Multiplication properties in pseudo-differential calculus with small regularity assumptions on the symbols
given at esi12 (16.10.12 15:45)
There are four important products which appear on the symbol levels in
pseudo-differential calculus. The composition of two Weyl quantizations
corresponds on the symbol level on the so called Weyl product.
The convolution appears when putting the theory of Toeplitz
operators (the same as localization operators) on functions on R^d
into the context of pseudo-differential calculus, using the fact that the
Weyl symbol of a Toeplitz operator is a convolution by the Toeplitz
symbol and an other convenient function. Finally, the twisted
convolution and ordinary multiplication appear when
applying the (symplectic) Fourier transform on Weyl products and
ordinary convolutions, respectively.
In the talk we establish Young and Hölder relations for such products
on Schatten-von Neumann symbols, Lebesgue spaces and modulation
spaces. We use the results to extend the class of possible window functions in
the definition of modulation spaces,
and to prove that any Schatten-p symbol in the Weyl calculus gives rise to
a Schatten-p Toeplitz operator. The Schatten-von Neumann classes here are
of the form I_p(H_1,H_2), where
H_1,H_2 are Hilbert spaces of (ultra-)modulation space type
which stay between the Gelfand-Shilov space Sigma_1(R^d)
and its dual Sigma_1'(R^d). Furthermore, the symbol classes of pseudo-differential
operators stay between Sigma_1(R^2d)$ and its dual Sigma_1'(R^2d).