\(L^p\)-\(L^q\) boundedness of continuous linear operators on smooth manifolds

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 \(L^p\)-\(L^q\)-boundedness of global operators. Our approach links the mapping properties of continuous linear operators on smooth manifolds with the \(L^p\)-estimates of eigenfunctions of operators including a variety of examples, harmonic oscillators, anharmonic oscillators, etc. First, we investigate \(L^p\)-boundedness of pseudo-multipliers in the setting of Hörmander–Mihlin type conditions. We also prove \(L^\infty\)-BMO estimates for pseudo-multipliers. Later, we concentrate our investigation to settle \(L^p\)-\(L^q\) boundedness of the Fourier multipliers and pseudo-multipliers operators for the range \(1<p \le 2 \le q<\infty .\) On the way to achieve our goal of \(L^p\)-\(L^q\) 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.