In the analysis of evolution equation, we will constantly treat this kind of function where , . Considering the difference of time and space variable, we will introduce this kind of norm

for

This kind of norm also has similar properties of norm.

here and are conjugate and so are .

, where and . All the powers are bigger than or equal 1.

From the proposition 1, LHS.

Define when , otherwise the proposition holds trivially.

Then and .