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Let $u=u(t,{\bf x},{\bf p})$ satisfy the transport equation $\frac {\partialu}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial u}{\partial{\bf x}}=f$, where$f=f(t,\bf x,\bf p)$ belongs to $ L^{p}((0,T)\times {\bf R}^{3}\times {\bfR}^{3})$ for $1<p<\infty$ and $\frac {\partial}{\partial t}+\frac {{\bfp}}{p_0}\frac{\partial}{\partial{\bf x}}$ is the relativistic-free transportoperator. We show the regularity of $\int_{{\bf R}^{3}}u(t, {\bf x}, {\bfp})d{\bf p}$ using the same method as given by Golse, Lions, Perthame andSentis. This average regularity is considered in terms of fractional Sobolevspaces and it is very useful for the study of the existence of the solution tothe Cauchy problem on the relativistic Boltzmann equation.

Average Regularity of the Solution to an Equation with the Relativistic-free Transport Operator