We prove that the multiplier algebra of the Drury–Arveson Hardy space $H_n^2$ on the unit ball in ${ℂ}^n$ has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov–Sobolev space $B_p^{\sigma }$ has the “baby corona property” for all $\sigma \ge 0$ and $1<p<\infty$. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

The reappearance of what is sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. We prove that such operators are unbounded on products of Lebesgue spaces but bounded on spaces of smooth functions (this is the exotic behavior referred to). In addition, by introducing a new way to approximate the product of two functions, estimates on a new paramultiplication are obtained.

We consider the problem of existence and global behavior of solitons for generalized Korteweg–de Vries equations (gKdV) with a slowly varying (in space) perturbation. We prove that such slowly varying media induce on the soliton dynamics large dispersive effects at large times. We also prove that, unlike the unperturbed case, there is no pure-soliton solution to the perturbed gKdV.