Multilinear pseudodifferential operators beyond Calderón–Zygmund theory

Abstract

We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander S m ρ,δ classes. These results are new in the case ρ < 1, that is, outwith the scope of multilinear Calderón-Zygmund theory.

On the boundedness of certain bilinear oscillatory integral operators

“…This amplitude class was first introduced by N. Michalowski, D. Rule and W. Staubach in [15]. Here we recall the definition of the class L p S m ρ (n, 1).…”

“…Equally, amplitudes associated to bilinear operators are denoted S m ρ,δ (n, 2) (see [15]). In particular, any function satisfying (2) belongs to the class S m 1,0 (n, 2).…”

“…The motivation for introducing the class L p S m ρ (n, 1) in [15] is that it proves useful in the study of bilinear operators. Indeed, the global regularity of bilinear Fourier integral operators with amplitudes that are neither compactly supported nor smooth in the first variable is studied in [16] in part by proving the boundedness of linear operators with amplitudes in the aforementioned classes.…”

“…The first is when either ξ or η is small. In this case we can write the bilinear operator as an iteration of linear operators, as in [15], and then apply techniques for linear operators. The second is when both ξ and η are large.…”

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

“…The nuclearity of pseudo-differential operators on R n has been treated in details by Aoki [2] and Rempala [54]. Multilinear pseudo-differential operators studied by several authors including Bényi, Maldonado, Naibo, and Torres, [3,4], Michalowski, Rule and Staubach, Miyachi and Tomita [41,42,43,44] and references therein. It is worth mentioning that the multilinear analysis for multilinear multipliers of the form T a (f )(x) = R nr e i2πx·(η 1 +···+ηr) a(η) f 1 (η 1 ) · · · f r (η r )dη, x ∈ R n , (1.6) born with the multilinear results by Coifman and Meyer (see [12,13]), where it was shown that the condition |∂ α 1 η 1 ∂ α 2 η 2 · · · ∂ αr ηr a(η 1 , η 2 , · · · , η r )| ≤ C α (|η 1 | + |η 2 | + · · · + |η r |) −|α| , (1.7)…”