1. As mentioned in the comments of [Itô's Formula for functions that are C2
 almost everywhere][1]

in ["Ito's Formula for Non-Smooth Functions"][2], they prove a version of Itô formula when $D_{x_{i}}D_{x_{j}}F\in L^{1}$, where the derivatives are understood in the sense of distributions.

 2. In the post https://math.stackexchange.com/questions/2521208/other-versions-of-a-weak-ito-formula, they also give a proof in the case of Sobolev function $F\in W_{\text{loc}}^{2,p}(\mathbb{R}^{n})$.


3. In the post [Generalized Ito's formula][3], they further give the following two references:

 -  Krylov's "Controlled Diffusion Processes" Ch 2 Section 10.
 - [H. F\"ollmer and Ph. Protter, On Itô's formula for multidimensional Brownian motion. Probab. Theory Relat. Fields 116, No.1, 1-20 (2000)][4]. *Consider a d-dimensional Brownian motion X = (X1,...,Xd ) and a function F which belongs **locally to the Sobolev space $W^{1,2}$**. We prove an extension of Ito’s formula ˆ where the usual second order terms are replaced by the quadratic covariations $[f_k (X), X_k ]$ involving the weak first partial derivatives $f_k$ of F*. 


  [1]: https://mathoverflow.net/questions/341453/itos-formula-for-functions-that-are-c2-almost-everywhere?_gl=1*6duvx5*_ga*NjI1Mzg0MzM3LjE3MjQ0NzQ0Mjg.*_ga_S812YQPLT2*MTcyNTE2MzA4OS4zMi4xLjE3MjUxNjQ2NTAuMC4wLjA.
  [2]: https://ems.press/content/serial-article-files/40449
  [3]: https://mathoverflow.net/questions/76609/generalized-itos-formula?_gl=1*o0w5e3*_ga*NjI1Mzg0MzM3LjE3MjQ0NzQ0Mjg.*_ga_S812YQPLT2*MTcyNTE2MzA4OS4zMi4xLjE3MjUxNjQ5MDAuMC4wLjA.
  [4]: https://link.springer.com/article/10.1007/PL00008719