A certain partial differential equation (Q909874)

Let P(D), \(D=(-i(\partial /\partial x_ 1),...,-i(\partial /\partial x_ n)\) be a differential operator with constant coefficients. Assume that a hypoelliptic polynomial \(P(\xi)\) with real coefficients has a nonempty set L of real roots and grad \(P(\xi)\neq 0\) for \(\xi\in L\). Then the equation \(P(D)u=f\) has a solution \(u\in L_ 2^{\gamma -1}\) for every \(f\in L^{\gamma}_ 2\), \(\gamma <1/2\), where \(L^{\gamma}_ 2\) denotes the set of all measurable functions u having the finite norm: \[ \| u\|_{\gamma}=(\int_{R^ n}(1+| x|^{\gamma})^ 2| u(x)|^ 2 dx)^{1/2}\quad with\quad x=(x^ 2_ 1+...+x^ 2_ n)^{1/2}. \]