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


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