Local integrability of parametrices of hypoelliptic operators

Following Hörmander's work, we investigate in this thesis whether the fundamental solutions of a hypoelliptic operator P(D) in R^n are always locally integrable. A fundamental solution of a hypoelliptic operator is known to be a smooth function outside the origin; hence, studying its local integrability essentially determines whether the notion of distribution in this context is merely convenient or truly essential. When P(D) is elliptic or semielliptic and E is a fundamental solution of P(D), we directly show that P^{(\alpha)}(D)E \in L^p_{loc}(R^n), for suitably chosen exponents p depending on the dimension n, the multi-index \alpha and the order of the symbol P(\xi). To study local integrability when P(D) is assumed to be merely hypoelliptic, we examine the inverse Fourier transform of parametrices of P(D). The behaviour of these distributions depends on the geometric properties of the two manifolds Z = { \zeta \in C^n | P(\zeta) = 0 }, Z_m = { \zeta \in C^n | P_m(\zeta) = 0 }, where P_m denotes the principal part of P. Using Puiseux series, convolution operators, and the stationary phase method, we explore the integrability properties of these distributions. In particular, the structure of the thesis is as follows: Chapter 1: we prove the integrability of parametrices of P(D) when n = 2. Chapter 2: we investigate the distribution P^{(\alpha)}(D)E in R^2, providing a sufficient condition for local integrability. Furthermore, we show that this condition is also necessary when |\alpha| = 1. Chapter 3: we construct an example demonstrating that for n > 13, there exist fundamental solutions which are not locally integrable. Chapter 4: we examine some consequences of the existence of parametrices with this property. This settles the question for the cases n = 2 and n > 13, highlighting that distributions are not merely convenient but essential in this context.