High order BV function has continuous representative

As known, bounded variation (BV) functions are not continuous, but evidently higher order BV functions on corresponding dimension have continuous representative. As a consequence, {W^{n,1}({\mathbb R}^n) \hookrightarrow C^0({\mathbb R}^n)}.

Definition A function {f \in BV_n({\mathbb R}^n)}, if {f \in W^{n-1,1}({\mathbb R}^n)}, and the nth order distributional derivative {D^n f} is a finite Radon measure.

Theorem If {f \in BV_n({\mathbb R}^n)}, then {f} has a continuous representative.

Proof: Note that {C_c^\infty({\mathbb R}^n)} is dense in {BV_n({\mathbb R}^n)}, i.e. for any {f \in BV_n({\mathbb R}^n)}, there exists a sequence {f_k \in C_c^\infty({\mathbb R}^n)} such that {\lim_{k \rightarrow \infty} f_k = f} in {W^{n-1,1}}, and {\lim_{k\rightarrow \infty} \| \nabla^n f_k\|_{L^1({\mathbb R}^n)} = \|D^n f \| ({\mathbb R}^n)}. (See for example, book by Evans and Gariepy). So it suffice to show for all {f \in C_c^\infty({\mathbb R}^n)},

\displaystyle \|f\|_{L^\infty({\mathbb R}^n)} \le \| \nabla^n f\|_{L^1({\mathbb R}^n)}

Indeed,

\displaystyle f(x_1, \cdots , x_n) = \int_{-\infty}^{x_1} \partial_1 f(s_1,x_2, \cdots , x_n)\,ds_1 = \int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_n} \partial_1 \cdots \partial_n f

This gives us the desire inequality. Then by density result we know that {BV_n({\mathbb R}^n)} admits continuous representative for every element. \Box

Corollary {W^{n,1}({\mathbb R}^n) \hookrightarrow C^0({\mathbb R}^n)}.

Remark If the dimension is greater than the order of BV function, then the above theorem will fail. For example, let {f(x) = |x|^{-\frac{1}{2}}} be defined in a neighborhood {\Omega} around 0, with smooth boundary in {{\mathbb R}^3}. {|\nabla f| \sim |x|^{-\frac{3}{2}}} and {|\nabla^2 f| \sim |x|^{-\frac{5}{2}}}. So {f \in W^{2,1}(\Omega)}. Then we can extend {f} on {{\mathbb R}^3}, since {\partial \Omega} is smooth (See for example, PDE book by Evans). But then {f \rightarrow \infty} as {|x| \rightarrow 0}, which doesn’t admit a continuous representative.