Continuity for pseudodifferential operators with symbols in weighted function spaces (Q2704046)

The author presents two new results, dealing with the spaces \(B_{p,k}\), \(1\leq p\leq\infty\), defined in terms of Fourier transforms and weights \(k(\eta):\mathbb{R}^n\to \mathbb{R}_+\); namely \(f\in B_{p,k}\) if \(k(\eta)\widehat f(\eta)\in L^p\). The first result concerns the boundedness of pseudodifferential operators with symbols \(a(x,\eta)\), smooth with respect to \(\eta\), and belonging to \(B_{p,k}\) with respect to \(x\). The second result refers to the semilinear equation NEWLINE\[NEWLINEP(D)u= F(x, Q_1(D)u,\dots, Q_M(D)u),NEWLINE\]NEWLINE where \(P(D)\) is an operator with constant coefficients, as well as \(Q_1(D),\dots, Q_M(D)\), which are of lower order, in the sense that \(\widetilde P(\eta)/\widetilde Q_j(\eta)\geq h(\eta)\geq C> 0\). The author gives a precise result of regularity for the solution \(u\) in terms of the spaces \(B_{p,k}\), deducing in particular \(u\in C^\infty\) when \(P(D)\) is hypoelliptic and \(h(\eta)\geq|\eta|^\lambda\), \(\lambda> 0\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].