# Generalization of Cauchy-Schwarz/Hölder inequality

For functions $$u,v \in L^2(\Omega)$$, with $$\Omega \subset \mathbb{R}^n$$, the Cauchy-Schwarz inequality as a special case of the Hölder inequality $$(p=q=2)$$ states that

(1) $$\Vert uv\Vert_{L^1} = \int_\Omega \vert uv \vert \le \sqrt{\int_\Omega \vert u\vert^2}\sqrt{\int_\Omega \vert v\vert^2}= \Vert u \Vert_{L^2} \Vert v\Vert_{L^2}$$.

I have come across inequalities of the following form using references to C.S. stating that for $$u,v\in H^1(\Omega)$$

(2) $$\int_\Omega \vert \nabla u \cdot \nabla v \vert \le \sqrt{\int_\Omega \vert \nabla u\vert^2}\sqrt{\int_\Omega \vert \nabla v \vert^2}= \Vert \nabla u \Vert_{L^2} \Vert \nabla v\Vert_{L^2}$$.

Is this formula correct? Or is there a constant missing on the right side:

The scalar product on the left side of (2) can be written as a sum $$\sum_{i=1}^n \partial_iu \partial_iv$$ and the triangle inequality can be applied. Then, (1) can be used on each subterm of the form $$\int_\Omega \vert \partial_i u \partial_i v \vert$$, leading to an inequality of the form of (2), but with an additional constant c=n on the right side.

The formula is correct; one applies first the Cauchy-Schwarz inequality in $$\mathbb{R}^n$$ to write $$|\nabla u\cdot \nabla v|\leq |\nabla u||\nabla v|,$$ and then we apply the integral version of C-S to the right hand side.