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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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##### References:
  Adams D., Studia Math. 48 pp 99– (1973)  Adams E., Indiana Univ. Math. Jour. 32 pp 477– (1983) · Zbl 0487.30027 · doi:10.1512/iumj.1983.32.32034  Chang S. Y. A., Some weighted norm inequalities concerning the Schrödinger operators · Zbl 0575.42025 · doi:10.1007/BF02567411  Chanillo S., Amer. Jour. Math., to appear  Coifman R. R., Studia Math 51 pp 241– (1974)  Dahlberg B. E. J., Indiana Univ. Math. Jour 28 pp 257– (1979) · Zbl 0413.31003 · doi:10.1512/iumj.1979.28.28018  Fabes E.B., Comm. P.D.E 7 pp 77– (1982) · Zbl 0498.35042 · doi:10.1080/03605308208820218  Fefferman C.L., Bull. Amer. Math. Soc 9 pp 129– (1983) · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6  Fefferman C.L., Amer. Jour. Math 93 pp 107– (1971) · Zbl 0222.26019 · doi:10.2307/2373450  Gatto A.E., Trans. Amer. Math. Soc  Gel’fand I.M., Academic Press (1968)  Hansson K., Math.Scand 45 pp 77– (1979) · Zbl 0437.31009 · doi:10.7146/math.scand.a-11827  Harboure E., Two-weighted Sobolev and Poincare inequalities and some applications  Kerman R., The fourier transform and carleson measures · Zbl 0564.35027  Krasnosel’skii, M.A. and Rutickii, Ya.B. 1961. ”Convex Functions and Orlicz Spaces”. The Netherlands, New York: P. Noordhoff Ltd, Gordon and Breach, Inc.  Stein E. M., Singular Integrals and Differentiability Properties of Functions (1970) · Zbl 0207.13501  E. Stredulinsky ”Weighted Inequalities and Degenerate Elliptic Partial Differential Equations,” Springer Lecture Notes No. 1074 1984. · Zbl 0541.35001  J.O. Sternberg and A. Torchinsky, ”Weighted Hardy Spaces,” to appear  Strömberg J.O., Trans. Amer. Math. Soc 270 pp 439– (1982) · Zbl 0491.42017 · doi:10.2307/1999855  J.O. Sternberg and R. L. Wheeden ”Fractional integrals on weighted HP and LP spaces,” Trans, Amer. Math. Soc, to appear  Lieb E., Rev. of Mod, Phys 48 (4) pp 553– (1976) · doi:10.1103/RevModPhys.48.553
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