## Category Archives: Schrodinger Eqn

### Mixed norm and basic property

In the analysis of evolution equation, we will constantly treat this kind of function $f(x,t)$ where $x\in \mathbb{R}^d$, $t\in \mathbb{R}$. Considering the difference of time and space variable, we will introduce this kind of norm

$\displaystyle ||f||_{L^q_t L^r_x}=\left(\int_\mathbb{R}\left(\int_{\mathbb{R}^d}|f|^rdx\right)^{\frac{q}{r}}dt\right)^{\frac 1q}$  for $q,r\geq 1$

This kind of norm also has similar properties of $L^p$ norm.

$\mathbf{Proposition 1:}$ $\displaystyle \iint |fg|dxdt \leq ||f||_{L^q_t L^r_x}||g||_{L^{q'}_t L^{r'}_x}$ here $q$ and $q'$ are conjugate  and so are $r,r'$.

$\mathbf{Proof:}$ $\displaystyle \iint |fg|dxdt =\int\left(\int |fg|dx\right)dt\leq \int\left(||f||_{L^r_x}||g||_{L^{r'}_x}\right)dt\leq||f||_{L^q_t L^r_x}||g||_{L^{q'}_t L^{r'}_x}$

$\mathbf{Corollary 1:}$ $\displaystyle ||f||_{L^q_t L^r_x}\leq ||f||_{L^{q_1}_t L^{r_1}_x}||f||_{L^{q_2}_t L^{r_2}_x}$, where $\displaystyle \frac 1q=\frac {1}{q_1}+\frac{1}{q_2}$ and $\displaystyle \frac 1r=\frac{1}{r_1}+\frac{1}{r_2}$. All the powers are bigger than or equal 1.

$\mathbf{Proposition 2:}$ $\displaystyle ||f||_{L^q_t L^r_x}=\sup_{||g||_{L^{q'}_t L^{r'}_x}}\iint fgdxdt$

$\mathbf{Proof:}$ From the proposition 1, LHS$\leq ||f||_{L^q_t L^r_x}$.

Define $\displaystyle g=sgn(f)|f|^{r-1}\left(\int |f|^r\right)^{\frac{q}{r}-1}||f||^{\frac{1}{q}-1}_{L^q_t L^r_x}$ when $||f||_{L^q_t L^r_x}\neq 0$, otherwise the proposition holds trivially.
Then $||g||_{L^{q'}_t L^{r'}_x}=1$ and $\displaystyle \iint fgdxdt=||f||_{L^q_t L^r_x}$.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$