What is the meaning of $l_p$ norm in this model for sparse channel estimation? The signal $y(n)$ can be modeled in the time-domain as

$$y(n) = \sum_{i=0}^{L-1}h(i) x(n-i) + v(t) \tag{1}$$

where $\mathbf{h} = [h_1,h_2,\ldots,h_L]^T$ is the coefficient of length $L-1$, and $v(t)$ is the additive White Gaussian noise with variance $\sigma^2_v$.  

In the Eq(20) found in the paper, http://www.eurasip.org/Proceedings/Eusipco/Eusipco2008/papers/1569101936.pdf

the eq has the term ${||h||}^p_p$ and the Authors mention that here $0&lt;p \le 1$.

Consider an array $h = [1,0.2,0,0.5]$. What is meant by ${||h||}^p_p$ Is it the difference between the elements of $h$? What would be the meaning when p =0, p=1, and p=2?

The Authors in Eq(24)present the derivative of the cost function computed in Eq(20). The derivative is taken with respect to $h$ parameter. In that, they use the term $p\lambda \tilde{h}$ where $\tilde{h}$ is the sparse coefficient. I think here $p$ may not be zero.
The symbol ${||h||}_p$ refers to the p-norm of the vector. It is calculated as

$${||h||}_p=\sqrt[p]{\left|h_1\right|^p+\left|h_2\right|^p+...+\left|h_n\right|^p}$$

When they write ${||h||}^p_p$, it is just that multiplied by itself $p$ times, so that the root disappears. Mathematically:

$${||h||}^p_p=\sum\limits_{i=1}^n \left|h_i\right|^p$$