It depends on the geometry of the domain. Consider e.g. the function $f(x,\,y) = xy$ on $\{|xy| < 1\}$, which is bounded with constant Hessian on this domain but has linearly growing gradient. Or take $f$ defined on a union of disjoint intervals in $\mathbb{R}$ of length $2^{-k}$ to be linear with slope $2^k$ on each and vanishing at the midpoints.

The result will hold locally by mollification and interpolation, and globally on domains where interpolation estimates between $C^0$ and $C^2$ hold, e.g. all of $\mathbb{R}^n$ (as Iosif shows) or bounded $C^2$ domains.

It depends on the geometry of the domain. Consider e.g. the function $f(x,\,y) = xy$ on $\{|xy| < 1\}$, which is bounded with constant Hessian on this domain but has linearly growing gradient. Or take $f$ defined on a union of disjoint intervals in $\mathbb{R}$ of length $2^{-k}$ to be linear with slope $2^k$ on each and vanishing at the midpoints.

The result will hold locally by mollification and interpolation, and globally on domains where interpolation estimates between $C^0$ and $C^2$ hold, e.g. convex domains (like $\mathbb{R}^n$, as Iosif shows) or bounded $C^2$ domains.

It depends on the geometry of the domain. Consider e.g. the function $f(x,\,y) = xy$ on $\{|xy| < 1\}$, which is bounded with constant Hessian on this domain but has linearly growing gradient. Or take $f$ defined on a union of disjoint intervals in $\mathbb{R}$ of length $2^{-k}$ to be linear with slope $2^k$ on each and vanishing at the midpoints.

The result will hold locally by mollification and interpolation, and globally on domains where interpolation estimates between $C^0$ and $C^2$ hold, e.g. bounded $C^2$ domains.

It depends on the geometry of the domain. Consider e.g. the function $f(x,\,y) = xy$ on $\{|xy| < 1\}$, which is bounded with constant Hessian on this domain but has linearly growing gradient. Or take $f$ defined on a union of disjoint intervals in $\mathbb{R}$ of length $2^{-k}$ to be linear with slope $2^k$ on each and vanishing at the midpoints.

The result will hold locally by mollification and interpolation, and globally on domains where interpolation estimates between $C^0$ and $C^2$ hold, e.g. all of $\mathbb{R}^n$ (as Iosif shows) or bounded $C^2$ domains.