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\(L^ p\) estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators. (English) Zbl 0578.46024
The principal theorem proved is a Sobolev embedding theorem which states that:
Theorem: Let \(\phi(t)\geq 0\) and increasing on \((0,\infty)\) which satisfies \(\int^{\infty}_{1} \frac{dt}{t[\phi(t)]^{p'-1}}<\infty\), \(1/p+1/p'=1\), \(1<p<\infty\). Suppose \(v(x)\geq 0\) is a function on \({\mathbb{R}}^ n\) for which we have for every cube \(q\subset {\mathbb{R}}^ n\) the condition \[ \int_{Q}\phi (| Q|^{p/n}v(x))v(x)dx\leq c| Q|^{1-p/n}. \] Then, for \(f\in C_ 0^{\infty}({\mathbb{R}}^ n)\) we have, \[ \int_{{\mathbb{R}}^ n}| f|^ pv(x)dx\leq c\int_{{\mathbb{R}}^ n}| \nabla f|^ pdx. \] The case \(\phi(t)=t^{r-1}\), \(r>1\) is due to C. Fefferman and D. Phong and the case \(p=2\) of the result above is due to Chang-Wilson-Wolff. The proof presented in this paper is elementary and generalizes to fractional integrals on weighted \(L^ p\) and \(H^ p\) spaces and to local versions.
The principal application of the theorem above is the following theorem also proved in the paper:
Theorem: Let \(n\geq 3\) and let \(L=-\sum \frac{\partial}{\partial x_ i}(a^{ij}(x)\frac{\partial}{\partial x_ j})-v(x)\), \(v\geq 0\), with \(\lambda | \xi |^ 2\leq a^{ij}(x)\xi_ i\xi_ j\leq \lambda^{-1}| \xi |^ 2\), \(\lambda >0\) a constant. (a) Suppose there exist N cubes \(\{Q_ i\}^ N_{i=1}\) such that the doubles of \(Q_ i\) are also disjoint and such that \[ \frac{1}{| Q_ i|}\int_{Q_ i}v\geq c| Q_ i|^{2/n},\quad i=1,2,...,N \] where \(c=c(n,\lambda)\) is some suitable large constant. Then L has at least N negative eigenvalues.
(b) Conversely if L has at least CN negative eigenvalues, \(C=C(n)\) and \(\phi\) is increasing with \(\int^{\infty}_{1}\frac{dt}{t\phi (t)}<\infty\). Then there exist N disjoint cubes \(\{Q_ i\}^ N_{i=1}\) such that, \[ \frac{1}{| Q_ i|}\int_{Q_ i}v\phi (| Q_ i|^{2/n}v(x))dx\geq c| Q_ i|^{2\quad /n} \] where \(c=c(n,\lambda).\)
It follows from (a) and (b) that the number of negative eigenvalues of L is at most \(c(n,\lambda) \int_{{\mathbb{R}}^ n}v^{n/2}(x)dx\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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