\label{main_thm}
Let $\mathcal{H}\in (\frac{1}{3},1)$, and let the initial data $u_0^{\epsilon},v_0^\epsilon $ be divergence-free, of zero mean, and such that
\begin{align*}
   \sup_{\epsilon\in (0,1)} \|u_0^{\epsilon}\|_{L^2_x}+\|\sqrt{\epsilon}v_0^{\epsilon}\|_{L^2_x}<+\infty.
\end{align*}
Assume that the operators $Q$ and $C$ satisfy the conditions stated in \autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\mathcal{H},\epsilon})$ to \eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \eqref{eq_Hgreat12} in the case $\alpha=1$ and $\mathcal{H}>\tfrac{1}{2}$, and to a probabilistically weak rough path solution of \eqref{eq_Hsmall12} in the case $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}$.

\begin{definition}\label{def_rough_sol}
    Let $\epsilon\in (0,1)$, $\frac1\H\in[N,N+1)\cap(1,3)$, $p\in (\frac{1}{\H},N+1)$ and assume $C,Q$ satisfy the condition of \autoref{sec_hp_noise}.\\
     A tuple $((\Omega^\epsilon, \mathcal{F}^\epsilon, (\mathcal{F}_t^\epsilon)_t,\mathbb{P}^{\epsilon}),W^{\epsilon}_t,u^\epsilon_t,r^\epsilon_t)$ is a probabilistically weak solution to equation \eqref{system_prelimit} if
    \begin{enumerate}
        \item $(\Omega^\epsilon, \mathcal{F}^\epsilon, (\mathcal{F}_t^\epsilon)_t,\mathbb{P}^{\epsilon})$ is a stochastic basis with complete and right-continuous filtration,
        \item $W^{\epsilon,\H}_t$ is a fractional Wiener process of Hurst parameter $\H$ and space covariance $Q$ and it is adapted to $(\Omega^\epsilon, \mathcal{F}^\epsilon, \mathcal{F}_t^\epsilon,\mathbb{P}^{\epsilon})$,
        \item $r^\epsilon\in L^2([0,T],H^{\gamma})$ $\mathbb{P}^{\epsilon}$-a.s., it is progressively measurable and satisfies $\mathbb{P}^{\epsilon}$-a.s.
       \begin{equation*}
       \begin{aligned}
    \partial_t r^{\epsilon}&=\epsilon^{-1}Cr^{\epsilon}+\nu \Delta(\epsilon^{-\beta}w^{\epsilon}+r^{\epsilon})\\&-b(u^{\epsilon}+\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon},\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon})
    \end{aligned}
       \end{equation*}
        in an analytically weak sense,
        \item $u^{\epsilon}\in L^{\infty}([0,T],L^2)\cap L^2([0,T],H^1)$ $\mathbb{P}^{\epsilon}$-a.s., it is progressively measurable and $\mathbb{P}^{\epsilon}$-a.s it holds, in a weak sense,
        \begin{equation*}
            u^\epsilon\st=\int_s^t \Delta u^\epsilon_\theta-b(u_\theta^\epsilon+r^\epsilon_\theta,u^\epsilon_\theta)\,d\theta+\sum_{k=1}^N \mathbb{A}^{k,\epsilon}\st u_s^\epsilon+u^{\natural,\epsilon}\st,
        \end{equation*}
        where $u^{\natural}\st\in C^{p/(N+1)\var}_{2,loc}(H^{-3})$ $\mathbb{P}^{\epsilon}$-a.s.
    \end{enumerate}
    \end{definition}

\begin{theorem}\label{main_thm}
Let $\H\in (\frac{1}{3},1)$, and let the initial data $u_0^{\epsilon},v_0^\epsilon $ be divergence-free, of zero mean, and such that
\begin{align*}
   \sup_{\epsilon\in (0,1)} \|u_0^{\epsilon}\|_{L^2_x}+\|\sqrt{\epsilon}v_0^{\epsilon}\|_{L^2_x}<+\infty.
\end{align*}
Assume that the operators $Q$ and $C$ satisfy the conditions stated in \autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\H,\epsilon})$ to \eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \eqref{eq_Hgreat12} in the case $\alpha=1$ and $\H>\tfrac{1}{2}$, and to a probabilistically weak rough path solution of \eqref{eq_Hsmall12} in the case $\H<\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\H$.
\end{theorem}

\begin{align}\label{e:equation1}
\begin{cases}
     du^{\epsilon}=\nu \Delta u^{\epsilon}dt-(u^{\epsilon}+v^{\epsilon})\cdot\nabla u^{\epsilon}dt+\nabla p^{\epsilon}\,dt,\\
    dv^{\epsilon}=\left(\nu\Delta v^{\epsilon}+\epsilon^{-1}Cv^{\epsilon}-(u^{\epsilon}+v^{\epsilon})\cdot\nabla v^{\epsilon}+\nabla q^{\epsilon}\right)\,dt+\epsilon^{-\alpha}dW^{\H}_t,\\
    \nabla\cdot u^{\epsilon}=0,\quad \nabla\cdot v^{\epsilon}=0,
\end{cases}
\end{align}

\begin{definition}\label{def_rough_lim_sol_Hsmall12}
    A tuple $((\Omega, \mathcal{F}, (\mathcal{F}_t)_t,\mathbb{P}),u_t)$ is a weak solution to equation \eqref{eq_Hsmall12} if
    \begin{enumerate}
        \item $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t,\mathbb{P})$ is a stochastic basis with complete and right-continuous filtration,

        \item $u\in C_w([0,T],L^2)\cap L^2([0,T],H^1)$ $\mathbb{P}$-a.s.,  it is progressively measurable and $\mathbb{P}$-a.s it holds, in a weak sense,
        \begin{equation*}
            u\st=\int_s^t \Delta u^\epsilon_\theta-b(u_\theta+\overline{r},u_\theta)\,d\theta,
        \end{equation*}
        where $\overline{r}\in H$.
    \end{enumerate}
    \end{definition}

\begin{lemma}\label{lem_ito_stokes}
   Let $\H\in (\frac{1}{4},\frac{1}{2}]$ denoting by
    \begin{align*}
        \overline{r}=\int_{L^2}(-C)^{-1}b(r,r)d\mu(r),\quad
        \begin{cases}
            dw&= Cwdt+dW_t\\
            w(0)&=0,\
        \end{cases}
    \end{align*}
    where $\mu$ is the centered Gaussian measure on $L^2$ with covariance operator
    \begin{align*}
        \overline{Q}=\sum_{i\geq 1} \frac{\sigma_i^2 e_i\otimes e_i}{2|\lambda_i|^{2\H}}\int_0^{+\infty}e^{-s} s^{2\H} ds
    \end{align*}
    then it holds
\begin{align}\label{convergence_ITO_STOKES_COND}
        \frac{1}{T}\int_0^T (-C)^{-1}b(w_s,w_s)ds\rightarrow \overline{r} \quad \text{in } L^2(\Omega;H).
    \end{align}
\end{lemma}

\begin{remark}\label{remark_d2}
    In dimension $d=2$, thanks to the uniqueness of both the prelimit and the limit equations, the above statement can be strengthened. Under the same assumptions as in \autoref{main_thm}, there exist probabilistically strong rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\H})$ to \eqref{system_prelimit} that converge in probability to the unique weak rough path solution of \eqref{eq_Hgreat12} in the case $\alpha=1$ and $\H>\tfrac{1}{2}$, and of \eqref{eq_Hsmall12} in the case $\H<\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\H$.
\end{remark}

\begin{definition}\label{def_rough_lim_sol_Hgreat12}
        Let $\frac1\H\in[N,N+1)\cap(1,3)$, $p\in (\frac{1}{\H},N+1)$ and assume $C,Q$ satisfy the condition of \autoref{sec_hp_noise}.\\
     A tuple $((\Omega, \mathcal{F}, (\mathcal{F}_t)_t,\mathbb{P}),\W^{\H}_t,u_t)$ is a probabilistically weak solution to equation \eqref{eq_Hgreat12} if
    \begin{enumerate}
        \item $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t,\mathbb{P})$ is a stochastic basis with complete and right-continuous filtration,
        \item $\W^{\H}_t$ is a $p$-rough path on $H^{\xi}$ $\mathbb{P}$-a.s.,
        \item $\mathbb{W}^{\H,1}_t$ is a fractional Wiener process of Hurst parameter $\H$ and space covariance $(-C)^{-1}Q(-C)^{-1}$ and it is adapted to $(\mathcal{F}_t)_t$,
        \item $u\in C_w([0,T],L^2)\cap L^2([0,T],H^1)$ $\mathbb{P}$-a.s., it is progressively measurable and $\mathbb{P}$-a.s it holds, in a weak sense,
        \begin{equation*}
            u\st=\int_s^t \Delta u_r-b(u_r,u_r)\,dr+\sum_{k=1}^N \mathbb{A}^{k}\st u_s+u^{\natural}\st,
        \end{equation*}
        where $u^{\natural}\st\in C^{p/(N+1)\var}_{2,loc}(H^{-3})$ $\mathbb{P}$-a.s., and $(\A^1,\A^2)$ is defined by
        \begin{equation}\label{def_limitURD}
            \A^{1}\st\phi=b(W^\H\st,\phi),\quad \A^2\st\phi=b^{(2)}(\mathbb{W}^{\H,2}\st,\phi).
        \end{equation}
    \end{enumerate}
    \end{definition}

\label{sec:introduction}
In recent years, slow-fast systems associated with fluid equations have attracted considerable attention; see, for instance, \cite{flandoli20212d, flandoli2022additive, debussche2024second,debussche2023rough}. The paradigmatic example to have in mind is provided by the stochastic Navier–Stokes equations on the three-dimensional torus, given by
\begin{align}\label{intro_syst_h12}
    \begin{cases}
    du^{\epsilon}=\nu \Delta u^{\epsilon}dt-(u^{\epsilon}+v^{\epsilon})\cdot\nabla u^{\epsilon}dt+\nabla p^{\epsilon}\,dt,\\
    dv^{\epsilon}=\left(\nu\Delta v^{\epsilon}+\epsilon^{-1}Cv^{\epsilon}-(u^{\epsilon}+v^{\epsilon})\cdot\nabla v^{\epsilon}+\nabla q^{\epsilon}\right)\,dt+\epsilon^{-1}dW_t,\\
    \nabla\cdot u^{\epsilon}=0,\quad \nabla\cdot v^{\epsilon}=0,
\end{cases}
\end{align}
where $W$ is an infinite-dimensional Brownian motion satisfying suitable spatial regularity assumptions, $\epsilon\ll 1$ is a small parameter governing the separation of scales between the \emph{large} scales $u^{\epsilon}$ and the \emph{small} scales $v^{\epsilon}$, and $C$ is a suitable damping operator. For simplicity, we restrict this heuristic discussion to the case $C=-I$. In the aforementioned works it was shown that, as $\epsilon\to 0$, the large-scale component $u^{\epsilon}$ converges in distribution to a process $u$, which is a weak solution to the following stochastic Navier–Stokes equations:
\begin{align}\label{intro_transport_h12}
    \begin{cases}
    du=\left(\nu \Delta u-u\cdot\nabla u-\overline{r}\cdot\nabla u\right) dt+d\nabla p-\circ dW\cdot\nabla u,\\
    \nabla\cdot u=0,
\end{cases}
\end{align}
where $\overline{r}$ is a deterministic vector field, referred to in the literature as the It\^o-Stokes drift (cf. \cite{memin2014fluid}, \cite{debussche2023consistent}), and is related to the spatial covariance structure of the noise $W$. Moreover, $p$ is a semimartingale, often called the turbulent pressure, and the stochastic integral is understood in the Stratonovich sense.

The interest in results of this type is twofold. On the one hand, they are relevant for stochastic model reduction techniques in climate dynamics; see, for example, \cite{majda2001mathematical, franzke2005low, franzke2006low}. On the other hand, they provide a rigorous justification for the emergence of transport-type noise in fluid equations such as \eqref{intro_transport_h12}. Alternative approaches to those developed in \cite{debussche2024second} can be found in \cite{holm2015variational, memin2014fluid}. Over the last decade, starting from \cite{flandoli2010well}, models like \eqref{intro_transport_h12} have received increasing attention due to their regularizing effects (cf. \cite{maurelli_et_al, flandoli2021high, coghi2023existence} and the references therein), as well as their ability to capture key features of turbulent flows, including enhanced and anomalous dissipation phenomena (see for istance \cite{MR1905858, flandoli2024quantitative, luo2024elementary, rowan2024anomalous, drivas2025anomalous}) and dynamo effects (cf. \cite{butori2024mean, butori2024background}).

From a modeling perspective, however, it is important to account for non-Markovian noise with memory effects when describing turbulent fluids; see, for instance, \cite{apolinario2022dynamical, apolinario2023linear, franzke2015stochastic, faranda2014modelling, lilly2017fractional}. This has led to a growing interest in the analysis of equations of the form \eqref{intro_transport_h12}, where the Brownian motion $W$ is replaced by a fractional Brownian motion, as in \cite{hofmanova2019navier, hofmanova2021rough, crisan2022solution, roveri2024well, galeati2026well, flandoli2023reduced, cifani2025diffusion}. Nevertheless, as pointed out in \cite{flandoli20212d, flandoli2022additive, debussche2024second, debussche2023rough}, such limit equations should be regarded as idealized models arising from the slow-fast system \eqref{intro_syst_h12}. A rigorous analysis of the original system when the driving noise $W$ is replaced by a fractional Brownian motion $W^{\mathcal{H}}$ with Hurst parameter $\mathcal{H}$ is still missing. Indeed, all the aforementioned works crucially rely on the Markovian nature of $W$ and on the availability of It\^o-type formulas for the system \eqref{intro_syst_h12}.

Furthermore, although a substantial body of literature on slow–fast systems driven by fractional noise has recently emerged (see, for instance, \cite{li2022slow, gehringer2022functional, pei2024almost, pei2023almost, li2026navierstokesfractionaltransportnoise} and the survey \cite{gehringer2019rough}), to the best of our knowledge none of these works addresses the specific issues discussed above. The present article initiates the study of this gap, see \autoref{main_thm} below for details.

Before presenting our main contributions, we briefly discuss the origin of the terms appearing in \eqref{intro_syst_h12}, as their justification plays a central role in our framework. We start from the incompressible Navier–Stokes equations
\begin{align*}
    \begin{cases}
        \partial_t \overline{u}&=\nu\Delta \overline{u}-\overline{u}\cdot\nabla \overline{u}+\nabla \overline{p},\\
        \nabla\cdot \overline{u}&=0.
    \end{cases}
\end{align*}
We decompose the initial condition into large and small scales according to a suitable scale-separation rule,
\begin{align*}
    \overline{u}(0)=u^{\epsilon}(0)+v^{\epsilon}(0),
\end{align*}
where $\epsilon>0$ is a parameter characterizing the separation of scales. Assuming that this separation persists for positive times, we introduce two velocity fields $u^{\epsilon}(t)$ and $v^{\epsilon}(t)$, describing the evolution of the large- and small-scale components, respectively. Their dynamics are then governed by
\begin{align}\label{system_intermediate_small}
    \begin{cases}
        \partial_t u^{\epsilon}+(u^{\epsilon}+v^{\epsilon})\cdot\nabla u^{\epsilon}+\nabla p^{\epsilon}&=0,\\
        \partial_t v^{\epsilon}+(u^{\epsilon}+v^{\epsilon})\cdot\nabla v^{\epsilon}+\nabla q^{\epsilon}&=0,\\
        \nabla\cdot u=\nabla\cdot v=0.
    \end{cases}
\end{align}
At this stage, system \eqref{system_intermediate_small} does not include either noise or damping at the level of the small scales. The introduction of an additive noise term in the small-scale equation is motivated by instabilities induced by boundary irregularities or internal obstacles, which are typically not explicitly modeled in mathematical formulations based on toroidal geometries or smooth bounded domains. We refer to \cite[Chapter~5]{flandoli2023stochastic} for a detailed discussion. However, the presence of additive noise alone leads, via It\^o’s formula, to an \emph{infinite} mean energy injection into the system as $\epsilon\to 0$. To counterbalance this effect, an additional damping operator must be introduced in the small-scale equation. This modification yields precisely the structure of \eqref{intro_syst_h12}, leaving only the scaling of the noise and damping terms to be justified.

The fact that the noise and damping operators scale in the same way can be motivated in at least two different manners in the literature for systems driven by standard Brownian motion. The first argument, inspired by Wong-Zakai-type approximations, is that this scaling leads the small-scale dynamics to approximate white noise. More precisely, consider the process
\begin{equation*}
dw^{\epsilon}_t = -\epsilon^{-1} w^{\epsilon}_t\,dt + dW_t.
\end{equation*}
It, formally, satisfies
\begin{align}\label{heuristic_convergence_noise}
    w^{\epsilon}_t = \epsilon^{-1} \int_0^t e^{-\epsilon^{-1}(t-s)}\, dW_s \rightarrow dW_t
\end{align}
because, upon integrating the equation for $w^{\epsilon}$ in time, we obtain
\begin{align*}
    W_t
    = \epsilon\, w^{\epsilon}_t + \int_0^t w^{\epsilon}_s\, ds
    \approx \int_0^t w^{\epsilon}_s\, ds.
\end{align*}
This heuristic can be made rigorous using, for instance, rough path techniques; see \cite{friz2014convergence}. Moreover, this argument is robust with respect to the choice of the Hurst parameter and therefore applies equally when $W$ is replaced by a fractional Brownian motion with Hurst parameter $\mathcal{H}$.

A second motivation is provided by multiscale analysis; see \cite[Section~2]{flandoli20212d} for the case $\mathcal{H}=\tfrac12$, and \autoref{sec:appendix_multiscale} for its extension to the general case considered here. In contrast to the Wong–Zakai argument, this approach crucially depends on the self-similarity properties of fractional Brownian motion, which vary with the Hurst parameter. As a consequence, from this perspective the scaling $\epsilon^{-1}$ in \eqref{intro_syst_h12} is not universal. For general values of $\mathcal{H}$, the multiscale analysis instead suggests the scaling $\epsilon^{-\frac12-\mathcal{H}}$. In particular, the Wong–Zakai and multiscale scalings coincide only in the classical Brownian case $\mathcal{H}=\tfrac12$.

In view of the discussion above, our primary objective is to rigorously analyze the asymptotic behavior, as $\epsilon \to 0$, of the system
\begin{align}\label{e:equation1}
\begin{cases}
     du^{\epsilon}=\nu \Delta u^{\epsilon}dt-(u^{\epsilon}+v^{\epsilon})\cdot\nabla u^{\epsilon}dt+\nabla p^{\epsilon}\,dt,\\
    dv^{\epsilon}=\left(\nu\Delta v^{\epsilon}+\epsilon^{-1}Cv^{\epsilon}-(u^{\epsilon}+v^{\epsilon})\cdot\nabla v^{\epsilon}+\nabla q^{\epsilon}\right)\,dt+\epsilon^{-\alpha}dW^{\mathcal{H}}_t,\\
    \nabla\cdot u^{\epsilon}=0,\quad \nabla\cdot v^{\epsilon}=0,
\end{cases}
\end{align}
focusing on the interaction between the \emph{large}- and \emph{small}-scale components under stochastic fractional perturbations. Here $\nu>0$, $p^{\epsilon}$ and $q^{\epsilon}$ are pressure fields, $u^{\epsilon},v^{\epsilon}\colon \mathbb{T}^d\times[0,T]\to\mathbb{R}^d$ denote the large- and small-scale velocity fields, respectively, and $W^{\mathcal{H}}$ is a divergence-free, $\mathcal{H}$-fractional Wiener process on $[L^2(\mathbb{T}^d)]^d$. The noise is spatially colored, with covariance operator $Q$, and has Hurst parameter $\mathcal{H}\in\left(\tfrac13,1\right)$. The operator $C$ is linear. Precise assumptions on $C$ and $Q$ are given in \autoref{sec_hp_noise}; here we only mention that $C$ and $Q$ are required to commute. Solutions to \eqref{e:equation1} are interpreted in the sense of unbounded rough drivers; see \cite{bailleul2017unbounded} and the recent review \cite{hocquet2025unbounded}. This notion is recalled in \autoref{sec_rough_formulation}.

As discussed above, there are in principle two relevant choices for the parameter $\alpha\in\{1,\tfrac12+\mathcal{H}\}$, corresponding to different noise intensities. For technical reasons (see the discussion following \autoref{main_thm}), for each fixed value of $\mathcal{H}$ we are only able to treat one of these two regimes. Specifically, we work with the choice $\alpha=1\wedge(\tfrac12+\mathcal{H})$, which yields a Wong–Zakai regime when $\mathcal{H}>\tfrac12$ and a multiscale regime when $\mathcal{H}<\tfrac12$. As already noted, the two regimes coincide in the critical case $\mathcal{H}=\tfrac12$, and this fact has important consequences for the structure of the limiting equation.

When $\mathcal{H}>\tfrac12$, we prove that $u^{\epsilon}$ converges in law to a weak solution of the Young-type PDE with transport noise
\begin{align}\label{eq_Hgreat12}
    \begin{cases}
    du=\left(\nu \Delta u-u\cdot\nabla u\right) dt+d\nabla p- d(-C)^{-1}W^{\mathcal{H}}\cdot\nabla u,\\
    \nabla\cdot u=0.
    \end{cases}
\end{align}
In this Wong–Zakai regime, $v^{\epsilon}$ converges, as expected, to a fractional white noise. However, in contrast to \cite{debussche2024second, debussche2023rough}, no It\^o–Stokes drift appears in the limit. This absence is not due to isotropy assumptions on the covariance $Q$, as in \cite{flandoli20212d, flandoli2022additive}, but rather to the specific scaling of the noise.

On the other hand, when $\mathcal{H}<\tfrac12$, the noise intensity is too weak to yield convergence to fractional white noise. Nevertheless, the quadratic nonlinearity $v^{\epsilon}\cdot\nabla v^{\epsilon}$ still produces a macroscopic effect in the limit dynamics. In this case, we show that $u^{\epsilon}$ converges in law to a weak solution of the deterministic PDE with an additional transport term,
\begin{align}\label{eq_Hsmall12}
    \begin{cases}
    \partial_t u=\nu \Delta u-\overline{r}\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
    \nabla\cdot u=0,
    \end{cases}
\end{align}
where $\overline{r}$ denotes the It\^o–Stokes drift and depends explicitly on $C$, $Q$, and $\mathcal{H}$; see \autoref{lem_ito_stokes} below.

It appears that only in the case $\mathcal{H}=\tfrac12$, where the Wong–Zakai and multiscale regimes coincide, do both effects, transport noise and It\^o–Stokes drift, simultaneously emerge in the limiting dynamics, as observed in \cite{debussche2024second, debussche2023rough}.

Finally, solutions to \eqref{eq_Hgreat12} (respectively, \eqref{eq_Hsmall12}) are understood in the sense of \autoref{def_rough_lim_sol_Hgreat12} (respectively, \autoref{def_rough_lim_sol_Hsmall12}).
\subsection{Methodologies and Main Results}\label{subsec_results}
Similarly to \cite{debussche2023rough}, we introduce the decomposition
\begin{align*}
    v^{\epsilon}=\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon}.
\end{align*}
After projecting onto divergence-free vector fields via the Leray projector $\Pi$, denoting by $A$ the Stokes operator $A:=\Pi\Delta$, and by $b(\cdot,\cdot)$ the Navier–Stokes nonlinearity as rigorously defined in \autoref{sec_notation}, the components evolve according to the following system:
\begin{align}\label{system_prelimit}
    \begin{cases}
        &\partial_t u^{\epsilon}=\nu A u^{\epsilon}-b(u^{\epsilon}+\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon},u^{\epsilon}),\\
        &dw^{\epsilon}=\epsilon^{-1}Cw^{\epsilon} dt+ \epsilon^{-\mathcal{H}}dW^\H_t, \\
        &\begin{aligned}\partial_t r^{\epsilon}&=\epsilon^{-1}Cr^{\epsilon}+\nu A(\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon})\\&-b(u^{\epsilon}+\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon},\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon}),
        \end{aligned}\\
        &w^{\epsilon}(0) =0,\quad r^{\epsilon}(0)=v_0^{\epsilon}.
    \end{cases}
\end{align}
Solutions to this system are understood in the sense of \autoref{def_rough_sol}. We do not provide the details of the construction of solutions for fixed $\epsilon\in(0,1)$, as these follow from classical arguments. In particular, the rescaled Ornstein–Uhlenbeck process $w^{\epsilon}$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\mathcal{H}}$, is uniquely determined, and is measurable with respect to $W^{\mathcal{H}}$. Moreover, for each fixed $\epsilon\in(0,1)$ there exists a probabilistically and analytically weak solution to \eqref{system_prelimit}. That is, for every $\epsilon\in(0,1)$ there exist a probability space $(\Omega_\epsilon,\mathcal F_\epsilon,\mathbb P_\epsilon)$, a $Q$-fractional Wiener process $W^{\epsilon,\mathcal{H}}$, and processes $(u^{\epsilon},r^{\epsilon},w^{\epsilon})$ solving \eqref{system_prelimit} in the analytically weak sense. The proof relies on a Galerkin approximation combined with stochastic compactness arguments. In dimension $d=2$, where uniqueness holds as in the deterministic case, probabilistically weak existence can in fact be strengthened to probabilistically strong existence: as for the Ornstein–Uhlenbeck process alone, the triple $(u^{\epsilon},w^{\epsilon},r^{\epsilon})$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\mathcal{H}}$.

As already mentioned, it is necessary to reformulate the system within the framework of rough path theory. This is carried out in \autoref{sec_rough_formulation} and \autoref{sec_unbounder_rough}, and allows us to control the singular term $b(\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon},u^{\epsilon})$ by introducing the process
\begin{align*}
    y^{\epsilon}_t=\epsilon^{-\alpha+\mathcal{H}}\int_0^t w^{\epsilon}_s ds
\end{align*}
together with its canonical lifts. Here, in contrast to \cite{debussche2023rough}, we cannot rely on It\^o-type formulas to characterize the lifts, and instead must appeal to the theory of Gaussian rough paths (see \cite{friz2010multidimensional, friz2014course}), especially in the case $\mathcal{H}<\tfrac12$. As already discussed, when $\mathcal{H}>\tfrac12$ no It\^o–Stokes drift appears in the limit, whereas for $\mathcal{H}<\tfrac12$ this drift is the only surviving contribution. This dichotomy stems from the convergence properties of $r^{\epsilon}$, which ultimately rely on an ergodic-type result for $\int_0^T b(w^{\epsilon}_s,w^{\epsilon}_s),ds$. While the Markovian structure makes this argument relatively standard when $\mathcal{H}=\tfrac12$ (cf. \cite[Section~5]{debussche2023rough}), in the fractional case we must instead rely on ad hoc computations exploiting the explicit structure of the Navier–Stokes nonlinearity; see \autoref{sec:stoc_conv}. With these preliminaries in place, we can now state our main result.

When $\mathcal{H}>\tfrac12$, we prove that $u^{\epsilon}$ converges in law to a weak solution of the Young-type PDE with transport noise
\begin{align}\label{eq_Hgreat12}
 \begin{cases}
 du=\left(\nu \Delta u-u\cdot\nabla u\right) dt+d\nabla p- d(-C)^{-1}W^{\mathcal{H}}\cdot\nabla u,\\
 \nabla\cdot u=0.
 \end{cases}
\end{align}
In this Wong–Zakai regime, $v^{\epsilon}$ converges, as expected, to a fractional white noise. However, in contrast to \cite{debussche2024second, debussche2023rough}, no It\^o–Stokes drift appears in the limit. This absence is not due to isotropy assumptions on the covariance $Q$, as in \cite{flandoli20212d, flandoli2022additive}, but rather to the specific scaling of the noise.

On the other hand, when $\mathcal{H}<\tfrac12$, the noise intensity is too weak to yield convergence to fractional white noise. Nevertheless, the quadratic nonlinearity $v^{\epsilon}\cdot\nabla v^{\epsilon}$ still produces a macroscopic effect in the limit dynamics. In this case, we show that $u^{\epsilon}$ converges in law to a weak solution of the deterministic PDE with an additional transport term,
\begin{align}\label{eq_Hsmall12}
 \begin{cases}
 \partial_t u=\nu \Delta u-\overline{r}\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
 \nabla\cdot u=0,
 \end{cases}
\end{align}
where $\overline{r}$ denotes the It\^o–Stokes drift and depends explicitly on $C$, $Q$, and $\mathcal{H}$; see \autoref{lem_ito_stokes} below.

Finally, solutions to \eqref{eq_Hgreat12} (respectively, \eqref{eq_Hsmall12}) are understood in the sense of \autoref{def_rough_lim_sol_Hgreat12} (respectively, \autoref{def_rough_lim_sol_Hsmall12}).
\subsection{Methodologies and Main Results}\label{subsec_results}
Similarly to \cite{debussche2023rough}, we introduce the decomposition
\begin{align*}
 v^{\epsilon}=\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon}.
\end{align*}
After projecting onto divergence-free vector fields via the Leray projector $\Pi$, denoting by $A$ the Stokes operator $A:=\Pi\Delta$, and by $b(\cdot,\cdot)$ the Navier–Stokes nonlinearity as rigorously defined in \autoref{sec_notation}, the components evolve according to the following system:
\begin{align}\label{system_prelimit}
 \begin{cases}
 &\partial_t u^{\epsilon}=\nu A u^{\epsilon}-b(u^{\epsilon}+\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon},u^{\epsilon}),\\
 &dw^{\epsilon}=\epsilon^{-1}Cw^{\epsilon} dt+ \epsilon^{-\mathcal{H}}dW^\H_t, \\
 &\begin{aligned}\partial_t r^{\epsilon}&=\epsilon^{-1}Cr^{\epsilon}+\nu A(\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon})\\&-b(u^{\epsilon}+\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon},\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon}+r^{\epsilon}),
 \end{aligned}\\
 &w^{\epsilon}(0) =0,\quad r^{\epsilon}(0)=v_0^{\epsilon}.
 \end{cases}
\end{align}
Solutions to this system are understood in the sense of \autoref{def_rough_sol}. We do not provide the details of the construction of solutions for fixed $\epsilon\in(0,1)$, as these follow from classical arguments. In particular, the rescaled Ornstein–Uhlenbeck process $w^{\epsilon}$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\mathcal{H}}$, is uniquely determined, and is measurable with respect to $W^{\mathcal{H}}$. Moreover, for each fixed $\epsilon\in(0,1)$ there exists a probabilistically and analytically weak solution to \eqref{system_prelimit}. That is, for every $\epsilon\in(0,1)$ there exist a probability space $(\Omega_\epsilon,\mathcal F_\epsilon,\mathbb P_\epsilon)$, a $Q$-fractional Wiener process $W^{\epsilon,\mathcal{H}}$, and processes $(u^{\epsilon},r^{\epsilon},w^{\epsilon})$ solving \eqref{system_prelimit} in the analytically weak sense. The proof relies on a Galerkin approximation combined with stochastic compactness arguments. In dimension $d=2$, where uniqueness holds as in the deterministic case, probabilistically weak existence can in fact be strengthened to probabilistically strong existence: as for the Ornstein–Uhlenbeck process alone, the triple $(u^{\epsilon},w^{\epsilon},r^{\epsilon})$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\mathcal{H}}$.

As already mentioned, it is necessary to reformulate the system within the framework of rough path theory. This is carried out in \autoref{sec_rough_formulation} and \autoref{sec_unbounder_rough}, and allows us to control the singular term $b(\epsilon^{-\alpha+\mathcal{H}}w^{\epsilon},u^{\epsilon})$ by introducing the process
\begin{align*}
 y^{\epsilon}_t=\epsilon^{-\alpha+\mathcal{H}}\int_0^t w^{\epsilon}_s ds
\end{align*}
together with its canonical lifts. Here, in contrast to \cite{debussche2023rough}, we cannot rely on It\^o-type formulas to characterize the lifts, and instead must appeal to the theory of Gaussian rough paths (see \cite{friz2010multidimensional, friz2014course}), especially in the case $\mathcal{H}<\tfrac12$. As already discussed, when $\mathcal{H}>\tfrac12$ no It\^o–Stokes drift appears in the limit, whereas for $\mathcal{H}<\tfrac12$ this drift is the only surviving contribution. This dichotomy stems from the convergence properties of $r^{\epsilon}$, which ultimately rely on an ergodic-type result for $\int_0^T b(w^{\epsilon}_s,w^{\epsilon}_s),ds$. While the Markovian structure makes this argument relatively standard when $\mathcal{H}=\tfrac12$ (cf. \cite[Section~5]{debussche2023rough}), in the fractional case we must instead rely on ad hoc computations exploiting the explicit structure of the Navier–Stokes nonlinearity; see \autoref{sec:stoc_conv}. With these preliminaries in place, we can now state our main result.

Since we have been able to pass to the limit in all the terms except the remainder, also $\overline{u}^{\epsilon,\sharp}$ converge $\overline{\mathbb{P}}-a.s$ to some object $\overline{u}^{\sharp}\st$ such that, for $\alpha=\frac{1}{2}+\H$,
\begin{align}\label{eqUtile}
 \overline{u}_{\st}=\int_s^t \Delta \overline{u}_\tau+b(\overline{u}_{\tau}+\overline{r}_\tau,\overline{u}_\tau)\,dr+\overline{u}^{\sharp}_{st} \quad \mathbb{\overline{P}}-a.s.,
\end{align}
while for $\alpha=1$,
\begin{align}
 \overline{u}_{\st}=\int_s^t \Delta \overline{u}_\tau+b(\overline{u}_{\tau},\overline{u}_\tau)\,dr+\sum_{k=1}^2 \mathbb{A}^{k}\st u_s+\overline{u}^{\sharp}_{st} \quad \mathbb{\overline{P}}-a.s..
\end{align}
Let us conclude by proving that $\overline{u}^{\sharp}_{st}\in C^{p/3\var}_{2,loc}(H^{-3}).$ Since the rough paths converge and, in particular, \eqref{est:UniformRegularityURDH<1/2} and \eqref{est:UniformRegularityURDH<1/2zeros} hold, then we can identify a uniform in $\epsilon$ random covering $\{J_k\}_k$ of $[0,T]$ by introducing the localization \[\sup_{\epsilon\in (0,1)}\omega_\epsilon^{1/p}(s,t)\le\frac{1}{2\lambda}.\]
Therefore thanks to
\eqref{stimaRemainder}, we have that on any $J_k$, for all $s,t \in J_k$ and $\phi \in H^3$, the following inequalities hold
 \begin{align} &|\overline{u}^{\sharp}\st(\phi)|=\lim_{\epsilon\to 0}|\overline{u}^{\epsilon,\sharp}\st(\phi)|\le\|\phi\|_{H^3}\liminf_{\epsilon\to 0}{\|\overline{u}^{\epsilon,\sharp}\|_{p/3\var,[s,t]}},\\
 &\|\overline{u}^{\epsilon,\sharp}\|_{p/3\var,[s,t]}\lesssim\sx(\omega_{\epsilon}(s,t)+\omega_{\mu^\epsilon}(s,t)^{p/2}\dx)^{3/p}\lesssim M(\omega),
 \end{align}
where $M(\omega)$ is a random constant that depends on \eqref{pathwise_estimate}.
Then for any $\mathbf{P}$ partition of $J_k$, we have that
\begin{equation}
 \sum_{(s,t)\in P}\|\overline{u}^{\sharp}\st\|_{H^{-3}}^{p/3}\lesssim \liminf_{\epsilon\to0}\sum_{(s,t)\in P}{\|\overline{u}^{\epsilon,\sharp}\st\|_{H^{-3}}^{p/3}}\lesssim \liminf_{\epsilon\to0} \|\overline{u}^{\epsilon,\sharp}\|_{p/3\var,J_k}^{p/3}< \infty.
\end{equation}
Therefore, we have shown that $\overline{u}^{\sharp}_{st}\in C^{p/3\var}_{2,loc}(H^{-3}).$ At last, we stress that in case of $\alpha=\frac12+\H$ we have that $\overline{u}^{\sharp}\equiv0$ since it is highly regular in time and it is given by the increments of a $1$-index function of time, see \eqref{eqUtile}.
\subsection*{Case $\H>\frac{1}{2}$}
The convergence in this final case is obtained by closely mirroring the proof from the preceding case.
The apriori estimates provided by \autoref{lemma_a_priori} combined with \autoref{proproughHgreat12} and the compactness criteria \cite[Lemma A.2]{hofmanova2019navier} imply that, in the case of \autoref{main_thm}, the family
\begin{align*}
 (u^{\epsilon},r^{\epsilon}, W^\H,\mathbb{Y}^{\epsilon,1})
\end{align*}
is tight in the space
\begin{align*}
 \mathcal{X}&=\mathcal{X}_{u,r,W}\times\mathcal{X}_{RP},\\
 \mathcal{X}_{u,r,W}&=(C_w([0,T];H)\cap L^2(0,T;H))\times L^2((0,T)\times \T^3)_w\times C([0,T];H),\\
 \mathcal{X}_{RP}&= C^{p\var}([0,T];H^{\xi}).
\end{align*}
 Therefore, by Jakubowski–Skorokhod’s representation theorem, see for example \cite[Section 2.7]{breit2018stochastically}, there exists a subsequence of $\epsilon_k\rightarrow 0$ which for the matter of notation we continue to denote by $\epsilon$ and an auxiliary probability space $\left(\overline{\Omega},\overline{\mathcal{F}},\overline{\mathbb{P}}\right)$ with $\mathcal{X}$ valued random variables $\sx(\overline{u}^{\epsilon},\overline{r}^{\epsilon}, \overline{W}^{\H,\epsilon},\overline{\mathbb{Y}}^{\epsilon,1}\dx)$ and $\sx(\overline{u},\overline{r}, \overline{W}^{\H},\overline{\mathbb{Y}}^{1}\dx)$ such that
\begin{align*}
 \sx(\overline{u}^{\epsilon},\overline{r}^{\epsilon}, \overline{W}^{\H,\epsilon},\overline{\mathbb{Y}}^{\epsilon,1}\dx)\stackrel{\mathcal{X}}{\rightarrow}\sx(\overline{u},\overline{r}, \overline{W}^{\H},\overline{\mathbb{Y}}^{1}\dx)\quad \overline{\mathbb{P}}-a.s..
\end{align*}
The convergence of all the terms as well as showing that the remainder $\overline{u}^{\sharp}\in C^{p/2\var}_{2,loc}(H^{-3})$ can be justified similarly to the previous case. Therefore, we need to identify $\overline{r}$ to conclude the proof of \autoref{main_thm}.
\begin{lemma}\label{identification_lemma_smoothcase}
 It holds
 \begin{align*}
 \overline{r}=0.
 \end{align*}
\end{lemma}
\begin{proof}
 Let us call by $\overline{w}^{\epsilon}$ the solution of
 \begin{align*}
d\overline{w}^{\epsilon}=\epsilon^{-1}C\overline{w}^{\epsilon}\,dt+\epsilon^{- \H}d\overline{W}^{\H,\epsilon}_t.
 \end{align*}
 For each $\psi\in C^{1}_c(0,T)$ it holds due to \autoref{def_rough_sol}
 \begin{align*}
 \int_0^T \psi_t \overline{r}^{\epsilon}_t dt&=\epsilon^{2\H-1}\int_0^T \psi_t(-C)^{-1}b(\overline{w}^{\epsilon}_t,\overline{w}^{\epsilon}_t)dt\\ & +\epsilon^{\H}\int_0^T\psi_t (-C)^{-1}\left(A\overline{w}^{\epsilon}_t+b(\overline{u}^{\epsilon}_t,\overline{w}^{\epsilon}_t)+b(\overline{w}^{\epsilon}_t,\overline{r}^{\epsilon}_t)+b(\overline{r}^{\epsilon}_t,\overline{w}^{\epsilon}_t)\right)dt\\ & +\epsilon\int_0^T\psi(-C)^{-1}\left(A\overline{r}^{\epsilon}_t+b(\overline{u}^{\epsilon}_t,\overline{r}^{\epsilon}_t)+b(\overline{r}^{\epsilon}_t,\overline{r}^{\epsilon}_t)\right)dt+\epsilon\int_0^T \partial_t \psi_t (-C)^{-1}\overline{r}^{\epsilon}_t dt\\ & =J_1^{\epsilon}+J_2^{\epsilon}+J_3^{\epsilon}+J_4^{\epsilon}.
 \end{align*}
 We will show that, up to subsequences, all the terms approach $0$ in $H^{-\xi}$ $\overline{\mathbb{P}}-a.s.$ . This is enough to conclude by uniqueness of the limit. The treatment of $J_2^\epsilon,\ J_3^\epsilon,\ J_4^\epsilon$ follows verbatim as in the first case of \autoref{identification_lemma} and we omit details. To treat $J_1^\epsilon$ we argue in a simpler way compared to the first case of \autoref{identification_lemma}. Indeed since $\H>\frac{1}{2}$, it holds
 \begin{align*}
 \mathbb{E}\left[\|I_1^{\epsilon}\|_{H^{-\xi}}\right] \lesssim \epsilon^{2\H-1}\int_0^T\mathbb{E}\left[\|\overline{w}^{\epsilon}_t\|^2 \right]dt\rightarrow 0
 \end{align*}
 by \autoref{hp_stoch_conv}. This completes the proof.
\end{proof}
\begin{remark}
 For the sake of simplicity, we have reported estimates showing that $\overline{u}^{\sharp}\in C^{p/2\var}_{2,loc}(H^{-3})$. Nevertheless, we stress that by refining \eqref{stimaExtraURD} we could have shown that
 $\overline{u}^{\sharp}\in C^{p/2\var}_{2,loc}(H^{-2})$ for $H\in (1/2,2/3]$ and $\overline{u}^{\sharp}\in C^{2p/(2p+1)\var}_{2,loc}(H^{-2})$ for $\H>2/3$. Therefore for all $\H\in (1/2,1)$, the integral term arising from the transport noise in the limiting equation can be constructed as a Young integral in $H^{-2}$ instead of $H^{-3}.$
\end{remark}
\begin{acknowledgements}
EL has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 949981). FT is supported by the Istituto Nazionale di Alta Matematica (IN$\delta$AM) through the GNAMPA 2025 project “Modelli stocastici in Fluidodinamica e Turbolenza” (CUP E5324001950001), and by the project
"Noise in fluid dynamics and related models" funded by the MUR Progetti di Ricerca
di Rilevante Interesse Nazionale (PRIN) Bando 2022 - grant 20222YRYSP. FT also acknowledges the warm hospitality of Universität Bielefeld, where this work started. Both authors thank Martina Hofmanová for valuable discussions at an early stage of the project.

\begin{remark}\label{remark_d2}
    In dimension $d=2$, thanks to the uniqueness of both the prelimit and the limit equations, the above statement can be strengthened. Under the same assumptions as in \autoref{main_thm}, there exist probabilistically strong rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\mathcal{H}})$ to \eqref{system_prelimit} that converge in probability to the unique weak rough path solution of \eqref{eq_Hgreat12} in the case $\alpha=1$ and $\mathcal{H}>\tfrac{1}{2}$, and of \eqref{eq_Hsmall12} in the case $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}$.
\end{remark}
\begin{remark}\label{remark_uniqueness}
    Note that even in the case $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}$, the limit object is, in principle, genuinely stochastic and supported on (possibly non-unique) weak solutions of \eqref{eq_Hsmall12}.
\end{remark}
We now discuss why the complementary regimes appear to be out of reach. In the case $\mathcal{H}>\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}>1$, neglecting the additional Navier–Stokes terms yields that the small scales satisfy
\begin{align*}
    v^{\epsilon}_t=\epsilon^{-(\mathcal{H}-1/2)}\epsilon^{-1}\int_0^t e^{-\epsilon^{-1}C(t-s)} dW^{\mathcal{H}}_s.
\end{align*}
In view of \eqref{heuristic_convergence_noise}, this leads to
\begin{align*}
    v^{\epsilon}_t\approx \epsilon^{-(\mathcal{H}-1/2)}(-C)^{-1}dW^{\mathcal{H}}_t\rightarrow +\infty,
\end{align*}
showing that the small-scale component diverges.
The case $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=1$ is more delicate. Here, the main difficulty is the lack of control of the quadratic nonlinearity in the equation for the small scales $v^{\epsilon}$. Nevertheless, for particularly simple choices of the noise, namely
\begin{align*}
    W^{\mathcal{H}}_t(x)=f(x)W^{\mathcal{H},1}_t,
\end{align*}
where $f$ is divergence-free, of zero mean, and satisfies $f\cdot\nabla f=0$\footnote{For instance, this holds when $f(x)=a_l \sin (l\cdot x),\ l,\ a_l\in \mathbb{R}^3,\ a_l\perp l$. } and $W^{\mathcal{H},1}$ is a real-valued fractional Brownian motion, we can still establish the following result.
\begin{proposition}\label{trivial_noise}
Let $\alpha=1$, $\mathcal{H}\in\left(\tfrac{1}{3},\tfrac{1}{2}\right)$, and let the initial data $u_0^{\epsilon}$ and $v_0^{\epsilon}$ be divergence-free, of zero mean, and satisfy
\begin{align*}
   \sup_{\epsilon\in (0,1)} \|u_0^{\epsilon}\|_{L^2_x}+\|\sqrt{\epsilon}v_0^{\epsilon}\|_{L^2_x}<+\infty.
\end{align*} Assume that $W^{\mathcal{H}}_t(x)=f(x)W^{\mathcal{H},1}_t$, with $f$ as above, and that $Q$ and $C$ satisfy the conditions stated in \autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\mathcal{H},\epsilon})$ to \eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \eqref{eq_Hgreat12}. An analogue of \autoref{remark_d2} holds in dimension $d=2$.
\end{proposition}
We conclude this subsection by mentioning the recent preprint \cite{li2026navierstokesfractionaltransportnoise}, which appeared during the final stages of preparation of this work. There, the authors study a fast–slow Navier–Stokes system similar to \eqref{e:equation1}, with $\mathcal{H}\in\left(\tfrac{1}{3},\tfrac{1}{2}\right)$ and $\alpha=1$, focusing on the Wong–Zakai regime. While the slow dynamics coincide with ours, the fast dynamics differ substantially, being governed in \cite{li2026navierstokesfractionaltransportnoise} by the linear equation
\begin{align}\label{eq_fast_1}
dv^{\epsilon}=\epsilon^{-1}Cv^{\epsilon}dt+\epsilon^{-1}dW^{\mathcal{H}}_t.
\end{align}
As already noted after \autoref{main_thm}, the impossibility to control the quadratic term in the fast dynamics prevents us from obtaining a Wong–Zakai convergence result analogous to \cite[Theorem~B]{li2026navierstokesfractionaltransportnoise} within our framework. If this difficulty is removed from the outset, either by adopting the simplified fast dynamics \eqref{eq_fast_1}, as in \cite{li2026navierstokesfractionaltransportnoise}, or by considering instead
\begin{align}\label{eq_fast_2}
dv^{\epsilon}=\nu Av^{\epsilon}dt+\epsilon^{-1}Cv^{\epsilon}dt-b(u^{\epsilon},v^{\epsilon})dt+\epsilon^{-1}dW^{\mathcal{H}}_t,
\end{align}
then our arguments readily extend to the regime $\mathcal{H}\in\left(\tfrac{1}{3},\tfrac{1}{2}\right)$ with $\alpha=1$, thereby recovering the results of \cite{li2026navierstokesfractionaltransportnoise}. We have chosen not to pursue this direction further; instead, we address the difficulty caused by the uncontrolled quadratic term in the fast dynamics by imposing a strong structural assumption on the covariance of the noise $W^{\mathcal{H}}$, as discussed in \autoref{trivial_noise}.
\subsubsection*{Plan of the paper}In \autoref{sec_prel} we rigorously introduce our assumptions on $Q$ and $C$, and we recall some rough path notation including the notion of solutions for \eqref{eq_Hgreat12}, \eqref{eq_Hsmall12}, \eqref{system_prelimit}. The analysis of the long-time behaviour of $w^{\epsilon}$ and of some of its nonlinear functionals is addressed in \autoref{sec:stoc_conv}. The core of our rough path analysis is \autoref{sec_unbounder_rough}, where we study the convergence properties of the canonical lift of $\int_0^{\cdot}e^{-\alpha+\mathcal{H}}w^{\epsilon}_s ds$. The last ingredients for the proof of \autoref{main_thm} are some a priori estimates obtained in \autoref{sec_apriori} exploiting the rough path formulation already mentioned. Combining these elements, the proofs of both \autoref{main_thm} and \autoref{trivial_noise} follow by standard arguments as discussed in \autoref{sec:end_proof}.

A proof outline is given in the “Plan of the paper” paragraph. The ingredients for proving Theorem~\ref{main_thm} are:
\begin{itemize}
\item In \autoref{sec_prel}, “we rigorously introduce our assumptions on $Q$ and $C$, and we recall some rough path notation including the notion of solutions for \eqref{eq_Hgreat12}, \eqref{eq_Hsmall12}, \eqref{system_prelimit}.”
\item In \autoref{sec:stoc_conv}, they analyze “the long-time behaviour of $w^{\epsilon}$ and of some of its nonlinear functionals.”
\item The “core of our rough path analysis” is in \autoref{sec_unbounder_rough}, where they “study the convergence properties of the canonical lift of $\int_0^{\cdot} e^{-\alpha+\mathcal{H}} w^{\epsilon}_s\, ds$.”
\item “The last ingredients for the proof of \autoref{main_thm} are some a priori estimates obtained in \autoref{sec_apriori} exploiting the rough path formulation already mentioned.”
\item “Combining these elements, the proofs of both \autoref{main_thm} and \autoref{trivial_noise} follow by standard arguments as discussed in \autoref{sec:end_proof}.”
\end{itemize}
Additionally, the text explains why complementary regimes are difficult: for $\mathcal{H}>\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}>1$, a heuristic shows “the small-scale component diverges,” while for $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=1$ “the main difficulty is the lack of control of the quadratic nonlinearity in the equation for the small scales $v^{\epsilon}$.”

A proof outline is given in the “Plan of the paper” paragraph. The ingredients for proving the main theorem are:
\begin{itemize}
\item Next, “we rigorously introduce our assumptions on $Q$ and $C$, and we recall some rough path notation including the notion of solutions” for
\begin{align}\label{eq_Hgreat12}
    \begin{cases}
    du=\left(\nu \Delta u-u\cdot\nabla u\right) dt+d\nabla p- d(-C)^{-1}W^{\H}\cdot\nabla u,\\
    \nabla\cdot u=0.
    \end{cases}
\end{align}
\begin{align}\label{eq_Hsmall12}
    \begin{cases}
    \partial_t u=\nu \Delta u-\overline{r}\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
    \nabla\cdot u=0,
    \end{cases}
\end{align}
and
\begin{align}\label{system_prelimit}
    \begin{cases}
        &\partial_t u^{\epsilon}=\nu A u^{\epsilon}-b(u^{\epsilon}+\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon},u^{\epsilon}),\\
        &dw^{\epsilon}=\epsilon^{-1}Cw^{\epsilon} dt+ \epsilon^{-\H}dW^\H_t, \\
        &\begin{aligned}\partial_t r^{\epsilon}&=\epsilon^{-1}Cr^{\epsilon}+\nu A(\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon})\\&-b(u^{\epsilon}+\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon},\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon}),
        \end{aligned}\\
        &w^{\epsilon}(0) =0,\quad r^{\epsilon}(0)=v_0^{\epsilon}.
    \end{cases}
\end{align}
\item Later, they analyze “the long-time behaviour of $w^{\epsilon}$ and of some of its nonlinear functionals.”
\item The “core of our rough path analysis” comes next, where they “study the convergence properties of the canonical lift of $\int_0^{\cdot} e^{-\alpha+\mathcal{H}} w^{\epsilon}_s\, ds$.”
\item “The last ingredients for the proof of the main theorem are some a priori estimates obtained later exploiting the rough path formulation already mentioned.”
\item “Combining these elements, the proofs of both the main theorem and the following proposition follow by standard arguments as discussed later.

We first record the proposition:
\begin{proposition}\label{trivial_noise}
Let $\alpha=1$, $\H\in\left(\tfrac{1}{3},\tfrac{1}{2}\right)$, and let the initial data $u_0^{\epsilon}$ and $v_0^{\epsilon}$ be divergence-free, of zero mean, and satisfy
\begin{align*}
   \sup_{\epsilon\in (0,1)} \|u_0^{\epsilon}\|_{L^2_x}+\|\sqrt{\epsilon}v_0^{\epsilon}\|_{L^2_x}<+\infty.
\end{align*} Assume that $W^{\H}_t(x)=f(x)W^{\H,1}_t$, with $f$ as above, and that $Q$ and $C$ satisfy the conditions stated in \autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\H,\epsilon})$ to \eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \eqref{eq_Hgreat12}. An analogue of \autoref{remark_d2} holds in dimension $d=2$.
\end{proposition}
”
\end{itemize}
Additionally, the text explains why complementary regimes are difficult: for $\mathcal{H}>\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}>1$, a heuristic shows “the small-scale component diverges,” while for $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=1$ “the main difficulty is the lack of control of the quadratic nonlinearity in the equation for the small scales $v^{\epsilon}$.”

\label{main_thm}
Let $\mathcal{H}\in (\frac{1}{3},1)$, and let the initial data $u_0^{\epsilon},v_0^\epsilon $ be divergence-free, of zero mean, and such that
\begin{align*}
   \sup_{\epsilon\in (0,1)} \|u_0^{\epsilon}\|_{L^2_x}+\|\sqrt{\epsilon}v_0^{\epsilon}\|_{L^2_x}<+\infty.
\end{align*}
Assume that the operators $Q$ and $C$ satisfy the stated conditions. Then there exist probabilistically weak rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\mathcal{H},\epsilon})$ to
\begin{align}\label{system_prelimit}
    \begin{cases}
        &\partial_t u^{\epsilon}=\nu A u^{\epsilon}-b(u^{\epsilon}+\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon},u^{\epsilon}),\\
        &dw^{\epsilon}=\epsilon^{-1}Cw^{\epsilon} dt+ \epsilon^{-\H}dW^\H_t, \\
        &\begin{aligned}\partial_t r^{\epsilon}&=\epsilon^{-1}Cr^{\epsilon}+\nu A(\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon})\\&-b(u^{\epsilon}+\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon},\epsilon^{-\alpha+\H}w^{\epsilon}+r^{\epsilon}),
        \end{aligned}\\
        &w^{\epsilon}(0) =0,\quad r^{\epsilon}(0)=v_0^{\epsilon}.
    \end{cases}
\end{align}
that converge in law to a probabilistically weak rough path solution of
\begin{align}\label{eq_Hgreat12}
    \begin{cases}
    du=\left(\nu \Delta u-u\cdot\nabla u\right) dt+d\nabla p- d(-C)^{-1}W^{\H}\cdot\nabla u,\\
    \nabla\cdot u=0.
    \end{cases}
\end{align}
in the case $\alpha=1$ and $\mathcal{H}>\tfrac{1}{2}$, and to a probabilistically weak rough path solution of
\begin{align}\label{eq_Hsmall12}
    \begin{cases}
    \partial_t u=\nu \Delta u-\overline{r}\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
    \nabla\cdot u=0,
    \end{cases}
\end{align}
in the case $\mathcal{H}<\tfrac{1}{2}$ and $\alpha=\tfrac{1}{2}+\mathcal{H}$.

Question:
Let \(\mathcal H\in(\tfrac13,1)\). For each \(\epsilon\in(0,1)\), let \(u_0^\epsilon\) and \(v_0^\epsilon\) be divergence-free, zero-mean initial data satisfying
\[
\sup_{\epsilon\in(0,1)}\Big(\|u_0^\epsilon\|_{L_x^2}+\|\sqrt{\epsilon}\,v_0^\epsilon\|_{L_x^2}\Big)<\infty.
\]
Assume that the operators \(Q\) and \(C\) satisfy the hypotheses required for the model. Let \(\Pi\) be the Leray projector onto divergence-free vector fields, \(A:=\Pi\Delta\) the Stokes operator, and \(b(\cdot,\cdot)\) the Navier--Stokes bilinear term. Let \(W_t^{\mathcal H}\) denote a \(Q\)-fractional Wiener process. Consider the prelimit system
\[
\begin{cases}
\partial_t u^{\epsilon}=\nu A u^{\epsilon}-b\big(u^{\epsilon}+\epsilon^{-\alpha+\mathcal H}w^{\epsilon}+r^{\epsilon},u^{\epsilon}\big),\\
dw^{\epsilon}=\epsilon^{-1}Cw^{\epsilon}\,dt+\epsilon^{-\mathcal H}dW_t^{\mathcal H},\\
\partial_t r^{\epsilon}=\epsilon^{-1}Cr^{\epsilon}+\nu A(\epsilon^{-\alpha+\mathcal H}w^{\epsilon}+r^{\epsilon})
-b\big(u^{\epsilon}+\epsilon^{-\alpha+\mathcal H}w^{\epsilon}+r^{\epsilon},\epsilon^{-\alpha+\mathcal H}w^{\epsilon}+r^{\epsilon}\big),\\
w^{\epsilon}(0)=0,\qquad r^{\epsilon}(0)=v_0^{\epsilon}.
\end{cases}
\]
Here \(\overline r\) denotes the It\^o--Stokes drift determined by \(C\), \(Q\), and \(\mathcal H\). A probabilistically weak rough path solution means a solution constructed on some probability space together with its driving rough path, rather than on a fixed prescribed probability space. Which existence-and-convergence statement holds for this family?

There exists a family of probabilistically weak rough path solutions \((u^{\epsilon},(-C)^{-1}dW^{\mathcal H,\epsilon})\) to the prelimit system such that, as \(\epsilon\to0\), the laws converge as follows: if \(\alpha=1\) and \(\mathcal H>\tfrac12\), then \(u^{\epsilon}\) converges in law to a probabilistically weak rough path solution of
\[
\begin{cases}
du=(\nu\Delta u-u\cdot\nabla u)\,dt+d\nabla p-d(-C)^{-1}W^{\mathcal H}\cdot\nabla u,\\
\nabla\cdot u=0;
\end{cases}
\]
if \(\mathcal H<\tfrac12\) and \(\alpha=\tfrac12+\mathcal H\), then \(u^{\epsilon}\) converges in law to a probabilistically weak rough path solution of
\[
\begin{cases}
\partial_t u=\nu\Delta u-\overline r\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
\nabla\cdot u=0.
\end{cases}
\]

Choices:
B.
There exists a family of probabilistically weak rough path solutions \((u^{\epsilon},(-C)^{-1}dW^{\mathcal H,\epsilon})\) to the prelimit system such that, as \(\epsilon\to0\), the laws converge as follows: if \(\alpha=1\) and \(\mathcal H\ge \tfrac12\), then \(u^{\epsilon}\) converges in law to a probabilistically weak rough path solution of
\[
\begin{cases}
du=(\nu\Delta u-u\cdot\nabla u)\,dt+d\nabla p-d(-C)^{-1}W^{\mathcal H}\cdot\nabla u,\\
\nabla\cdot u=0;
\end{cases}
\]
if \(\mathcal H\le \tfrac12\) and \(\alpha=\tfrac12+\mathcal H\), then \(u^{\epsilon}\) converges in law to a probabilistically weak rough path solution of
\[
\begin{cases}
\partial_t u=\nu\Delta u-\overline r\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
\nabla\cdot u=0.
\end{cases}
\]
C.
There exists a family of probabilistically weak rough path solutions \((u^{\epsilon},(-C)^{-1}dW^{\mathcal H,\epsilon})\) to the prelimit system such that, as \(\epsilon\to0\), for \(\alpha=1\) and \(\mathcal H>\tfrac12\), the processes \(u^{\epsilon}\) converge in law to a probabilistically weak rough path solution of
\[
\begin{cases}
du=(\nu\Delta u-u\cdot\nabla u)\,dt+d\nabla p-d(-C)^{-1}W^{\mathcal H}\cdot\nabla u,\\
\nabla\cdot u=0.
\end{cases}
\]
D.
There exists a family of probabilistically strong rough path solutions \((u^{\epsilon},(-C)^{-1}dW^{\mathcal H})\) to the prelimit system on any prescribed probability space carrying the given fractional Wiener process such that, as \(\epsilon\to0\), the laws converge as follows: if \(\alpha=1\) and \(\mathcal H>\tfrac12\), then \(u^{\epsilon}\) converges in law to a probabilistically strong rough path solution of
\[
\begin{cases}
du=(\nu\Delta u-u\cdot\nabla u)\,dt+d\nabla p-d(-C)^{-1}W^{\mathcal H}\cdot\nabla u,\\
\nabla\cdot u=0;
\end{cases}
\]
if \(\mathcal H<\tfrac12\) and \(\alpha=\tfrac12+\mathcal H\), then \(u^{\epsilon}\) converges in law to a probabilistically strong rough path solution of
\[
\begin{cases}
\partial_t u=\nu\Delta u-\overline r\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
\nabla\cdot u=0.
\end{cases}
\]
E.
There exists a family of probabilistically weak rough path solutions \((u^{\epsilon},(-C)^{-1}dW^{\mathcal H,\epsilon})\) to the prelimit system such that, as \(\epsilon\to0\), the laws converge as follows: if \(\alpha=1\) and \(\mathcal H>\tfrac12\), then \(u^{\epsilon}\) converges in law to a probabilistically weak rough path solution of
\[
\begin{cases}
du=(\nu\Delta u-u\cdot\nabla u-\overline r\cdot\nabla u)\,dt+d\nabla p-d(-C)^{-1}W^{\mathcal H}\cdot\nabla u,\\
\nabla\cdot u=0;
\end{cases}
\]
if \(\mathcal H<\tfrac12\) and \(\alpha=1\), then \(u^{\epsilon}\) converges in law to a probabilistically weak rough path solution of
\[
\begin{cases}
\partial_t u=\nu\Delta u-\overline r\cdot\nabla u-u\cdot\nabla u+\nabla p,\\
\nabla\cdot u=0.
\end{cases}
\]

dropped the H<1/2, alpha=1/2+H convergence regime

probabilistically weak existence replaced by strong existence on a fixed probability space

absence of Ito-Stokes drift for H>1/2 and correct scaling alpha in the H<1/2 regime


[{see Theorem \ref{theorem l2 del delbar lemma}}]\label{theorem intro l2 del delbar lemma}
    Let $(M,g)$ be a complete K\"ahler manifold such that the self-adjoint Laplacian $\Delta$ has a spectral gap in $L^2\Lambda^{k}_\C$. Given a smooth $L^2$-form $\alpha\in L^2A^k_\C$ which satisfies $\partial\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
    \begin{enumerate}
        \item $\alpha=\partial\delbar\beta$, for $\beta\in L^2 A^{k-2}_\C$;
        \item $\alpha=\partial\gamma$, for $\gamma\in L^2 A^{k-1}_\C$;
        \item $\alpha=\delbar\zeta$, for $\zeta\in L^2 A^{k-1}_\C$;
        \item $\alpha=\partial\eta+\delbar\theta$, for $\eta,\theta\in L^2 A^{k-1}_\C$;
        \item $\alpha=d\lambda$, for $\lambda\in L^2 A^{k-1}_\C$.
    \end{enumerate}

\begin{theorem}[{see Theorem \ref{theorem l2 del delbar lemma}}]\label{theorem intro l2 del delbar lemma}
    Let $(M,g)$ be a complete K\"ahler manifold such that the self-adjoint Laplacian $\Delta$ has a spectral gap in $L^2\Lambda^{k}_\C$. Given a smooth $L^2$-form $\alpha\in L^2A^k_\C$ which satisfies $\del\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
    \begin{enumerate}
        \item $\alpha=\del\delbar\beta$, for $\beta\in L^2 A^{k-2}_\C$;
        \item $\alpha=\del\gamma$, for $\gamma\in L^2 A^{k-1}_\C$;
        \item $\alpha=\delbar\zeta$, for $\zeta\in L^2 A^{k-1}_\C$;
        \item $\alpha=\del\eta+\delbar\theta$, for $\eta,\theta\in L^2 A^{k-1}_\C$;
        \item $\alpha=d\lambda$, for $\lambda\in L^2 A^{k-1}_\C$.
    \end{enumerate}
\end{theorem}

On compact K\"ahler manifolds, the $\partial\delbar$-Lemma \cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\alpha\in A^k_\C$ which is $\partial$- and $\delbar$-closed, \textit{i.e.}, $\partial\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
\begin{enumerate}
 \item $\alpha=\partial\delbar\beta$, for $\beta\in A^{k-2}_\C$;
 \item $\alpha=\partial\gamma$, for $\gamma\in A^{k-1}_\C$;
 \item $\alpha=\delbar\zeta$, for $\zeta\in A^{k-1}_\C$;
 \item $\alpha=\partial\eta+\delbar\theta$, for $\eta,\theta\in A^{k-1}_\C$;
 \item $\alpha=d\lambda$, for $\lambda\in A^{k-1}_\C$,
\end{enumerate}
where the exterior derivative decomposes as $d=\partial+\delbar$.
The validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\partial\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \cite{AT}.

The main purpose of this paper is to generalise the $\partial\delbar$-Lemma to an $L^2$-$\partial\delbar$-Lemma involving square integrable forms on a class of complete K\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\partial^*,\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\partial,\delbar$, and by $\Delta_D:=DD^*+D^*D$ for $D\in\{d,\partial,\delbar\}$ the Hodge, $\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\Lambda^\bullet_\C:=\oplus_{k\in\N}L^2\Lambda^k_\C$, \textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\infty)$. We say that such an operator has a \emph{spectral gap} if its spectrum is contained in $\{0\}\cup[C,+\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed.

On the other hand, if the Hermitian metric $g$ is K\"ahler, \textit{i.e.}, its fundamental $2$-form is closed, then by K\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\Delta_d=2\Delta_\delbar=2\Delta_\partial$. As a consequence, if the metric is both K\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\Delta$.

Besides compact K\"ahler manifolds, where the spectrum is discrete, examples of complete K\"ahler manifolds where $\Delta$ has a spectral gap on the whole algebra of forms $L^2\Lambda^\bullet_\C$ are given by complete K\"ahler $d$-bounded manifolds \cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\omega$ is not only closed but also exact $\omega=d\eta$, and $\eta$ is bounded in the pointwise norm. Explicit examples of complete K\"ahler $d$-bounded manifolds are: complete simply connected K\"ahler manifolds with sectional curvature bounded from above by a negative constant \cite[0.1.B]{G} \cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \cite[0.1.C'']{G} \cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \cite[0.3.A(b)]{G} \cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\C^n$ with the Bergman metric \cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\C^n$ \cite[Theorem 1]{KO}.
We expect this $L^2$-$\partial\delbar$-Lemma to be useful for further studies of the above classes of complete K\"ahler manifolds. For some recent results related to complete K\"ahler $d$-bounded manifolds we refer to \cite{PT1,BDET,H}. The notion of an $L^2$-$\partial\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \cite{HP,BP}.

We use the notation $L^2A^k_\C$ to denote the space of smooth $L^2$-forms $L^2\Lambda^k_\C\cap A^k_\C$. The following theorem is the main result of the paper.

\begin{theorem}\label{theorem l2 del delbar lemma}
    Let $(M,g)$ be a complete K\"ahler manifold such that $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{k}_\C$. Given a smooth $L^2$-form $\alpha\in L^2A^k_\C$ which satisfies $\del\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
    \begin{enumerate}
        \item $\alpha=\del\delbar\beta$, for $\beta\in L^2 A^{k-2}_\C$;
        \item $\alpha=\del\gamma$, for $\gamma\in L^2 A^{k-1}_\C$;
        \item $\alpha=\delbar\zeta$, for $\zeta\in L^2 A^{k-1}_\C$;
        \item $\alpha=\del\eta+\delbar\theta$, for $\eta,\theta\in L^2 A^{k-1}_\C$;
        \item $\alpha=d\lambda$, for $\lambda\in L^2 A^{k-1}_\C$.
    \end{enumerate}
\end{theorem}

\begin{theorem}\label{theorem closed image del delbar}
Let $(M,g)$ be a complete K\"ahler manifold. If $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{p,q}$, then for $b\in\{s,w\}$ it follows that
$\im\del\delbar_b$ and $\im\delbar^*\del^*_b$ are closed in $L^2\Lambda^{p,q}$.
\end{theorem}
\begin{proof}
By Theorem \ref{theorem spectral gap kodaira spencer laplacians}, $\tilde\Delta_{BC,bw}$ has a spectral gap in $L^2\Lambda^{p,q}$. By Lemma \ref{lemma spectral gap equiv}, this is equivalent to the closure of the image of $\tilde\Delta_{BC,bw}$. Therefore we can orthogonally decompose
\begin{align*}
L^2\Lambda^{p,q}&=\ker\tilde\Delta_{BC,bw}\overset{\perp}{\oplus}\im\tilde\Delta_{BC,bw}\\
&\subseteq \ker\tilde\Delta_{BC,bw}\overset{\perp}{\oplus}\left(\im\del\delbar_b + \im \delbar^*\del^*_s+\im \del^*\delbar_s+\im\delbar^*\del_s+\im(\del^*\oplus\delbar^*)_s\right)\\
&\subseteq \ker\tilde\Delta_{BC,bw}\overset{\perp}{\oplus}\im\del\delbar_b \overset{\perp}{\oplus} (\im\del^*_s+\im\delbar^*_s)\subseteq L^2\Lambda^{p,q}.
\end{align*}
The first inclusion follows by definition of $\tilde\Delta_{BC,bw}$, while the second inclusion is justified by \emph{(2)} of Proposition \ref{proposition properties strong weak} and the fact that
\[
\im(\del^*\oplus\delbar^*)_s\subseteq\c{\im(\del^*\oplus\delbar^*)_s}=\c{\im\del^*_s}+\c{\im\delbar^*_s}=\im\del^*_s+\im\delbar^*_s,
\]
where the first equality is equation \eqref{equation kernel image del* e delbar*} and the second is obtained via Lemmas \ref{lemma spectral gap equiv 2} and \ref{lemma im closed} using the spectral gap of $\Delta_{\delbar,sw}=\Delta_{\del,sw}$ in $L^2\Lambda^{p,q}$.
The orthogonality of the above decompositions is easy to check.
The closure of $\im\del\delbar_b$ now follows from the orthogonality of the decomposition of $L^2\Lambda^{p,q}$ in the last line. Using the spectral gap of $\tilde\Delta_{A,sb}$ we similarly find the closure of $\im\delbar^*\del^*_b$ in $L^2\Lambda^{p,q}$.
\end{proof}

\begin{theorem}\label{theorem l2 del delbar lemma}
 Let $(M,g)$ be a complete K\"ahler manifold such that $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{k}_\C$. Given a smooth $L^2$-form $\alpha\in L^2A^k_\C$ which satisfies $\del\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
 \begin{enumerate}
 \item $\alpha=\del\delbar\beta$, for $\beta\in L^2 A^{k-2}_\C$;
 \item $\alpha=\del\gamma$, for $\gamma\in L^2 A^{k-1}_\C$;
 \item $\alpha=\delbar\zeta$, for $\zeta\in L^2 A^{k-1}_\C$;
 \item $\alpha=\del\eta+\delbar\theta$, for $\eta,\theta\in L^2 A^{k-1}_\C$;
 \item $\alpha=d\lambda$, for $\lambda\in L^2 A^{k-1}_\C$.
 \end{enumerate}
\end{theorem}
\begin{proof}
 Since $\Delta_{\delbar,sw}=\Delta_{\del,sw}=\frac12\Delta_{d,sw}$ has a spectral gap in $L^2\Lambda^{k}_\C$, by Theorem \ref{theorem spectral gap kodaira spencer laplacians} also $\tilde\Delta_{BC,sw}$ and $\tilde\Delta_{A,sw}$ have a spectral gap in $L^2\Lambda^{k}_\C$.
 We define the orthogonal projection
 \[
 P_{BC}:L^2\Lambda^{k}_\C\to\ker \tilde\Delta_{BC,sw}
 \]
 and the Green operator
 \begin{align*}
 G_{BC}:&\im \tilde\Delta_{BC,sw}\to \D(\tilde\Delta_{BC,sw})\\
 & \tilde\Delta_{BC,sw}\beta\mapsto\beta.
 \end{align*}
 Being $\im \tilde\Delta_{BC,sw}$ closed, and recalling the orthogonal decomposition
 \[
 L^2\Lambda^{k}_\C=\ker \tilde\Delta_{BC,sw}\oplus\im \tilde\Delta_{BC,sw},
 \]
 we can extend the Green operator on the whole space $L^2\Lambda^{k}_\C$ simply by setting $G_{BC}=0$ on $\ker \tilde\Delta_{BC,sw}$. Given any form $\alpha\in L^2\Lambda^{k}_\C$, it decomposes as
 \[
 \alpha=P_{BC}\alpha+(id-P_{BC})\alpha,
 \]
 where $(id-P_{BC})\alpha=\tilde\Delta_{BC,sw}\beta$. Now $G_{BC}\alpha=\beta$, therefore
 \[
 \alpha=P_{BC}\alpha+\tilde\Delta_{BC,sw}G_{BC}\alpha.
 \]
 If, in particular, $\alpha$ is smooth and so $\alpha\in L^2A^k_\C$, then by elliptic regularity (see, \textit{e.g.}, \cite[Theorem 2.1]{HP}) also $G_{BC}\alpha=\beta\in L^2A^k_\C\cap \D(\tilde\Delta_{BC,sw})$ is smooth. In particular, by \emph{(5)} of Proposition \ref{proposition properties strong weak} we obtain
 \[
 \tilde\Delta_{BC,sw}G_{BC}\alpha=\tilde\Delta_{BC}G_{BC}\alpha.
 \]
 The same holds for analogue operators $P_\delta$ and $G_\delta$ for $\delta\in\{A,\del,\delbar,d\}$.
 Therefore, if $\alpha\in L^2A^k_\C$, for $\delta\in\{A,BC\}$ and $\epsilon\in\{\del,\delbar,d\}$ we deduce decompositions
 \[
 \alpha=P_{\delta}\alpha+\tilde\Delta_{\delta}G_{\delta}\alpha=P_\epsilon\alpha+\Delta_{\epsilon}G_{\epsilon}\alpha,
 \]
 with smooth forms $G_{\delta}\alpha,G_{\epsilon}\alpha\in L^2A^k_\C$. Recall that by Theorem \ref{theorem complete kahler equality harmonic} the kernels of $\Delta_{\delbar,sw}=\Delta_{\del,sw}=\frac12\Delta_{d,sw}$, $\tilde\Delta_{BC,sw}$ and $\tilde\Delta_{A,sw}$ coincide, thus $P_{BC}\alpha=0$ iff $P_\delta=0$ for all $\delta\in\{A,\del,\delbar,d\}$.

Now assume that $\alpha\in L^2A^k_\C$ satisfies $\del\alpha=\delbar\alpha=0$. We know $\ker\del_w\cap\ker\delbar_w\subseteq L^2A^k_\C$ is $L^2$-orthogonal to
 \begin{align*}
 &\im d_s^*, &&\im\del_s^*, & &\im\delbar_s^*, &\im(\del^*\oplus\delbar^*)_s, &&\im\del\delbar_s^*,
 \end{align*}
 as in the proof of Theorem \ref{theorem reduced l2 del delbar lemma}. Thanks to this and to \emph{(5)} of Proposition \ref{proposition properties strong weak}, we obtain
 \begin{align*}
 &\alpha=\tilde\Delta_{A}G_{A}\alpha &&\iff && \alpha=(\del\del^*+\delbar\delbar^*)G_{A}\alpha,\\
 &\alpha=\tilde\Delta_{BC}G_{BC}\alpha &&\iff && \alpha=\del\delbar\delbar^*\del^*G_{BC}\alpha,\\
 &\alpha=\Delta_{\del}G_{\del}\alpha &&\iff && \alpha=\del\del^*G_{\del}\alpha,\\
 &\alpha=\Delta_{\delbar}G_{\delbar}\alpha &&\iff && \alpha=\delbar\delbar^*G_{\delbar}\alpha,\\
 &\alpha=\Delta_{d}G_{d}\alpha &&\iff && \alpha=dd^*G_{d}\alpha.
 \end{align*}
 It is then enough to set $\beta:=\delbar^*\del^*G_{BC}\alpha$, $\gamma:=\del^*G_{\del}\alpha$, $\zeta:=\delbar^*G_{\delbar}\alpha$, $\eta:=\del^*G_{A}\alpha$, $\theta:=\delbar^*G_{A}\alpha$, $\lambda:=d^*G_{d}\alpha$, which lie in $L^2\Lambda^\bullet_\C$ since the image of each Green operator is the domain of the corresponding Laplacian.
 This concludes the proof.
\end{proof}

\begin{remark}\label{remark bounded geometry}
 Suppose that, in addition to the assumptions of Theorem \ref{theorem l2 del delbar lemma}, the manifold $(M,g)$ is of bounded geometry and the elliptic Laplacians $\Delta_{\delbar}=\Delta_{\del}=\frac12\Delta_{d}$, $\tilde\Delta_{BC}$ and $\tilde\Delta_{A}$ are $\cinf$-bounded and uniformly elliptic, with the bundles of differential forms which are of bounded geometry (cf. \cite{Sh}). Examples of such manifolds are given by the universal coverings of K\"ahler hyperbolic manifolds \cite{G}. Then, by \cite[Proposition 1.1]{Sh} the domains of the above operators coincide with the Sobolev spaces
 \[
 \D(\Delta_{\delbar,sw})=\D(\Delta_{\del,sw})=\D(\Delta_{d,sw})=W^2_2(M,\Lambda^\bullet_\C)
 \]
 and
 \[
 \D(\tilde\Delta_{A,sw})=\D(\tilde\Delta_{BC,sw})=W^4_2(M,\Lambda^\bullet_\C),
 \]
 with the notation for Sobolev spaces as in \cite{Sh}. Therefore, since the image of each Green operator is one of the above domains, and thus the image is a Sobolev space, in the statement of Theorem \ref{theorem l2 del delbar lemma} we can choose $\beta\in W^2_2(M,\Lambda^{k-2}_\C)$ and $\gamma,\zeta,\eta,\theta,\lambda\in W^1_2(M,\Lambda^{k-1}_\C)$. In particular, $\del\beta,\delbar\beta\in W^1_2(M,\Lambda^{k-1}_\C)\subseteq L^2\Lambda_\C^{k-1}$.
\end{remark}

The proof of Theorem \ref{theorem intro l2 del delbar lemma} relies primarily on new spectral properties of some self-adjoint operators introduced in \cite{HP} as part of the development of the Aeppli and Bott-Chern versions of the $L^2$-Hodge theory of general Hermitian manifolds and of complete K\"ahler manifolds. More precisely, we prove that if $\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity. Since this constitutes the core of the proof, we now discuss these Aeppli and Bott–Chern Laplacians in more detail.

Given a compact Hermitian manifold, there is a unique choice for the second-order Hodge, $\partial$- and Dolbeault Laplacians $\Delta_D=DD^*+D^*D$, for $D\in\{d,\partial,\delbar\}$, such that the corresponding kernels are isomorphic to de Rham, $\partial$- and Dolbeault cohomology, via classical Hodge theory. In the Aeppli and Bott-Chern cases the situation is slightly different: there are multiple possible choices for the associated Laplacians. We recall the definitions of Aeppli and Bott-Chern cohomology via the related differential complex. We denote the spaces of $(p,q)$-forms by $A^{p,q}$, so that $A^k_\C=\oplus_{p+q=k}A^{p,q}$, and the Hilbert space of $L^2$-$(p,q)$-forms by $L^2\Lambda^{p,q}$. For any choice of integers $(p,q)$ we consider the complex
\[
\dots\longrightarrow A^{p-1,q-2}\oplus A^{p-2,q-1}\overset{\delbar\oplus\partial}{\longrightarrow} A^{p-1,q-1}\overset{\partial\delbar}{\longrightarrow} A^{p,q}\overset{\partial+\delbar}{\longrightarrow} A^{p+1,q}\oplus A^{p,q+1}{\longrightarrow}\dots
\]
where $\delbar\oplus\partial$ operates on $A^{p-1,q-2}\oplus A^{p-2,q-1}$ as $\delbar$ on $A^{p-1,q-2}$ plus $\partial$ on $A^{p-2,q-1}$. The Aeppli and Bott-Chern cohomology spaces are defined as
\begin{align*}
H^{p-1,q-1}_A:=\frac{\ker\partial\delbar\cap A^{p-1,q-1}}{\im\delbar\oplus\partial},&&H^{p,q}_{BC}:=\frac{\ker(\partial+\delbar)\cap A^{p,q}}{\im\partial\delbar}.
\end{align*}
The \lq\lq natural" Aeppli and Bott-Chern Laplacians are then defined as
\begin{align*}
\Delta_A&:=\delbar^*\partial^*\partial\delbar+(\delbar\oplus\partial)(\delbar\oplus\partial)^*=\delbar^*\partial^*\partial\delbar+\partial\partial^*+\delbar\delbar^*,\\
\Delta_{BC}&:=\partial\delbar\delbar^*\partial^*+(\partial+\delbar)^*(\partial+\delbar)=\partial\delbar\delbar^*\partial^*+\partial^*\partial+\delbar^*\delbar.
\end{align*}
The kernels of these operators are isomorphic to the Aeppli and Bott-Chern cohomology spaces, but they are not elliptic \cite[Proposition 2.1]{S}. The first elliptic fourth-order operators whose kernels were shown to be isomorphic to the Aeppli and Bott-Chern cohomologies were defined in \cite{KS} as
\begin{align*}
\tilde\Delta_{A} &:=
\partial\delbar\delbar^*\partial^*+
\delbar^*\partial^*\partial\delbar+
\partial\delbar^*\delbar\partial^*+\delbar\partial^*\partial\delbar^*+
\partial\partial^*+\delbar\delbar^*,\\
\tilde\Delta_{BC} &:=
\partial\delbar\delbar^*\partial^*+
\delbar^*\partial^*\partial\delbar+\partial^*\delbar\delbar^*\partial+\delbar^*\partial\partial^*\delbar
+\partial^*\partial+\delbar^*\delbar.
\end{align*}
We will refer to these operators as the Kodaira-Spencer Laplacians. Another pair of elliptic fourth-order operators whose kernels are isomorphic to the Aeppli and Bott-Chern cohomologies were defined in \cite{V} as
\begin{align*}
\square_{A}&:=\delbar^*\partial^*\partial\delbar+((\delbar\oplus\partial)(\delbar\oplus\partial)^*)^2=\delbar^*\partial^*\partial\delbar+(\partial\partial^*+\delbar\delbar^*)^2,\\
\square_{BC}&:=\partial\delbar\delbar^*\partial^*+((\partial+\delbar)^*(\partial+\delbar))^2=\partial\delbar\delbar^*\partial^*+(\partial^*\partial+\delbar^*\delbar)^2.
\end{align*}
We will refer to these operators as the Varouchas Laplacians.

In \cite[Corollary 8.12]{HP} it was proven that on a complete K\"ahler manifold, if $\Delta$ has a spectral gap in $L^2\Lambda^\bullet_\C$, then there are self-adjoint extensions of $\Delta_A$ and $\Delta_{BC}$ having a spectral gap. Being $\Delta_A$ and $\Delta_{BC}$ non-elliptic, this was not sufficient to prove Theorem \ref{theorem intro l2 del delbar lemma}. In fact, to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians, which is shown in Theorem \ref{theorem spectral gap kodaira spencer laplacians}. We also prove a spectral gap of the Varouchas Laplacians in Theorem \ref{theorem spectral gap varouchas laplacians}. We remark that the spectral gap assumption of $\Delta$ is essential for a statement which directly generalises the classical $\partial\delbar$-Lemma such as Theorem \ref{theorem intro l2 del delbar lemma} does. Indeed, in the absence of a spectral gap, the images of the closed extensions of $d$, $\partial$, $\delbar$, and $\partial\delbar$ fail to be closed. For a weaker statement on complete K\"ahler manifolds without assuming a spectral gap, see Theorem \ref{theorem reduced l2 del delbar lemma} (cf. \cite[Corollary 8.6]{HP}).

The paper is organised as follows. In Section \ref{section preliminaries}, we present preliminaries on Hilbert complexes, self-adjoint operators, spectral gaps, minimal and maximal closed extensions of differential operators, differential complexes on complex manifolds, self-adjoint extensions of second and fourth-order complex Laplacians, $L^2$ Hodge theory on complete K\"ahler manifolds. In Section \ref{section spectral gap}, we prove the spectral gaps of the elliptic Aeppli and Bott-Chern Laplacians. In Theorem \ref{theorem closed image del delbar} we also present a new proof of the closure of the image of the closed extensions of the operator $\partial\delbar$, originally proved in \cite[Theorem 8.10]{HP}. Finally, in section \ref{section l2 del delbar lemma}, we prove our main result Theorem \ref{theorem intro l2 del delbar lemma}. In Remark \ref{remark bounded geometry} we also point out an improvement of Theorem \ref{theorem intro l2 del delbar lemma} when the manifold is of bounded geometry.

The post-theorem introduction says that “The proof of Theorem \ref{theorem intro l2 del delbar lemma} relies primarily on new spectral properties of some self-adjoint operators introduced in \cite{HP}.” More precisely, they “prove that if $\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity,” and this is described as “the core of the proof.”

Because the “natural” Aeppli and Bott-Chern Laplacians $\Delta_A,\Delta_{BC}$ have kernels matching Aeppli/Bott--Chern cohomology but “are not elliptic,” the text explains that the spectral gap for self-adjoint extensions of $\Delta_A,\Delta_{BC}$ from \cite[Corollary 8.12]{HP} “was not sufficient to prove Theorem \ref{theorem intro l2 del delbar lemma}.” Instead, “to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians,” establishing it in “Theorem \ref{theorem spectral gap kodaira spencer laplacians},” and they “also prove a spectral gap of the Varouchas Laplacians in Theorem \ref{theorem spectral gap varouchas laplacians}.”

Finally, the introduction remarks that the “spectral gap assumption of $\Delta$ is essential” for a direct generalisation like Theorem \ref{theorem intro l2 del delbar lemma}, since “in the absence of a spectral gap, the images of the closed extensions of $d$, $\partial$, $\delbar$, and $\partial\delbar$ fail to be closed,” and it points to a weaker variant without a spectral gap (Theorem \ref{theorem reduced l2 del delbar lemma}).

The post-theorem introduction says that “The proof of the main theorem relies primarily on new spectral properties of some self-adjoint operators introduced in \cite{HP}.” More precisely, they “prove that if $\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity,” and this is described as “the core of the proof.”

Because the “natural” Aeppli and Bott-Chern Laplacians $\Delta_A,\Delta_{BC}$ have kernels matching Aeppli/Bott--Chern cohomology but “are not elliptic,” the text explains that the spectral gap for self-adjoint extensions of $\Delta_A,\Delta_{BC}$ from \cite[Corollary 8.12]{HP} “was not sufficient to prove the main theorem.” Instead, “to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians,” establishing it in the following theorem.

\begin{theorem}\label{theorem spectral gap kodaira spencer laplacians}
Let $(M,g)$ be a complete K\"ahler manifold. If $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{p,q}$, then for $b\in\{s,w\}$ the operators $\tilde\Delta_{A,sb}$ and $\tilde\Delta_{BC,bw}$ have a spectral gap in $L^2\Lambda^{p,q}$.
\end{theorem}

They “also prove a spectral gap of the Varouchas Laplacians” in the following theorem.

\begin{theorem}\label{theorem spectral gap varouchas laplacians}
    Let $(M,g)$ be a complete K\"ahler manifold.
    If $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{p,q}$, $L^2\Lambda^{p-1,q}$ and $L^2\Lambda^{p,q-1}$, then $\square_{A,sw}$ has a spectral gap in $L^2\Lambda^{p,q}$. If $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{p,q}$, $L^2\Lambda^{p+1,q}$ and $L^2\Lambda^{p,q+1}$, then $\square_{BC,sw}$ has a spectral gap in $L^2\Lambda^{p,q}$.
\end{theorem}

Finally, the introduction remarks that the “spectral gap assumption of $\Delta$ is essential” for a direct generalisation like the main theorem, since “in the absence of a spectral gap, the images of the closed extensions of $d$, $\partial$, $\delbar$, and $\partial\delbar$ fail to be closed,” and it points to a weaker variant without a spectral gap given by the following theorem.

\begin{theorem}\label{theorem reduced l2 del delbar lemma}
Given a complete K\"ahler manifold $(M,g)$, in both the spaces $L^2\Lambda^k_\C$ and $L^2\Lambda^{p,q}$ the following equalities hold true
\begin{align*}
\c{\im \del\delbar_s}
&=\c{\im \del_s}\cap\ker\delbar_w=\c{\im \delbar_s}\cap\ker\del_w\\
&=(\c{\im\del_s}+\c{\im\delbar_s})\cap\ker\del_w\cap\ker\delbar_w\\
&=\c{\im d_s}\cap\ker\del_w\cap\ker\delbar_w.
\end{align*}
\end{theorem}

\begin{theorem}\label{theorem l2 del delbar lemma}
    Let $(M,g)$ be a complete K\"ahler manifold such that $\Delta_{\delbar,sw}$ has a spectral gap in $L^2\Lambda^{k}_\C$. Given a smooth $L^2$-form $\alpha\in L^2A^k_\C$ which satisfies $\del\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
    \begin{enumerate}
        \item $\alpha=\del\delbar\beta$, for $\beta\in L^2 A^{k-2}_\C$;
        \item $\alpha=\del\gamma$, for $\gamma\in L^2 A^{k-1}_\C$;
        \item $\alpha=\delbar\zeta$, for $\zeta\in L^2 A^{k-1}_\C$;
        \item $\alpha=\del\eta+\delbar\theta$, for $\eta,\theta\in L^2 A^{k-1}_\C$;
        \item $\alpha=d\lambda$, for $\lambda\in L^2 A^{k-1}_\C$.
    \end{enumerate}
\end{theorem}\label{theorem intro l2 del delbar lemma}
    Let $(M,g)$ be a complete K\"ahler manifold such that the self-adjoint Laplacian $\Delta$ has a spectral gap in $L^2\Lambda^{k}_\C$. Given a smooth $L^2$-form $\alpha\in L^2A^k_\C$ which satisfies $\partial\alpha=\delbar\alpha=0$, then the following conditions are equivalent:
    \begin{enumerate}
        \item $\alpha=\partial\delbar\beta$, for $\beta\in L^2 A^{k-2}_\C$;
        \item $\alpha=\partial\gamma$, for $\gamma\in L^2 A^{k-1}_\C$;
        \item $\alpha=\delbar\zeta$, for $\zeta\in L^2 A^{k-1}_\C$;
        \item $\alpha=\partial\eta+\delbar\theta$, for $\eta,\theta\in L^2 A^{k-1}_\C$;
        \item $\alpha=d\lambda$, for $\lambda\in L^2 A^{k-1}_\C$.
    \end{enumerate},

Question: Let \((M,g)\) be a complete K\"ahler manifold such that the self-adjoint \(\bar\partial\)-Laplacian \(\Delta_{\bar\partial,sw}\) on \(L^2\) complex-valued \(k\)-forms has a spectral gap, i.e. its spectrum is contained in \(\{0\}\cup [C,\infty)\) for some \(C>0\). Let \(\alpha\in L^2A^k_{\mathbb C}\) be a smooth \(L^2\) \(k\)-form with \(\partial\alpha=\bar\partial\alpha=0\), where \(L^2A^j_{\mathbb C}\) denotes the smooth complex-valued \(j\)-forms of finite \(L^2\)-norm and \(d=\partial+\bar\partial\). Which statement is equivalent to both \(\alpha=\partial\bar\partial\beta\) for some \(\beta\in L^2A^{k-2}_{\mathbb C}\) and \(\alpha=\partial\gamma\) for some \(\gamma\in L^2A^{k-1}_{\mathbb C}\)?

\(\alpha=d\lambda\) for some \(\lambda\in L^2A^{k-1}_{\mathbb C}\).

Choices:
B. \(\alpha=d\lambda\) for some \(\lambda\in L^2A^{k-1}_{\mathbb C}\), and this remains equivalent under the weaker assumption that \(\Delta_{\bar\partial,sw}\) merely has closed range on \(L^2\Lambda^{k}_{\mathbb C}\) away from \(\ker\Delta_{\bar\partial,sw}\), without requiring a spectral gap.
C. \(\alpha=\partial\eta+\bar\partial\theta\) for some \(\eta,\theta\in L^2A^{k-1}_{\mathbb C}\).
D. \(\alpha=d\lambda\) for some \(\lambda\in L^2A^{k-1}_{\mathbb C}\), provided \(\Delta_{\bar\partial,sw}\) has a spectral gap on each bidegree summand \(L^2\Lambda^{p,q}_{\mathbb C}\) with \(p+q=k\), as well as on all adjacent summands \(L^2\Lambda^{p-1,q}_{\mathbb C}\) and \(L^2\Lambda^{p,q-1}_{\mathbb C}\).
E. \(\alpha=\partial\bar\partial\beta\) for some \(\beta\in L^2A^{k-2}_{\mathbb C}\), and in fact one may choose \(\beta\in L^2A^{k-2}_{\mathbb C}\) so that \(\partial\beta,\bar\partial\beta\in L^2A^{k-1}_{\mathbb C}\) without any additional geometric assumptions.