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Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. (English) Zbl 0834.31006
The trace inequality for Bessel potentials is considered and the connection with applications in partial differential equations is studied.
Authors’ abstract: Some new characterizations of the class of positive measures \(\gamma\) on \(\mathbb R^n\) such that \(H_p^l\subset L_p(\gamma)\) are given where \(H_p^l\) \((1<p<\infty, \, 0<l<\infty)\) is the space of Bessel potentials This imbedding as well as the corresponding trace inequality \[ \| J_lu \|_{L_p(\gamma)}\leq C\,\| u\|_{L_p} \] for Bessel potentials \(J_l =(1-\Delta)^{-l/2}\) is shown to be equivalent to one of the following conditions
(a)\(\quad J_l (J_l\gamma)^p \leq C\,J_l\gamma\) a. e.
(b)\(\quad M_l (M_l\gamma)^{p'} \leq C\,M_l\gamma\) a. e.
(c)\(\quad\) For all compact subsets \(E\) of \(\mathbb R^n\) \[ \int_E(J_l\gamma)^p\,dx\leq C \text{cap}(E\,H_l^p) \] where \(1/p+1/p'=1\), \(M_l\) is the fractional maximal operator and \(\text{cap}(H_p^l)\) is the Bessel capacity. In particular it is shown that the trace inequality for a positive measure \(\gamma\) holds if and only if it holds for the measure \((J_l\gamma)^{p'}\, dx\). Similar results are proved for the Riesz potentials \(I_l\gamma=|x|^{l-n}*\gamma\).
These results are used to get a complete characterization of the positive measures on \(\mathbb R^n\) giving rise to bounded pointwise multipliers \(M(H_p^m \to H_p^{-l})\). Some applications to elliptic partial differential equations are considered including coercive estimates for solutions of the Poisson equation and existence of positive solutions for certain linear and semilinear equations.

MSC:
31C15 Potentials and capacities on other spaces
26A33 Fractional derivatives and integrals
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[1] Adams, D. R., On the existence of capacitary strong type estimates inR^n, Ark Mat, 14, 125-140 (1976) · Zbl 0325.31008
[2] Adams, D R andHedberg, L IFunction Spaces and Potential Theory, Springer Verlag, Berlin-Heidelberg-New York, to appear
[3] Adams, D. R.; Pierre, M., Capacitary strong type estimates in semilinear problems, Ann Inst Fourier (Grenoble), 41, 117-135 (1991) · Zbl 0741.35012
[4] Agmon, S.; Greco, D., On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, Methods of Functional Analysis and Theory of Elliptic Equations, 19-52 (1983), Naples: Liguori, Naples
[5] Baras, P.; Pierre, M., Critère d’existence de solutions positives pour des équations semi linéaires non monotones, Ann Inst H Poincaré Anal Non Linéaire, 2, 185-212 (1985) · Zbl 0599.35073
[6] Chanillo, S.; Sawyer, E. T., Unique continuation for Δ+v and the C Feffer man-Phong class, Trans Amer Math Soc, 318, 275-300 (1990) · Zbl 0702.35034
[7] Chung, K. L.; Li, P.; Rota, G. C., Comparison of probability and eigenvalue methods for the Schrödinger equation, Probability, Statistical Mechanics, and Number Theory, 25-34 (1986), Orlando, Fla: Academic Press, Orlando, Fla
[8] Dahlberg, B. E J., Regularity properties of Riesz potentials, Indiana Univ Math J, 28, 257-268 (1979) · Zbl 0413.31003
[9] Fefferman, C., The uncertainty principle, Bull Amer Math Soc, 9, 129-206 (1983) · Zbl 0526.35080
[10] Hansson, K., Imbedding theorems of Sobolev type in potential theory, Math Scand, 45, 77-102 (1979) · Zbl 0437.31009
[11] Hansson, K, Continuity and compactness of certain convolution operators,Inst Mittag Leffler Report9 (1982)
[12] Hedberg, L. I., On certain convolution inequalities, Proc Amer Math Soc, 36, 505-510 (1972) · Zbl 0283.26003
[13] Jerison, D.; Kenig, C. E., Unique continuation and absence of positive eigen values for Schrödinger operators, Ann of Math, 121, 463-494 (1985) · Zbl 0593.35119
[14] Kerman, R.; Sawyer, E. T., The trace inequality and eigenvalue estimates for Schrödinger operators, Ann Inst Fourier (Grenoble), 36, 207-228 (1986) · Zbl 0591.47037
[15] Khas’minsky, R. Z., On positive solutions of the equation Δu +V u=0, Theory Probab Appl, 4, 309-318 (1959) · Zbl 0089.34501
[16] Khavin, V. P.; Maz’ya, V. G., Nonlinear potential theory, Uspekhi Mat Nauk, 27, 6, 67-138 (1972)
[17] Landkof, N. S., Foundations of Modern Potential Theory (1966), Moscow: Nauka, Moscow · Zbl 0253.31001
[18] Maz’ya, V. G., On the theory of multidimensional Schrödinger operator, Izv Akad Nauk SSSR Ser Mat, 28, 1145-1172 (1964) · Zbl 0148.35602
[19] Maz’ya, V. G., Capacity estimates for “fractional” norms, Zap Nauchn Sem Lenin grad Otdel Mat Inst Steklov (LOMI), 70, 161-168 (1977) · Zbl 0433.46032
[20] Maz’ya, V. G., Sobolev Spaces (1985), Berlin-New York: Springer Verlag, Berlin-New York · Zbl 0727.46017
[21] Maz’ya, V. G.; Netrusov, Yu, Some counterexamples for the theory of Sobolev spaces on bad domains, Potential Anal, 4, 47-65 (1995) · Zbl 0819.46023
[22] Maz’ya, V. G.; Shaposhnikova, T. O., The Theory of Multipliers in Spaces of Differentiable Functions (1985), New York: Pitman, New York · Zbl 0645.46031
[23] Muckenhoupt, B.; Wheeden, R. L., Weighted norm inequalities for fractional integrals, Trans Amer Math Soc, 192, 261-274 (1974) · Zbl 0289.26010
[24] Sawyer, E. T.; Herz, C.; Rigelhot, R., Weighted norm inequalities for fractional maximal operators, 1980Seminar on Harmonic Analysis (Montréal, Que, 1980), 283-309 (1981), Providence, R I: Amer Math Soc, Providence, R I
[25] Sawyer, E. T.; Wheeden, R. L., Weighted norm inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer J Math, 114, 813-874 (1992) · Zbl 0783.42011
[26] Verbitsky, I. E., Weighted norm inequalities for maximal operators and Pisier’s theorem on factorization throughL^p∞, Integral Equations Operator Theory, 15, 124-153 (1992) · Zbl 0782.47027
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