Schatten-von Neumann properties in the Weyl calculus

作者：Buzano Ernesto; Toft Joachim^*

来源：Journal of Functional Analysis, 2010, 259(12): 3080-3114.

DOI：10.1016/j.jfa.2010.08.021

• Hormander symbols
• Schatten-von Neumann classes
• Embeddings
• Necessary conditions
• Sufficient conditions

Let Op(t) (a), for t is an element of R, be the pseudo-differential operator
f (x) --> (2 pi)(-n)integral integral a((1 - t)x + ty,xi)f (y)e(i(x-y,xi)) dyd xi
and let I(p), be the set of Schatten-von Neumann operators of order p is an element of [1, infinity] on L(2). We are especially concerned with the Weyl case (i.e. when / = 1/2). We prove that if m and g are appropriate metrics and weight functions respectively, h(g) is the Planck's function, h(g)(k/2) m is an element of L(p) for some k >= 0 and a is an element of S(m, g), then Op(t) (a) is an element of I(p) , iff a is an element of L(p). Consequently, if 0 <= 8 < rho <= 1 and a is an element of S(rho,infinity)(r), then Op(t) (a) is bounded on L(2), iff a is an element of L(infinity).

• 出版日期2010-12-15