Anisotropic classes of inhomogeneous pseudodifferential symbols
The authors introduce a class of pseudo-differential operators in the anisotropic setting induced by an expansive dilation extending some classical isotropic classes. This is a continuation of [the authors, Stud. Math. 200, No. 1, 41--66 (2010; Zbl 1215.47038)]. For a given expansive matrix \(A\) and denoting by \( \rho_{A^*}\) the canonical quasi-norm associated to \(A^*\), the authors define the class \(S^m_{\gamma, \delta}(A)\) as those symbols \(\sigma(x,\xi)\) satisfying \[ \partial ^\alpha_x \partial^\beta_\xi( \sigma(A^{-{k_1}}\cdot, (A^*)^{-k_2}\cdot)(A^{k_1}x, A^{k_2}\xi)\leq C_{\alpha,\beta}(1+ \rho_{A^*}(\xi))^m \] for all \(\alpha, \beta\) and \(k_1, k_2\) depending on \(\gamma, \delta\) and \(A\). Their main result establishes that, for symbols \(\sigma\in S^0_{\delta,\delta}\) with \(0\leq \delta\leq 1\), the pseudo-differential operator \(T_\sigma f(x)=\int_{\mathbb R^n} \sigma(x,\xi)\hat f (\xi) e^{i\xi x}\,d\xi\) is bounded on \(L^2(\mathbb R^n)\). They present several examples of anisotropic symbols in the above classes.
scientific article; zbMATH DE number 6175263
scientific article