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\\mathbf{R}^d\n  \\label{main eqn} & u(0,x)=0, \\end{align} where $d \\in \\mathbb{N}$, $p \\in (1,\\infty]$, $\\gamma,m \\in (0,\\infty)$, $\\Lambda_{\\gamma+m}$ is the Lipschitz space on $\\mathbf{R}^d$ whose order is $\\gamma+m$, $f \\in L_p\\left((0,T) , \\Lambda_{\\gamma} \\right)$, and $\\psi(t,i\\nabla)$ is a time measurable pseudo-differential 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