## Coercive inequalities on weighted Sobolev spaces.(English)Zbl 0827.46028

Let $$P_j= (P_{j1}, \dots, P_{jk})$$ $$(j=1, \dots, N)$$ be scalar differential operators of order $$m$$, acting on vector-valued functions $$f= (f_1, \dots, f_k)$$: $P_j f=\sum_{i=1}^k P_{ji} f_i, \qquad P_{ji} g(x)= \sum_{|\alpha|\leq m} a_{\alpha, j,i} (x) Dg(x).$ Denote by $$P^m_j$$ the principle part of $$P_j$$, involving differentiations of highest order, and by $$P^0_j$$ the part involving differentiations of order less than $$m$$.
By $$W^{m,p} (\Omega)$$ we denote the weighted Sobolev spaces: $W^{m,p} (\Omega):= \{f\in {\mathcal D}' (\Omega):\;D^\alpha f\in L^p_\rho (\Omega),\;|\alpha|\leq m\}$ with norm $$|f|_{W_\rho^{m,p} (\Omega)}:= \sum_{|\alpha|\leq m} |D^\alpha f|_{L^p_\rho (\Omega)}$$, and $$\Omega$$ be an open subset of $$\mathbb{R}^n$$, $$\rho\geq 0$$ a locally integrable function.
The main result of this paper is the following theorem:
Let $$\Omega$$ be a bounded domain with cone property, $$\rho\in A_p$$, $$1\leq p\leq \infty$$ (where $$A_p$$ is the class of Muckenhoupt type weights), and let $$\{P_j\}_{j= 1,\dots, N}$$ be a family of differential operators of order $$m$$ acting on vector-valued functions $$f= (f_1, \dots, f_k)$$ such that
(i) the coefficients of $$P_j^m$$ are continuous in $$\Omega$$, and those of $$P^0_j$$ are bounded in $$\Omega$$,
(ii) the matrix $$\{P_{ij} (x, i\xi) \}^{j= 1,\dots, N}_{i= 1,\dots, k}$$ has rank $$k$$ for any $$\xi\neq (0, \dots, 0)$$ with complex $$\xi_i$$ and $$x\in \Omega$$.
Then there exists a constant $$C$$ such that for any $$f\in W_\rho^{m,p} (\Omega)$$, $|\nabla^m f|_{L^p_\rho (\Omega)}\leq C \Biggl\{ |f|_{L^p_\rho (\Omega)}+ \sum_{j=1}^N |P_j f|_{L^p_\rho (\Omega)} \Biggr\}.$

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35B45 A priori estimates in context of PDEs
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