Strichartz's conjecture on the Poisson transform over noncompact symmetric spaces

Let $X=G/K$ be a Riemannian symmetric space of noncompact type, with Iwasawa decomposition $G=KAN$, Furstenberg boundary $B=K/M$, and rank $r$. Let $obreak\textstyle\mathfrak{a}_{\mathrm{reg}}^*$ denote the regular part of the complexified dual of the Lie algebra of $A$, and let $obreak\mathcal{E}_{\lambda}(X)$ be the joint eigenspace of $G$-invariant differential operators with spectral parameter $obreak\lambda$. For $R>0$, let $B(0,R)$ be the open ball in $X$ centered at $0$ with radius $R$, and let $P_\lambda$ be the Poisson transform from $L^2(B)$ to joint eigenfunctions.

Strichartz's conjecture. (i) If $\lambda\in\mathfrak{a}_{\mathrm{reg}}^*$ and $F\in\mathcal{E}_{\lambda}(X)$, then $F=P_\lambda f$ for some $f\in L^2(B)$ if and only if

$$\sup_{R>0}\frac{1}{R^r}\int_{B(0,R)}|F(x)|^2\,\mathrm{d}x<+\infty.$$

Moreover, for such $f$,

$$\lim_{R\rightarrow+\infty}\frac{1}{R^r}\int_{B(0,R)}|P_\lambda f(x)|^2\,\mathrm{d}x=2^{-r/2}\Gamma(r/2+1)^{-1}|c(\lambda)|^2\|f\|^2_{L^2(B)},$$

and there is a positive constant $C$ such that

$$C^{-1}|c(\lambda)|^2\|f\|^2_{L^2(B)}\leq\sup_{R>0}\frac{1}{R^r}\int_{B(0,R)}|P_\lambda f(x)|^2\,\mathrm{d}x\leq C|c(\lambda)|^2\|f\|^2_{L^2(B)}.$$

(ii) If $F_\lambda=\mathbb{P}_\lambda F$ for $F\in L^2(X)$, then

$$C^{-1}\|F\|^2_{L^2(X)}\leq\int_{\mathfrak{a}^{*}_{+}}\sup_{R>0}\frac{1}{R^r}\int_{B(0,R)}|F_\lambda(x)|^2\,\mathrm{d}x\,\mathrm{d}\lambda\leq C\|F\|^2_{L^2(X)},$$

and

$$\|F\|^2_{L^2(X)}=2^r\pi^{r/2}\Gamma(r/2+1)\lim_{R\rightarrow+\infty}\int_{\mathfrak{a}^{*}_{+}}\frac{1}{R^r}\int_{B(0,R)}|F_\lambda(x)|^2\,\mathrm{d}x\,\mathrm{d}\lambda.$$

Conversely, any almost-everywhere family $F_\lambda\in\mathcal{E}_{\lambda}(X)$ for which the right-hand side of either of the two displayed norm conditions is finite is of the form $F_\lambda=\mathbb{P}_\lambda F$ for some $F\in L^2(X)$.