Weak type operator Lipschitz and commutator estimates for commuting tuples

Let $f:{\mathbb{R}}^d\to \mathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if $\{A_k{\}}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]\in L_1(H),$ then $$\mathrm{\parallel }[f(A_1,\cdots ,A_d),B]{\mathrm{\parallel }}_{1,\mathrm{\infty }}\le c(d)\mathrm{\parallel }\mathrm{\nabla }(f){\mathrm{\parallel }}_{\mathrm{\infty }}\underset{1\le k\le d}{\max }\mathrm{\parallel }[A_k,B]{\mathrm{\parallel }}_1,$$ where $c(d)$ is a constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{1,\mathrm{\infty }}$ denotes the weak $L_1$-norm. If $\{X_k{\}}_{k=1}^d$ (respectively, $\{Y_k{\}}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_k\in L_1(H),$ then $$\mathrm{\parallel }f(X_1,\cdots ,X_d)-f(Y_1,\cdots ,Y_d){\mathrm{\parallel }}_{1,\mathrm{\infty }}\le c(d)\mathrm{\parallel }\mathrm{\nabla }(f){\mathrm{\parallel }}_{\mathrm{\infty }}\underset{1\le k\le d}{\max }\mathrm{\parallel }{X}_k-Y_k{\mathrm{\parallel }}_1\mathrm{.}$$