Relative boundedness-compactness inequalities for a second order differential operator. (Q2711478)

If \(C\) and \(D\) are linear operators having domains \({\mathcal D}(C)\) and \({\mathcal D}(D)\) in a Hilbert space \({\mathcal H}\) with the norm \(| | \cdot| | \), then \(D\) is said to be relatively bounded with respect to \(C\) if \({\mathcal D} (C)\in{\mathcal D}(D)\) and there exist constants \(\alpha\) and \(\beta\) such that for all \(y\in{\mathcal D}(C)\), \(| | Dy| | \leq \alpha | | y| | +\beta| | Cy| | \). The operator \(D\) is said to be relatively compact with respect to \(C\) if \({\mathcal D} (C)\in{\mathcal D}(D)\) and \(D\) is a compact operator acting on on the graph of \(C\) with graph norm \(| | y| | +| | Cy| | \). The authors consider the differential expression of the form NEWLINE\[NEWLINE L[y](x)=-x^{-\gamma}[x^{\alpha}y^{\prime}(x)]^{\prime}. NEWLINE\]NEWLINE The operator acts in the Hilbert space \({\mathcal L}^2(x^{\gamma},I)\) of functions \(f\) satisfying \(\int_{I}x^{\gamma}| f(x)| ^2dx<\infty\). Here, \(I=[c,\infty)\), \(c>0\) or \(I=(0,d]\), \(d>0\). The maximal operator \(L_1\) associated with \(L\) has the domain NEWLINE\[NEWLINE {\mathcal D}(L_1)=\{y\in{\mathcal L}^2(x^{\gamma}, I): x^{\alpha}y^{\prime} \in AC_{loc}(I), L[y]\in{\mathcal L}^2(x^{\gamma}, I)\}. NEWLINE\]NEWLINE Let \(A_1\) and \(B_1\) be the maximal operators associated with the expressions NEWLINE\[NEWLINE A[y]=x^{-\gamma}ay, \;\;\;B[y]=x^{-\gamma}by^{\prime},NEWLINE\]NEWLINE where \(a\) and \(b\) are real- or complex-valued functions on \(I\). The authors investigate the conditions on \(a\) and \(b\) under which \(A_1\) and \(B_1\) are relatively bounded with respect to \(L_1\) and conditions under which \(A_1\) and \(B_1\) are relatively compact with respect to \(L_1\).