On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators

AbstractCommutators [a(M), b(D)] of a multiplication (a(M)u)(x) = a(x) u(x) and a convolution b(D) = F−1b(M)F (F = Fourier transform) are L2-compact if only the continuous functions a and b are bounded and for c = a and c = b we have lim¦x¦→∞sup{¦ c(x + h) − c(x)¦ : ¦ h ¦ ⩽ 1} = 0. An improvement of a result by Calderon and Vaillancourt of boundedness of pseudodifferential operators is discussed (including an independent proof). Similar results on Lp-compactness and Lp-boundedness, 1 < p < ∞, using the Hoermander-Mihlin boundedness theorem on Rn-Fourier-multipliers, and with conditions and proofs different from the case of L2