Averages of diagonal Elliott-Halberstam problem twisted by Möbius function with Sobolev and Hölder-Zygmund weights

Recalling that the so-called Elliott-Halberstam conjecture twisted by the M\"obius function $\mu(n)$ claims that $$\sum_{q\leq N^{\theta}}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{{\scriptstyle n\equiv a\,\mod\,q}}{n\leq y}}\Lambda(n)\mu\left(N-n\right)-\frac{1}{\varphi\left(q\right)}\sum_{n\leq y}\Lambda(n)\mu\left(N-n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}}$$ for every $A>0$, where $0<\theta<1$ is fixed, and also recalling that the validity of this conjecture, in combination with the validity of the classical Elliott-Halberstam for suitable $\theta$, proves the binary Goldbach conjecture, in this paper we study weighted average variants of this problem. We will show that, under Generalized Riemann Hypothesis, a weak version of the Gonek-Hejhal conjecture and working with weights belonging to the Sobolev space $W^{2,1}$ or in the H\"older-Zygmund spaces $\mathcal{C}^{\delta}$ for suitable range of $\delta$, the bound of the average is consistent with the bound of the ``diagonal versions'' of this conjecture (that is, taking $y=N$ and taking $n\equiv N\mod q)$. In particular, in the case of weights in Sobolev space, the consistent upper bound holds for the whole $0<\theta<1$ and, in the case of weights in the H\"older-Zygmund class $\mathcal{C}^{\delta}$, for $\theta$ that depends on the choice of $\delta$ but still not below the $1/2-2\varepsilon$ threshold.