Multipliers in weighted Sobolev spaces

Let $X_1$ and $X_2$ be a pair of Banach spaces of functions in $\Omega\subset\mathbb R^n$. A multiplier from $X_1$ into $X_2$ is a function $\gamma$ on $\Omega$ such that $\gamma X_1=\{\gamma f,\,f\in X_1\}\subset X_2$. By the norm $\|\gamma\|=\|\gamma\|_{M(X_1\to X_2)}$ one means the norm of the operator $T(u)=\gamma u$, $u\in X_1$. Conditions ensuring that a function $\gamma$ belongs to the multiplier classes $M(W_1\to W_2)$ and $M(W\to L)$ are found, where $W$ and $L$ are Sobolev and Lebesgue weighted spaces, respectively. Estimates of the norms of multipliers free from capacity characteristics are found. Special local maximal operators are introduced and significantly used.