Roe–Strichartz theorem for iterated Laplacian sequences on symmetric spaces

Let $X$ be the Riemannian symmetric space under consideration, let $K/M$ be its Furstenberg boundary, and let $q'$ satisfy $1/q+1/q'=1$. Fix $q\in(1,2)$, and let $\{f_j\}_{j\in\mathbb N}$ be an infinite sequence of measurable functions on $X$ such that, for every $j\in\mathbb N$,

$$\Delta f_j=\frac{4\rho^2}{qq'}f_{j+1}$$

and, for a fixed $p\geq1$, there is a constant $C_p>0$ depending only on $p$ such that

$$\|f_j(\cdot a)\|_{L^p(K)}\leq C_p\phi_{i\gamma_q\rho}(a)$$

for all $a\in\overline{A^+}$. Here $\gamma_q=2/q-1$ and $P_{i\gamma_q\rho}$ denotes the Poisson transform. Roe–Strichartz conjecture. Then

$$\Delta f_0=-\frac{4\rho^2}{qq'}f_0.$$

Moreover, if $p>1$, then $f_0(x)=P_{i\gamma_q\rho}F(x)$ for some $F\in L^p(K/M)$, while if $p=1$, then $f_0=P_{i\gamma_q\rho}\mu(x)$ for some signed measure $\mu$ on $K/M$. This expected result would extend the known Roe–Strichartz phenomenon from established cases to the stated sequence and growth setting; the supplied source does not establish its resolution.