I have heard that a Lindeloff-type bound for Hecke Maass form L functions exist on average. However, I am unable to find it in literature nor understand the precise statement. Is it something like $$\frac{1}{M}\sum\limits_{T \leq t_j \leq T+M} \frac{L(u_j,1/2+it)}{|| u_j ||^2 \cosh \pi t_j} \ll (1+t_j+|t|)^{\epsilon}\ \ \quad ?$$
Also, what is the precise summand in that result? Is it $\left| \frac{L(u_j,1/2+it)}{|| u_j ||^2 \cosh \pi t_j}\right|$ or the second moment $\left| \frac{L(u_j,1/2+it)}{|| u_j ||^2 \cosh \pi t_j}\right|^2$ by any chance? Please mention the relevant reference and kindly specify how the Lindeloff follows from that reference theorem or Corollary?
Here u_j is an orthogonal basis for the space of Maass Cusp forms of weight 0 with respect to $SL(2,\mathbb{Z})$ and $t_j=\sqrt{\lambda_j-1/4}$ is the spectral parameter of $u_j$ and $\lambda_j$ is $u_j$-s Hyperbolic Laplacian eigenvalue.
Recall that the generalised Lindelöf hypothesis states that $$L\left(\frac{1}{2} + it,f\right) \ll_{\varepsilon} ((1 + |t_f + t|)(1 + |t_f - t|))^{\varepsilon}$$ for $f \in \mathcal{B}_0$, where $\mathcal{B}_0$ denotes an orthonormal basis of Hecke-Maass cusp forms of weight $0$, level $1$; here $t_f$ denotes the spectral parameter of $f$. Since the Weyl law states that $$\#\{f \in \mathcal{B}_0 : T - U \leq t_f \leq T + U\} \asymp TU$$ for $T^{\varepsilon} \leq U \leq T$ (more precisely, this is asymptotic to a constant multiple of $TU$), a Lindelöf-on-average bound for the $k$-th moment of $L(1/2 + it,f)$ would be something of the form $$\sum_{\substack{f \in \mathcal{B}_0 \\ T - U \leq t_f \leq T + U}} \frac{\left|L\left(\frac{1}{2} + it,f\right)\right|^k}{L(1,\operatorname{ad}f)} \ll_{k,\varepsilon} TU ((1 + |T + t|)(1 + |T - t|))^{\varepsilon},$$ as this is the bound we would get simply by bounding each term individually assuming the generalised Lindelöf hypothesis and then summing over these bounds using the Weyl law.
As an aside, we note that $L(1,\operatorname{ad}f)$ is a special value of an $L$-function that is positive and satisfies the bounds $t_f^{-\varepsilon} \ll_{\varepsilon} L(1,\operatorname{ad}f) \ll_{\varepsilon} t_f^{\varepsilon}$ and is such that $$\int_{\Gamma \backslash \mathbb{H}} |f(z)|^2 \, \frac{dx \, dy}{y^2} = \frac{2 |\rho_f(1)|^2 L(1,\operatorname{ad}f)}{\cosh \pi t_f}$$ with $\rho_f(1)$ the first Fourier-Whittaker coefficient of $f$, so that $$f(z) = \sum_{\substack{n = -\infty \\ n \neq 0}}^{\infty} \rho_f(n) W_{0,it_f}(4\pi|n|y) e(nx).$$
In general (without any conditions on $k,t,U$), a Lindelöf-on-average bound for the $k$-th moment of $L(1/2 + it,f)$ is not known. However, many special cases are known.
If $k = 2$, then we can use the approximate functional equation (Theorem 5.3 of Iwaniec-Kowalski) to roughly write $$L\left(\frac{1}{2} + it,f\right) \approx \sum_{n \leq ((1 + |t_f + t|)(1 + |t_f - t|))^{1/2}} \frac{\lambda_f(n)}{n^{\frac{1}{2} + it}}.$$ From here, the spectral large sieve (due to Deshouillers and Iwaniec; see this paper for a general version: https://doi.org/10.5802/jtnb.887) implies that if $|t| \leq TU$, then $$\sum_{\substack{f \in \mathcal{B}_0 \\ T - U \leq t_f \leq T + U}} \frac{\left|L\left(\frac{1}{2} + it,f\right)\right|^2}{L(1,\operatorname{ad}f)} \ll_{\varepsilon} TU ((1 + |T + t|)(1 + |T - t|))^{\varepsilon},$$ as desired. However, if $|t| \geq (TU)^{1 + \delta}$ for some $\delta > 0$, then this no longer holds (the length of the Dirichlet polynomial is too long).
Similarly, if $k = 4$, then one can use the approximate functional equation for $|L(1/2 + it,f)|^2$ to see that if $|t| \leq T$, then $$\sum_{\substack{f \in \mathcal{B}_0 \\ 0 \leq t_f \leq T}} \frac{\left|L\left(\frac{1}{2} + it,f\right)\right|^4}{L(1,\operatorname{ad}f)} \ll_{\varepsilon} T^{2 + \varepsilon}.$$ This method does not give strong enough bounds if $U \leq T^{1 - \delta}$ for some $\delta > 0$ or if $|t| \geq T^{1 + \delta}$ for some $\delta > 0$, however.
Improving the former result for $|t|$ large or the latter result for either $|t|$ large or $U$ small is difficult, but is known in certain cases. It is much harder than simply using approximate functional equations and the spectral large sieve. Some good references for this are these papers:
https://doi.org/10.2298/PIM0476041J
https://doi.org/10.1007/BF02588051
Finally, there are also bounds for integral moments in the $t$-aspect rather than discrete moments in the $t_f$-aspect. One can investigate the $k$-th integral moment $$\int_{T - U}^{T + U} \left|L\left(\frac{1}{2} + it,f\right)\right|^k \, dt,$$ where $T^{\varepsilon} \leq U \leq T$. A Lindelöf-on-average bound for this would be something of the form $O_{k,\varepsilon}(U ((1 + |T + t_f|)(1 + |T - t_f|))^{\varepsilon})$.
For $k = 2$, one can use approximate functional equations together with the Montgomery-Vaughan mean value estimate (i.e. Gallagher's large sieve) to obtain this bound so long as $U = T$ and $t_f \leq T$. In other ranges, this is a hard problem, but modern methods can obtain results in certain ranges.