Abstract
We develop a potential theory for a Riesz type kernel in a homogeneous space and characterize the compact sets K with capacity zero as the sets K for which every continous function f on K is the restriction to K of a continuous potential \(U^{\sigma _{f}}_{k}\) of an absolutely continuous measure σ f supported in an arbitrarily small neighbourhood of K. The measure σ f can be choosen as a suitable restriction of a single measure σ that only depends on the set K and the kernel k.
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References
Adams, D.R., Hedberg, L.-I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften 314. Springer, Berlin (1996)
Björn, A., Björn, J.: Nonlinear potential theory in metric spaces. Tracts in Mathematics 17. European Mathematical Society (2012)
Carleson, L.: Selected Problems on Exceptional Sets. Mathematical Studies. Van Nostrand, Princeton (1967)
Christ, M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. LX/LXI (1990)
Coifman, R.R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogenes, Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)
Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, 1966. Springer, Berlin (2009)
Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin (1984)
Evans, G.C.: On potentials of positive mass, Part II. Trans. Am. Math. Soc. 38, 201–236 (1935)
Fuglede, B.: On the theory of potentials in locally compact spaces. Acta Math. 103, 139–215 (1960)
Frostman, O.: Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Thèse (1935)
Hajibayov, M.G.: Continuity properties of potentials on spaces of homogeneous type. Int. J. Math. Anal. 2, 315–328 (2008)
Ja, H.S.: A universal potential for compact sets of capacity zero, and other approximation characteristics of sparse sets. Izw. Akad. Nauk Armjan. SSR Ser. Mat. 10(4), 356–372 (Russian) (1975)
Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126, 1–33 (2012)
Hytönen, T., Martikainen, T.H.: Non–homogeneous Tb theorems and random dyadic cubes on metric measure spaces, to appear in J. Geom. Anal.
Landkof, N.S.: Foundations of Modern Potential Theory, Grundlehren der Mathematischen Wissenschaften, Band 180. Springer, Berlin (1972)
Macias, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)
Simmons, G.F.: Introduction to Topology and Modern Analysis. International Student Edition. McGraw-Hill, New York (1963)
Sjödin, T.: Bessel potentials and extension of continuous functions on compact sets. Ark. Mat. 13(2), 263–271 (1974)
Sjödin, T.: Capacities of compact sets in linear subspaces of R n. Pac. J. Math. 78(1), 261–266 (1978)
Sjödin, T.: A note on capacity and Hausdorff measure in homogeneous spaces. Potential Anal. 6, 87–97 (1997). 1
Wallin, H.: Continuous functions and potential theory. Ark. Mat. 5(1), 55–84 (1963)
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Sjödin, T. Continuous Functions and Riesz Type Potentials in Homogeneous Spaces. Potential Anal 43, 495–511 (2015). https://doi.org/10.1007/s11118-015-9483-4
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DOI: https://doi.org/10.1007/s11118-015-9483-4
Keywords
- Homogeneous space
- Doubling measure
- Kernel
- Potential
- Energy
- Capacity
- Capacitary potential
- Approximate identity
- Dyadic cubes