Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces

Abstract

LetP be a complex polynomial inn variables of degree 2 andP(D) the corresponding partial differential operator with constant coefficients. It is shown thatP (D) :C ∞(ℝ n ) →C ∞(ℝ n ) admits a continuous linear right inverse if and only if after a separation of variables and up to a complex factor for some c ∈ ℂ the polynomialP has the form $$P(x_1 ,...,x_n ) = Q(x_1 ,...,x_r ) + L(x_{r + 1} ,...,x_n ) + c$$ where eitherr=1 andL≡0 orr>1,Q andL are real andQ is indefinite. The proof of this characterization is based on the general solution of the right inverse problem for such operators and the fact that for each operatorP(D) of the given form and each characteristic vectorN there exists a fundamental solution forP(D) supported by {x ∈ ℝ n : 〈x, N〉 ⪰ 0 #x007D;, which can be constructed explicitely using partial Fourier transform. The existence of sufficiently many fundamental solutions with support in closed half spaces implies that some right inverse can be given by a concrete formula. An example shows that the present characterization is restricted to operators of order 2.