A pseudodifferential operator multiplies a function's Fourier transform by a symbol sigma(x, xi) before inverting; a symbolic calculus says what happens to that symbol when operators are transposed or composed. This concept covers the calculus for symbol classes in which the usual Hormander bounds hold with constants that themselves decay as |x| + |xi| grows, so the class is closed under transposition and the adjoint symbol admits an asymptotic expansion with remainders of ever-lower order. Because compactness criteria of T(1) type impose symmetric hypotheses on an operator and its transpose, a student learns how transposition-invariance of a symbol class converts into compactness of the associated Calderon-Zygmund operators and of their commutators with multiplication operators.
A pseudodifferential operator T_sigma acts by multiplying the Fourier transform of a function by a symbol sigma(x, xi) before inverting. The Hoermander class S^s_{1,0} consists of symbols whose deriv…