The author of the present note proves theorems on the compactness of imbeddings in weighted abstract anisotropic spaces. He also considers estimates of the approximation numbers and the Kolmogorov widths of operators of imbedding in weighted spaces; such estimates are useful in investigating the spectral properties of boundary value problems for degenerate equations.
Theorems are proved below on the location of the spectrum for operators generated by boundary value problems for degenerate differential-operator equations.
Definition 1. A linear operator A acting in a Hilbert space H is said to be positive if \(\overline{D(A)}=H\) and \[ \| (A+s)^{- 1}\|_{B(H)}\leq c(1+s)^{-1},\quad s\geq 0,\quad A+s=A+sI=A_ s, \] where I is the identity operator on H, and B(H) is the space of bounded linear operators on H.
Definition 2. \[ H(A^{\theta})=\{u;u\in D(A^{\theta}),\| u\| \quad p_{H(A^{\theta})}=\| Au\| \quad p_ H\quad +\| u\| \quad p_ H<\infty,\quad 1\leq p<\infty \}. \] Let g(x) be a positive measurable function on \(\Omega\subset R^ n\). Denote by \(L_{p,g}(\Omega;H)\) the space of strongly measurable functions with values in H, with the norm \[ \| f\|_{L_{p,g}(\Omega;H)}=(\int_{\Omega}\| f(x)\| \quad p_ H\cdot g(x)dx)^{1/p},\quad 1\leq p<\infty. \] Let \(g_ k(x)\), \(k=0,1,...,n\) be positive measurable functions on \(\Omega\), and \(H_ 0\) and H Hilbert spaces with \(H_ 0\) continuously and densely imbedded in H. Denote by \(D'(\Omega;H_ 0)\) the space of distributions with values in \(H_ 0\), and let \(D_ j^{\ell_ j}=\partial^{\ell_ j}/\partial x_ j^{\ell_ j}\), \(j=1,...,n\), where the derivative is understood in the sense of \(D'(\Omega;H_ 0).\)
Definition 3.
\(W^{\ell}_{p,g,g_ 0}(\Omega;H_ 0,H)=\{f:f\in L_{p,g_ 0}(\Omega;H_ 0),g_ j(x)\cdot\) \(D_ j^{\ell_ j}f\in L_{p,g_ 0}(\Omega;H),\| f\|\) \(p_{W^{\ell}_{p,g,g_ 0}(\Omega;H_ 0,H)}=\int_{\Omega}(\| f(x)\|^ p_{H_ 0}g_ 0(x)+\sum^{n}_{j=1}g\) \(p_ j(x)\| D_ j^{\ell_ j}f\|^ p_ H\) \(g_ 0(x))dx<\infty\), \(\ell =(\ell_ 1,...,\lambda_ n)\), \(1\leq p<\infty \}.\)
Let \(Q_ Y=\{x;x\in \Omega,| x_ j-y_ j| <g_ j(y)\), \(y\in \Omega\), \(j=1,...,n\}\). We consider the linear mapping \(T_ y\) defined by \(T_ yx=\xi\), \(\xi_ j=(x_ j-y_ j)g_ j^{-1}(y)+y_ i\), \(j=1,...,n\), \(x=(x_ 1,...,x_ n)\), \(\xi =(\xi_ 1,...,\xi_ n)\), \(y=(y_ 1,...,y_ n)\). The image of \(Q_ y\) under \(T_ y\) is denoted by \(Q\) \(*_ y\). Let \(u\) \(*_ y(\xi)=u(T_ y^{-1}(\xi)).\)
Condition 1. Let \(\Omega\) be a bounded domain satisfying the strong \(\ell\)-horn condition, [O. V. Besov, V. P. Il’in, S. M. Nikol’skij, Integral representations of functions and imbedding theorems, Moscow, (1975; Zbl 0352.46023)]. Let \(g_ i(x)\), \(i=0,1,...,n\), be positive continuous functions on \(\Omega\) with \[ \int_{\Omega}g_ i^{-1/p-1}(x)dx<\infty,\quad \int_{\Omega}g_ 0^{1/p}(x)\cdot g_ j^{-1}(x)dx<\infty, \] \(i=0,1,...,n\), \(j=1,...,n\), \(1<p<\infty\). Suppose that for any \(x\in Q_ y\) \(c^{- 1}\leq g_ i(x)\cdot g_ i^{-1}(y)\leq c\), \(c>0\), \(i=0,1,...,n\). Denote by \(\sigma_{\infty}(H)\) the space of compact operators on H. Let \(m=(m_ 1,...,m_ n)\), \(\ell =(\ell_ 1,...,\ell_ n)\), \(| m:\ell | =\sum^{n}_{1}m_ j/\ell_ j\), and D \(m=D_ 1^{m_ 1}...D_ n^{m_ n}.\)
Condition 2. Suppose that A is a positive operator acting in H, \(A^{- 1}\in \sigma_{\infty}(H)\), and the following inequality holds for any \(y\in \Omega\), \(| m:\ell | +\mu <1\), for some \(\mu,\eta,\kappa >0\), and for any \(u\in W_ p^{\ell}(Q_ y;H(A),H):\) \[ (1)\quad \| A^{1-| m:\ell | -\mu}D\quad mu\quad *_ y\|_{L_ p(Q\quad *_ y;H)}\leq c\{\eta^{\kappa +\mu}=u\quad *_ y\|_{W_ p^{\ell}(Q\quad *_ y;H(A),H)}+\eta^{-(1-\kappa -\mu)}\| u\quad *_ y\quad \|_{L_ p(Q\quad *_ y;H)}\}. \] The author obtains:
Theorem. Assume conditions 1 and 2. Then the imbedding D \(mW^{\ell}_{p,g,g_ 0}(\Omega;H(A),H)L_{p,g_ m}(\Omega;H(A^{1- | m:\ell | -m}))\), \(g_ m=g_ m(x)=g_ 0(x)\cdot \prod^{n}_{j=1}g_ i^{(m_ j/\ell_ j)p}\), is compact.