Pointwise estimates of pseudo-differential operators

As a new technique it is shown how general pseudo-differential operators can be estimated at arbitrary points in Euclidean space when acting on functions $u$ with compact spectra. The estimate is a factorisation inequality, in which one factor is the Peetre–Fefferman–Stein maximal function of $u$, whilst the other is a symbol factor carrying the whole information on the symbol. The symbol factor is estimated in terms of the spectral radius of $u$, so that the framework is well suited for Littlewood–Paley analysis. It is also shown how it gives easy access to results on polynomial bounds and estimates in $L_p$, including a new result for type $1,1$-operators that they are always bounded on $L_p$-functions with compact spectra.