Dimension of the graph of a function $\varphi : \mathbb R^2 \to \mathbb R$

Question

Let $\varphi : \mathbb{R}^2 \to \mathbb{R}$ be a continuous function, and let $G(\varphi)$ be the graph of $\varphi$. Denote $R:=\{(x,0) \in \mathbb{R}^2 | x \in \mathbb{R}\}$ as the real line in $\mathbb{R}^2$. If the restriction of $G(\varphi)$ to $R$ has Hausdorff dimension larger than 1, i.e., $\dim_{\mathbb{H}} G(\varphi|_R) > 1$, then can we deduce that $G(\varphi)$ has Hausdorff dimension larger than 2, i.e., $\dim_{\mathbb{H}} G(\varphi) > 2$? If not, can we deduce that the box dimension of $G(\varphi)$ is larger than 2, i.e., $\dim_{\mathbb{B}} G(\varphi) > 2$?

Answer 1

I don't believe it necessary that the dimension of the graph of $\varphi$ be larger than 2. An example is provided by examining Poisson's integral formula for the upper half plane:

$$ u(x,y) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y}{(x-\chi)^2+y^2}f(\chi)d\chi. $$

This function is known to be differentiable off the $x$-axis, continuous on the closed upper half-plane, and satisfy $u(x,0)=f(x)$. It could be extended to be continuous on the whole plane via reflection.

The forumula holds as long as $f$ is continuous and $$ \int_{-\infty}^{\infty} \frac{|f(x)|}{1+x^2} \, dx $$ converges. In particular, you may take $f$ to be a continuous, bounded function whose graph is known to have Hausdorff dimension larger than 1.

Answer 2

This answer is not complete (I am not sure that the function below is smooth).

It seems $\dim_{\mathbb{H}} G(\varphi)$ may be exactly $2$.

Let $f:\mathbb{R}\to\mathbb{R}$ be any continuous function, and let $\phi:\mathbb{R}\to\mathbb{R}$ be a bump function supported in $[-1,1]$. Then the graph $\{(x,y,z)\in\mathbb{R}^3;z=\varphi(x,y)\}$ of the function $\varphi:\mathbb{R}^2\to\mathbb{R}$; $\varphi(x,y)=\varphi\left(\frac{x}{y}\right)$ likely has Hausdorff dimension $\leq2$. To show this it would be enough if the restriction of $\varphi$ to $\{(x,y)\in\mathbb{R}^2;y\neq0\}$ was smooth (or at least locally Lipschitz), as the graph of a locally Lipschitz function $\mathbb{R}^2\to\mathbb{R}$ has Hausdorff dimension $2$.

The same argument should work with box-counting dimension if you consider bounded subsets of the graph (I have only seen the definition of box-counting dimensions for bounded sets).