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Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft (2007)

Studia Mathematica

Let WF⁎ be the wave front set with respect to C ∞ , quasi analyticity or analyticity, and let K be the kernel of a positive operator from C ₀ ∞ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution u ∗ B φ ( x ) = ∫ u ( x - y ) φ ( y ) B ( x , y ) d y , where B ∈ C ∞ is appropriate, and prove that if ( u ∗ B φ , φ ) ≥ 0 for every φ ∈ C ₀ ∞ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

Weighted estimates for commutators of linear operators

Josefina Alvarez, Richard Bagby, Douglas Kurtz, Carlos Pérez (1993)

Studia Mathematica

We study boundedness properties of commutators of general linear operators with real-valued BMO functions on weighted L p spaces. We then derive applications to particular important operators, such as Calderón-Zygmund type operators, pseudo-differential operators, multipliers, rough singular integrals and maximal type operators.

Weyl product algebras and classical modulation spaces

Anders Holst, Joachim Toft, Patrik Wahlberg (2010)

Banach Center Publications

We discuss continuity properties of the Weyl product when acting on classical modulation spaces. In particular, we prove that M p , q is an algebra under the Weyl product when p ∈ [1,∞] and 1 ≤ q ≤ min(p,p’).

When is a pseudo-differential equation solvable ?

Nicolas Lerner (2000)

Annales de l'institut Fourier

This paper begins with a broad survey of the state of the art in matters of solvability for differential and pseudo-differential equations. Then we proceed with a Hilbertian lemma which we use to prove a new solvability result.

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