map $$L^2(X \times X) \rightarrow  L^2(X \times X): (a_{x,y})_{x,y \in X} \mapsto (\psi(x,y) a_{x,y})_{x,y \in X},$$ which canonically determines a bounded map on ${\mathcal{S}}_2(L^2 X)$. If it maps ${\mathcal{S}}_2(L^2 X)\cap {\mathcal{S}}_p(L^2 X)$ to ${\mathcal{S}}_p(L^2 X)$ and it extends boundedly to ${\mathcal{S}}_p(L^2X)$, it will be called a (Schur) multiplier of ${\mathcal{S}}_p(L^2X)$.

We will use the following result, which is a minor modification of [@LafSal Theorem 1.19].

\[thm=mult\_de\_symb\_continu\] Let $\mu$ be a Radon measure on a locally compact space $X$, and $\psi:X \times X \to \C$ a continuous function. Let $1 \leq p
  \leq \infty$ and $K>0$. The following are equivalent:

(i) \[asser=norm\_locCompact\] $\psi$ defines a bounded multiplier on ${\mathcal{S}}_p(L^2(X,\mu))$ with norm less than $K$.

(ii) \[asser=norm\_parties\_semifinies\] For every $\sigma$-finite measurable subset $X_0$ in $X$, $\psi$ restricts to a bounded multiplier on ${\mathcal{S}}_p(L^2( X_0, \mu))$ with norm less than $K$.

(iii) \[asser=norm\_parties\_finies\] For any finite subset $F=\{x_1,\dots,x_N\}$ in $X$ belonging to the support of $\mu$, the multiplier $(\psi(x_i,x_j))$ is bounded on ${\mathcal{S}}_p(\ell^2 F)$ with norm less than $K$.

The equivalence of (\[asser=norm\_locCompact\]) and (\[asser=norm\_parties\_finies\]) of Theorem \[thm=mult\_de\_symb\_continu\] was proved in [@LafSal Theorem 1.19] under the additional assumption that $\mu$ is $\sigma$-finite. In the current theorem, this implies the equivalence of (\[asser=norm\_parties\_semifinies\]) and 