## Arkiv för Matematik

• Ark. Mat.
• Volume 33, Number 1 (1995), 81-115.

### Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers

#### Abstract

Some new characterizations of the class of positive measures γ on Rn such that H ${}_{p}^{l}$ ∉Lp(γ) are given where H ${}_{p}^{l}$ (1< p<∞ 0< l∞) is the space of Bessel potentials This imbed ding as well as the corresponding trace inequality $||J_l u||_{L_p (\gamma )} \leqslant C||u||_{L_p }$ for Bessel potentials Jl=(1-Δ)-1/2 is shown to be equivalent to one of the following conditions

1. Jl(Jlγ)pCJ a e
2. Ml(Mlγ)p’CM a e
3. For all compact subsets E of Rn
$\int_E {(J_{l\gamma } )^p dx} \leqslant C{\text{ }}cap (E H_p^l )$ where 1/p+1/p'=1Ml is the fractional maximal operator and cap (H ${}_{p}^{l}$ ) is the Bessel capacity In particular it is shown that the trace inequality for a positive measure \gg holds if and only if it holds for the measure (Jl\gg)p'dx Similar results are proved for the Riesz potentials Ilγ=|x|l-n* γ

These results are used to get a complete characterization of the positive measures on Rn giving rise to bounded pointwise multipliers M(H ${}_{p}^{m}$ →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 semi linear equations

#### Article information

Source
Ark. Mat., Volume 33, Number 1 (1995), 81-115.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898306

Digital Object Identifier
doi:10.1007/BF02559606

Mathematical Reviews number (MathSciNet)
MR1340271

Zentralblatt MATH identifier
0834.31006

Rights

#### Citation

Maz'ya, Vladimir G; Verbitsky, Igor E. Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat. 33 (1995), no. 1, 81--115. doi:10.1007/BF02559606. https://projecteuclid.org/euclid.afm/1485898306

#### References

• Adams, D R, On the existence of capacitary strong type estimates in Rn, Ark Mat 14 (1976), 125–140
• Adams, D R and Hedberg, L IFunction Spaces and Potential Theory, Springer Verlag, Berlin-Heidelberg-New York, to appear
• Adams, D R and Pierre, M, Capacitary strong type estimates in semilinear problems, Ann Inst Fourier (Grenoble) 41 (1991), 117–135
• Agmon, S, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in Methods of Functional Analysis and Theory of Elliptic Equations (Greco, D, ed), pp 19–52, Liguori, Naples, 1983
• Baras, P and Pierre, M, Critère d'existence de solutions positives pour des équations semi linéaires non monotones, Ann Inst H Poincaré Anal Non Linéaire 2 (1985), 185–212
• Chanillo, S and Sawyer, E T, Unique continuation for Δ+v and the C Feffer man-Phong class, Trans Amer Math Soc 318 (1990), 275–300
• Chung, K L and Li, P, Comparison of probability and eigenvalue methods for the Schrödinger equation, in Probability, Statistical Mechanics, and Number Theory (Rota, G C, ed) pp 25–34, Adv Math Suppl Stud 9, Academic Press, Orlando, Fla, 1986
• Dahlberg, B E J, Regularity properties of Riesz potentials, Indiana Univ Math J 28 (1979), 257–268
• Fefferman, C, The uncertainty principle, Bull Amer Math Soc 9 (1983), 129–206
• Hansson, K, Imbedding theorems of Sobolev type in potential theory, Math Scand 45 (1979), 77–102
• Hansson, K, Continuity and compactness of certain convolution operators, Inst Mittag Leffler Report 9 (1982)
• Hedberg, L I, On certain convolution inequalities, Proc Amer Math Soc 36 (1972), 505–510
• Jerison, D and Kenig, C E, Unique continuation and absence of positive eigen values for Schrödinger operators, Ann of Math 121 (1985), 463–494
• Kerman, R and Sawyer, E T, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann Inst Fourier (Grenoble) 36 (1986), 207–228
• Khas'minsky R Z, On positive solutions of the equation Δu +V u=0, Theory Probab Appl 4 (1959), 309–318
• Khavin, V P and Maz'ya, V G, Nonlinear potential theory, Uspekhi Mat Nauk 27: 6 (1972), 67–138 (Russian) English transl: Russian Math Surveys 27 (1972), 71–148
• Landkof, N S, Foundations of Modern Potential Theory, Nauka, Moscow, 1966 (Russian) English transl: Springer Verlag, Berlin-Heidelberg, 1972
• Maz'ya, V G, On the theory of multidimensional Schrödinger operator, Izv Akad Nauk SSSR Ser Mat 28 (1964), 1145–1172 (Russian)
• Maz'ya, V G, Capacity estimates for “fractional” norms, Zap Nauchn Sem Lenin grad Otdel Mat Inst Steklov (LOMI) 70 (1977), 161–168 (Russian) English transl: J Soviet Math 23 (1983), 1997–2003
• Maz'ya, V G, Sobolev Spaces, Springer Verlag, Berlin-New York, 1985
• Maz'ya, V G and Netrusov, Yu, Some counterexamples for the theory of Sobolev spaces on bad domains, Potential Anal 4 (1995), 47–65
• Maz'ya, V G and Shaposhnikova, T O, The Theory of Multipliers in Spaces of Differentiable Functions, Pitman, New York 1985
• Muckenhoupt B and Wheeden, R L, Weighted norm inequalities for fractional integrals, Trans Amer Math Soc 192 (1974), 261–274
• Sawyer, E T, Weighted norm inequalities for fractional maximal operators, in 1980Seminar on Harmonic Analysis (Montréal, Que, 1980) (Herz, C and Rigelhot, R, eds), pp 283–309, CMS Conf Proc 1, Amer Math Soc, Providence, R I, 1981
• Sawyer, E T and Wheeden, R L, Weighted norm inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer J Math 114 (1992), 813–874
• Verbitsky, I E, Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through Lp∞, Integral Equations Operator Theory 15 (1992), 124–153