Mixed Lebesgue norm
= Mixed Lebesgue norm
{title2=$\|f\|_{L_x^pL_v^q}$}
A <mixed Lebesgue norm> first takes the <Lp norm> in one variable and then the norm in another. For finite $p,q$, $\|f\|_{L_x^pL_v^q}=(\int(\int|f(x,v)|^q\,dv)^{p/q}dx)^{1/p}$; an infinite exponent replaces that integral norm by an <essential supremum>. The order matters: generally $L_x^pL_v^q$ and $L_v^qL_x^p$ are different spaces. This distinction is crucial in <free transport dispersion>.