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In this paper we study the boundedness of global pseudo-differentialoperators on smooth manifolds. By using the notion of global symbol we extend aclassical condition of H\"ormander type to guarantee the$L^p$-$L^q$-boundedness of global operators. First we investigate$L^p$-boundedness of pseudo-differential operators in view of theH\"ormander-Mihlin condition. We also prove $L^\infty$-$BMO$ estimates forpseudo-differential operators. Later, we concentrate our investigation tosettle $L^p$-$L^q$ boundedness of the Fourier multipliers andpseudo-differential operators for the range $1<p \leq 2 \leq q<\infty.$ On theway to achieve our goal of $L^p$-$L^q$ boundedness we prove two classicalinequalities, namely, Paley inequality and Hausdorff-Young-Paley inequality forsmooth manifolds. Finally, we present the applications of our boundednesstheorems to the well-posedness properties of different types of the nonlinearpartial differential equations.

$L^p$-$L^q$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations