Abstract
Abstract In this paper, we study the boundedness of global continuous linear operators on smooth manifolds. Using the notion of a global symbol, we extend a classical condition of Hörmander type to guarantee the $$L^p$$ L p - $$L^q$$ L q -boundedness of global operators. Our approach links the mapping properties of continuous linear operators on smooth manifolds with the $$L^p$$ L p -estimates of eigenfunctions of operators including a variety of examples, harmonic oscillators, anharmonic oscillators, etc. First, we investigate $$L^p$$ L p -boundedness of pseudo-multipliers in the setting of Hörmander–Mihlin type conditions. We also prove $$L^\infty$$ L ∞ -BMO estimates for pseudo-multipliers. Later, we concentrate our investigation to settle $$L^p$$ L p - $$L^q$$ L q boundedness of the Fourier multipliers and pseudo-multipliers operators for the range $$1 1 < p ≤ 2 ≤ q < ∞ . On the way to achieve our goal of $$L^p$$ L p - $$L^q$$ L q boundedness, we prove two classical inequalities, namely, Paley inequality and Hausdorff–Young–Paley inequality for smooth manifolds. Finally, we present some examples about the well-posedness of abstract non-linear equations.