For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.

We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators, generalized Radon transforms and singular oscillatory integrals.

We consider the equation $u_t\mathrm{-}i\mathrm{\Lambda }u=0$, where $\mathrm{\Lambda }=\lambda \left(D_x\right)$ is a first order pseudo-differential operator with real symbol $\lambda \left(\xi \right)$. Under a suitable convexity assumption on $\lambda$ we find the decay properties for $u\left(t,x\right)$. These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem $u_t\mathrm{-}i\mathrm{\Lambda }u=f\left(u\right)$, $u\left(0,x\right)=g\left(x\right)$. If $f\left(u\right)$ is a smooth function which satisfies $f\left(u\right)\simeq {\left|u\right|}^p$ near $u=0$, and $g$ is small in suitably Sobolev norm, we prove global existence theorems provided $p$ is greater than a critical exponent.

Mathematics Subject Classification: 35CXX, 26A33, 35S10The well known Duhamel principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for corresponding homogeneous equations. In the paper one of the possible generalizations of the classical Duhamel principle to the time-fractional pseudo-differential equations is established.* This work partially supported by NIH grant P20 GMO67594.

Pseudodifferential operators $P(x,D)\sim {\sum }_{j=0}^{+\infty }{P}_{m-j}(x,D)$ are studied, from the viewpoint of local solvability and under the assumption that, micro-locally, the principal symbol factorizes as $P_m=Q{L}^2$ with $Q$ elliptic, homogeneous of degree $m-2$, and $L$ homogeneous of degree one, satisfying the following condition : there is a point $(x_0,{\xi }^0)$ in the characteristic variety $L=0$ and a complex number $z$ such that $d_{\xi }\mathrm{Re}\left(zL\right)\ne 0$ at $(x_0,{\xi }^0)$ and such that the restriction of $\mathrm{Im}\left(zL\right)$ to the bicharacteristic strip of $\mathrm{Re}\left(zL\right)$ vanishes of order $k\<+\infty$ at $(x_0,{\xi }^0)$, changing sign there from minus to...