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As a physics major student, I'm not familiar what regularity means in the scope of PDE. Based on my understanding, regularity of a real-valued function expresses how smooth the function is. But I have some questions about the details. From one paper (in appendix of this papper), I read the following content:

Suppose that one is given a linear equation like $Lf=g$: knowing the regularity of $g$, what can be said of the regularity of $f$ ? Of course the answer depends of the type of the equation: if it is elliptic, then typically second derivatives of $f$ have the same regularity as $g$, etc. In the case then $f$ is a priori $\gamma$ degrees less smooth than g, one says that the equation loses $\gamma$ derivatives.

I don't understand why $f$ would be less smooth than $g$? For example: $$ f(x)= \begin{cases} x^2, \quad x \geq 0\\ 0,\,\,\, \quad x <0 \end{cases}, \qquad g(x) = {\partial^2 \over \partial x^2}f= \begin{cases} 2, \quad x \geq 0\\ 0, \quad x <0 \end{cases} $$

Shouldn't $f$ be more smooth than $g$? Do I have some misunderstanding?

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I guess that your first sentence is the important point. This is somewhat technical and the paper that you refer to is not really the best source. Since, again, this is technical I prefer to refer to a wonderful source, such as the introduction of Kohn's paper in Ann. of Math. 162 (2005), 943-986.

I also would like to recall Lewy's dramatic example showing there exists a linear partial differential equation (with $L$ as yours linear) with no solutions for some $C^\infty$ right-hand side! See wikipedia (or better say Pazy's book).

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    Yes your right, my source is just a brief introduction about regularity written for physicists. But since I'm not a mathematics major, Kohn's paper seem too difficult for me to understand. Is there simpler introduction or simple example of this topic? For example, if $L=\partial_{xx}+\partial_{yy}$, what would the relation of regularity of $f$ and $g$? – 2017-01-13
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    I think that I understood that (I looked at your link), that's why I referred explicitly to the introduction of his paper. :) You can then follow the references there. Basically you really want too much, and the truth is that "loss of derivatives" sometimes doesn't mean simply "less derivatives" but some specific bounds or lack of solutions of certain kind (in which case it is simply impossible to take derivatives). – 2017-01-13
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    I partly understand. The term "loss of derivatives" seems far more complex than I previously thought. Thanks :) – 2017-01-13