Let $L_{p_i,\phi_i}$ $(i=1,2,3)$ be Morrey spaces. 

A function $g$ is called a pointwise multiplier 

from $L_{p_1,\phi_1}$ to $L_{p_2,\phi_2}$, 

if the pointwise product $fg$ belongs to $L_{p_2,\phi_2}$ 

for each $f\in L_{p_1,\phi_1}$. 

We denote by $\PWM(L_{p_1,\phi_1}, L_{p_2,\phi_2})$ the set of all pointwise multipliers from $L_{p_1,\phi_1}$ to $L_{p_2,\phi_2}$.

A sufficient condition on $p_i$ and $\phi_i$ ($i=1,2,3$)

for $\PWM(L_{p_1,\phi_1}, L_{p_2,\phi_2})=L_{p_3,\phi_3}$

was given in \cite{Nakai1997b}.

In this paper, we give a necessary condition.

In connection with these conditions, 

we also give sufficient conditions 

for $\PWM(L_{p_1,\phi_1}, L_{p_2,\phi_2})=\{0\}$.