On the Hörmander Classes of Bilinear Pseudodifferential Operators

Abstract

Abstract. Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calderón-Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz' rule are presented as well.

On the boundedness of certain bilinear oscillatory integral operators

“…The change of the inequality in (4) to the strict inequality in (7) is typical when considering amplitudes only measurable in x satisfying (5) rather than smooth amplitudes satisfying (2). Of direct relevance to this paper is another result in [16] which proved a corresponding boundedness result for oscillatory integral operators.…”

“…However, given that the amplitude is assumed to be smooth and satisfy (2) one would expect such a result to hold up to and including the end-point m = 0. The main goal of this paper is to confirm that this is the case.…”

Boundedness and Compactness for the Commutators of Bilinear Operators on Morrey Spaces

“…Moreover, Calderón-Zygmund operators are "essentially the same" as pseudodifferential operators with symbols in the subclass BS 0 1,δ , 0 ≤ δ < 1, see [10]. On bilinear pseudodifferential operators, one can see [6,7,9,12,13,15,26,27,49].…”

$$L^p$$-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)…”