Fractional Klein-Gordon equation with singular mass

Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case

“…The present paper is the extension and improvement of the results obtained in the first part [9] which was devoted to the classical Klein-Gordon equation. In fact, the setting of [9] was the Equation (1) for G = R d and R = − being the positive Laplacian on the Euclidean space.…”

“…We note that in Definition 3 in the previous paper [9], the word 'regularisation' needs to be understood, in general, as an approximation not necessarily depending on the classical convolution and specific mollifiers. In this case, our definition of the uniqueness of the very weak solutions here includes also the version in Definition 3 in [9], but Definition 4.6 makes it more rigorous. To clarify this further, we can take, for example, m to be a regularisation of m by a convolution (if m is a distribution), and takẽ…”

“…We note that in Theorem 4 in the paper [9], one wrote the assumption that m ∈ L ∞ (R d ) in the consistency result. This may be not sufficient in general.…”

“…The wave type equations with time-dependent coefficients on graded Lie groups were analysed in [7] for Hölder coefficients, and in [8] for distributional time-dependent coefficients, using the notion of very weak solutions. All these works deal with the time-dependent equations and in the recent papers [9][10][11][12], the authors start to develop the notion of very weak solutions for equations with (irregular) space-depending coefficients.…”

“…In fact, the setting of [9] was the Equation (1) for G = R d and R = − being the positive Laplacian on the Euclidean space. Consequently, the results here contain the results of [9] as a special case, and we also use this chance to slightly correct the consistency statement given in that paper, see Remark 5.3, as well as a clarifying Remark 4.7.…”