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We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱL^p{\left({ℝ}^d\right)}_{comp}$, 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱL^p{\left({ℝ}^d\right)}_{comp}$ if the mapping $x\mapsto {\nabla }_x\Phi (x,\eta )$ is constant on the fibres, of codimension r, of an affine...

Given an embeddable $CR$ manifold $M$ and a non-characteristic hypersurface $S\subset M$ we present a necessary condition for the tangential Cauchy-Riemann operator ${\overline{\partial }}_M$ on $M$ to be locally solvable near a point $x_0\in S$ determined by $S$.