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The trace inequality and eigenvalue estimates for Schrödinger operators. (English) Zbl 0591.47037
Suppose $$\Phi$$ is a nonnegative, locally integrable, radial function on $$R^ n$$, which is nonincreasing in $$| x|$$. Set $(Tf)(x)=\int_{R^ n}\Phi (x-y)f(y)dy$ when $$f\geq 0$$ and $$x\in R^ n$$. Given $$1<p<\infty$$ and $$v\geq 0$$, we show there exists $$C>0$$ so that $\int_{R^ n}(Tf)(x)^ pv(x)dx\leq C\int_{R^ n}f(x)^ pdx$ for all $$f\geq 0$$. If and only if $$C'>0$$ exists with $\int_{Q}T(x_ Qv)(x)^{p'}dx\leq C'\int_{Q}v(x)dx<\infty$ for all dyadic cubes Q, where $$p'=p/(p-1)$$. This result is used to refine recent estimates of C. L. Fefferman and D. H. Phong on the distribution of eigenvalues of Schrödinger operators.

##### MSC:
 47F05 General theory of partial differential operators 47A10 Spectrum, resolvent 26D10 Inequalities involving derivatives and differential and integral operators 35J10 Schrödinger operator, Schrödinger equation
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##### References:
 [1] D. R. ADAMS, A trace inequality for generalized potentials, Studia Math., 48 (1973), 99-105. · Zbl 0237.46037 [2] D. R. ADAMS, On the existence of capacitary strong type estimates in rn, Ark. Mat., 14 (1976), 125-140. · Zbl 0325.31008 [3] D. R. ADAMS, Lectures on lp-potential theory (preprint), Univ. of Umeä, 2 (1981). [4] N. ARONSZAJN and K. T. SMITH, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. · Zbl 0102.32401 [5] S. Y. A. CHANG, J. M. WILSON and T. H. WOLFF, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helv., 60 (1985), 217-246. · Zbl 0575.42025 [6] S. CHANILLO and R. L. WHEEDEN, Lp estimates for fractional integrals and Sobolev inequalities, with applications to Schrödinger operators, Comm. Partial Differential Equations, 10 (1985), 1077-1116. · Zbl 0578.46024 [7] R. COIFMAN and C. FEFFERMAN, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. · Zbl 0291.44007 [8] B. DAHLBERG, Regularity properties of Riesz potentials, Ind. U. Math. J., 28 (1979), 257-268. · Zbl 0413.31003 [9] E. FABES, C. KENIG and R. SERAPIONI, The local regularity of solutions of degenerate elliptic equations, Comm. in P.D.E., 7 (1982), 77-116. · Zbl 0498.35042 [10] C. L. FEFFERMAN, The uncertainty principle, Bull. A.M.S., (1983), 129-206. · Zbl 0526.35080 [11] M. De GUZMAN, Differentiation of integrals in rn, Lecture Notes in Math., vol. 481, Springer-Verlag, Berlin and New York, 1975. · Zbl 0327.26010 [12] K. HANSSON, Continuity and compactness of certain convolution operators, Institut Mittage-Leffler, Report No. 9, (1982). [13] R. KERMAN and E. SAWYER, Weighted norm inequalities for potentials with applications to Schrödinger operators, Fourier transforms and Carleson measures, announcement in Bull. A.M.S., 12 (1985), 112-116. · Zbl 0564.35027 [14] V. G. MAZ’YA, On capacitary estimates of the strong type for the fractional norm, Zap. Sen. LOMI Leningrad, 70 (1977), 161 - 168. · Zbl 0433.46032 [15] B. MUCKENHOUPT and R. L. WHEEDEN, Weighted norm inequalities for fractional integrals, Trans. A.M.S., 192 (1974), 251-275. · Zbl 0289.26010 [16] M. REED and B. SIMON, Methods of mathematical physics, Vol. I, Academic Press, New York and London, 1972. · Zbl 0242.46001 [17] E. SAWYER, Weighted norm inequalities for fractional maximal operators, C.M.S. Conf. Proc., 1 (1980), 283-309. · Zbl 0546.42018 [18] E. SAWYER, A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75 (1982), 1-11. · Zbl 0508.42023 [19] E. M. STEIN, The characterization of functions arising as potentials I, Bull. Amer. Math. Soc., 67 (1961), 102-104, II (IBID), 68 (1962), 577-582. · Zbl 0127.32002 [20] E. M. STEIN, Singular integrals and differentiability properties of functions, 2nd edition, Princeton University Press, 1970. · Zbl 0207.13501 [21] J.-O. STRÖMBERG and R. L. WHEEDEN, Fractional integrals on weighted hp and lp spaces, Trans. Amer. Math., Soc., 287 (1985), 293-321. · Zbl 0524.42011
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