(1) $ \ Bex \ Sen {d ^ K f }_{ \ dot B ^ s _ {P, Q }}\ Sim \ Sen {f }_{\ dot B ^ {S + k }_{ P, Q }}. \ EEx $
(2) $ \ beex \ Bea & \ quad S> 0, \ Q \ In [1, \ infty], \ quad P_1, R_1 \ In [1, \ infty], \ cfrac {1} {P }=\ cfrac {1} {P_1} + \ cfrac {1} {P_2 }=\ cfrac {1} {R_1} + \ cfrac {1} {R_2 }\\\& \ Ra \ Sen {FG }_{\ dot B ^ s _ {P, Q }}\ Leq C \ sex {\ Sen {f }_{ L ^ {P_1 }}\ Sen {g }_{ \ dot B ^ s _ {P_2, Q }}+ \ Sen {g }_{ L ^ {R_1 }}\ Sen {f }_{ \ dot B ^ s _ {r_2, Q }}}. \ EEA \ eeex $
(3) $ \ beex \ Bea & \ quad S_1, S_2 \ Leq \ cfrac {n} {p }, \ quad S_1 + S_2> 0 \ & \ Ra \ Sen {FG }_{ \ dot B ^ {S_1 + S_2-\ frac {n} {P }}_{ P, 1 }}\ Leq C \ Sen {f }_{ \ dot B ^ {S_1 }_{ P, 1 }}\ Sen {g }_{\ dot B ^ {S_2 }_{ P, 1 }}. \ EEA \ eeex $
(4) $ \ beex \ Bea & \ quad-\ cfrac {n} {p}-1 <s \ Leq \ cfrac {n} {p} \ & \ Ra \ Sen {[U, \ lap_q] W }_{ L ^ p} \ Leq c_q 2 ^ {-Q (S + 1 )} \ Sen {u }_{ \ dot B ^ {-\ frac {n} {p} + 1 }_{ P, 1 }}\ Sen {W }_{ \ dot B ^ s _ {P, 1 }}\ quad \ sex {\ sum _ {Q \ in {\ BF z} c_q \ Leq 1 }. \ EEA \ eeex $
(5) $ \ beex \ Bea & \ quad S, S_1> 0, S = \ TT S_1, 0 <\ TT <1 \ & \ Ra \ Sen {f }_{ \ dot B ^ s _ {2, 1 }}\ Leq C \ Sen {f }_{ \ dot B ^ {S_1 }_{ 2, 1 }}^ \ TT \ Sen {f }_{ L ^ 2} ^ {1-\ TT }. \ EEA \ eeex $
(6) [To be determined... the definition of Triebel-lizorkin space $ \ dot f ^ s _ {\ infty, Q} $ for $1 \ Leq q <\ infty $...] $ \ Bex \ Sen {f }_{ BMO} \ Leq C \ sex {\ Sen {\ n f }_{ BMO} + \ Sen {f }_{ L ^ 2 }}. \ EEx $
(7) $ \ Bex \ Sen {f }_{ L ^ \ infty} \ Leq C \ Sen {f }_{ L ^ 2} ^ \ frac {1} {4 }\ sen {\ lap F }_{ L ^ 2} ^ \ frac {3} {4 }. \ EEx $ see [D. chae, J. lee, on the blow-up criterion and small data global existence for the hall-magnetohydrodynamics, J. differential Equations, 256 (2014), 3835--3858].