Mixed Lebesgue norm - Knowlpedia

For a jointly measurable $u(t,x)$ on a product of sigma-finite measure spaces, the mixed Lebesgue norm $L^q_tL^p_x$ is

\|u\|_{L^q_tL^p_x} =\left(\int \|u(t,\cdot)\|_{L^p_x}^{q}\,dt\right)^{1/q}, \qquad 1\le p,q<\infty.

Replace an integral norm by the corresponding essential supremum when its exponent is infinite. For vector fields use the Euclidean magnitude inside the spatial norm.

Order and examples

The order matters when $p\ne q$ . If $p=q<\infty$ , Tonelli's theorem identifies the mixed norm with the product-space $L^p$ norm. For a separated function $u(t,x)=a(t)b(x)$ , it equals $\|a\|_q\|b\|_p$ . In particular, $L^\infty_tL^2_x$ gives a bound on spatial energy for almost every time, not automatically a chosen representative at every time.

Banach-valued notation

The notation $L^q(I;L^p(X))$ also commonly denotes a Bochner space. For $p=\infty$ , strong measurability of the map into $L^\infty(X)$ is an additional issue; joint scalar measurability alone does not always give that property.