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For $0p\le1$, let $h^p(\mathbb{R}^n)$denote the local Hardy space. Let $\mathcal{F}$be a Fourier integral operator defined by the oscillatory integral $$ \mathcal{F}f(x)=\iint_{\mathbb{R}^{2n}}\exp(2\pi\mathrm{i}(\phi(x,\xi)-y\cdot\xi))b(x,y,\xi)f(y)\,\mathrm{d} y\,\mathrm{d}\xi, $$ where $\phi$is a $\mathcal{C}^\infty$non-degenerate real phase function, and $b$is a symbol of order $\mu$and type $(\rho,1-\rho)$, $\sfrac12\rho\le1$, vanishing for $x$outside a compact set of $\mathbb{R}^n$. We show that when $p\le1$and $\mu\le-(n-1)(1/p-1/2)$then $\mathcal{F}$initially defined on Schwartz functions in $h^p(\mathbb{R}^n)$extends to a bounded operator $\mathcal{F}:h^p(\mathbb{R}^n)\rightarrow h^p(\mathbb{R}^n)$. The range of $p$and $\mu$is sharp. This result extends to the local Hardy spaces the seminal result of Seeger \et for the $L^p$spaces. As immediate applications we prove the boundedness of smooth Radon transforms on hypersurfaces with non-vanishing Gaussian curvature on the local Hardy spaces. Finally, we prove a local version for the boundedness of Fourier integral operators on local Hardy spaces on smooth Riemannian manifolds of bounded geometry. [ABSTRACT FROM AUTHOR]

We study the L p–boundedness properties of convolution operators whose convolution kernels are obtained by adapting product kernels to curves in the plane. [ABSTRACT FROM AUTHOR]