**ABSTRACT:**

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 HÃ¶lder 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).