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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 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 and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

Subalgebras to a Wiener type algebra of pseudo-differential operators

Joachim Toft — 2001

Annales de l’institut Fourier

We study general continuity properties for an increasing family of Banach spaces S w p of classes for pseudo-differential symbols, where S w = S w was introduced by J. Sjöstrand in 1993. We prove that the operators in Op ( S w p ) are Schatten-von Neumann operators of order p on L 2 . We prove also that Op ( S w p ) Op ( S w r ) Op ( S w r ) and S w p · S w q S w r , provided 1 / p + 1 / q = 1 / r . If instead 1 / p + 1 / q = 1 + 1 / r , then S w p w * S w q S w r . By modifying the definition of the S w p -spaces, one also obtains symbol classes related to the S ( m , g ) spaces.

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