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Abstract: We discuss operators of the form:
Af(x)=\int\int b(x,y,\xi)f(y)exp[2pi*i\psi(x,y,\xi)] dyd\xi,
where b is a symbol and \psi is a phase function.
The main result reads as follows. Assume A is a linear operator on L^2(R^d) of the form above, where p \in [1,2], and c is a second class Fourier integral operator slice permutation. For certain values of p_1,...,p_6d, If the kernel of A, b*exp[2pi*i\psi(x,y,\xi)] \in M(c)^{p_1,...,p_6d}, the mixed modulation space, then A belongs to the Schatten p-class.
The proof of this theorem is based on Shannon Bishop’s PhD thesis and one corollary of that thesis.