Integral Inequality for Functions on a Cube

Abstract

Let Q _d be an n-dimensional cube with edge length d and with sides parallel to coordinate axes. Let p≥1 and k, l be integers, 0≤k≤l. We denote a function in $W^{l}_{p}(Q_{d})$, p≥1, by u.

The inequality

$$\|u\|_{L_q(Q_d)} \le C \sum^l_{j=k+1}d^{j-1}\| \nabla_j u\|_{L_p(Q_d)}$$

(14.0.1)

with q in the same interval as in the Sobolev embedding theorem often turns out to be useful. This inequality occurs repeatedly in the following chapters. Obviously, (14.0.1) is not valid for all $u \in W^{l}_{p}(Q_{d})$, but it holds provided u is subject to additional requirements.