinear operators and commutators (Q2845436)

The authors study the notion of compactness in the multilinear setting, which can be traced back to [\textit{A.~Calderón}, Stud. Math. 24, 113--190 (1964; Zbl 0204.13703)]. Their main result shows that if \(T\) is a bilinear Calderón-Zygmund operator, \(b\in \) CMO (the closure of \({\mathcal C}^\infty_c\) in BMO), \(1/p+1/q=1/r\), \(1<p,q<\infty\) and \(1\leq r<\infty\), then the commutator \([T,b]_1:L^p\times L^q\rightarrow L^r\) is compact, where NEWLINE\[NEWLINE [T,b]_1(f,g)=T(bf,g)-bT(f,g). NEWLINE\]NEWLINE A similar result follows for \([T,b]_2\).NEWLINENEWLINEAlso, a simple characterization of the compactness property, for general bilinear operators \(T\), is also given in terms of classical results, like the existence of convergent subsequences for the sequence \(\{T(x_n,y_n)\}_n\), when \(\{(x_n,y_n)\}_n\) is a bounded sequence. Finally, examples of some concrete bilinear operators \(T\) are given satisfying different properties, like NEWLINE\[NEWLINET(f,g)(x)=\int_0^xf(t)g(t)\,dt,NEWLINE\]NEWLINE defined on \(X\times Y\) (with \(X=Y=({\mathcal C}(0,1),\|\,\|_1\)) and taking values in \(Z=({\mathcal C}(0,1),\|\,\|_\infty)\), which is separately compact but not compact (not even bounded).