The Marcinkiewicz multiplier condition for bilinear operators (Q2717712)

The authors study bilinear translation invariant operators of the form NEWLINE\[NEWLINE T(f,g) = \int \hat f(\xi) \hat g(\eta) m(\xi,\eta) \exp(2\pi i x \cdot(\xi+\eta))\;d\xi d\etaNEWLINE\]NEWLINE where the symbol \(m\) satisfies the product symbol estimates NEWLINE\[NEWLINE |\partial_\xi^\alpha \partial_\eta^\beta m(\xi,\eta)|\leq C_{\alpha,\beta} |\xi|^{-|\alpha|} |\eta|^{-|\beta|}.NEWLINE\]NEWLINE This is less restrictive than the usual paraproducts of Coifman-Meyer type, in which the right-hand side of the symbol estimates is \(C_{\alpha,\beta} (|\xi|+|\eta|)^{-|\alpha|-|\beta|}\). These operators are essentially bounded linear combinations of operators of the form \((\Delta_j f) (\Delta_k g)\), where \(\Delta_j\) is a Littlewood-Paley projection to frequencies \(\sim 2^j\). Somewhat surprisingly, these bilinear multipliers do not have a good \(L^p\) theory, and the authors construct an example of these multipliers which has a logarithmic divergence near the axes \(\xi=0\) and \(\eta=0\). Conversely, if one damps the symbol condition by a logarithmic factor on these axes then one does obtain the expected \(L^p\) mapping properties (down to, but not quite including, \(L^1 \times L^1 \to L^{1/2}\)).