Abstract
First introduced by J. Deny, the classical principle of positivity of mass states that if \(\kappa _{\alpha }\mu \leqslant \kappa _{\alpha }\nu \) everywhere on \(\mathbb {R}^n\), then \(\mu (\mathbb {R}^n)\leqslant \nu (\mathbb {R}^n)\). Here \(\mu ,\nu \) are positive Radon measures on \(\mathbb {R}^n\), \(n\geqslant 2\), and \(\kappa _{\alpha }\mu \) is the potential of \(\mu \) with respect to the Riesz kernel \(|x-y|^{\alpha -n}\) of order \(\alpha \in (0,2]\), \(\alpha <n\). We strengthen Deny’s principle by showing that \(\mu (\mathbb {R}^n)\leqslant \nu (\mathbb {R}^n)\) still holds even if \(\kappa _{\alpha }\mu \leqslant \kappa _{\alpha }\nu \) is fulfilled only on a proper subset A of \(\mathbb {R}^n\) that is not inner \(\alpha \)-thin at infinity; and moreover, this condition on A cannot in general be improved. Hence, if \(\xi \) is a signed measure on \(\mathbb {R}^n\) with \(\int 1\,d\xi >0\), then \(\kappa _{\alpha }\xi >0\) everywhere on \(\mathbb {R}^n\), except for a subset which is inner \(\alpha \)-thin at infinity. The analysis performed is based on the author’s recent theories of inner Riesz balayage and inner Riesz equilibrium measures (Potential Anal., 2022), the inner equilibrium measure being understood in an extended sense where both the energy and the total mass may be infinite.
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Notes
In B. Fuglede’s terminology [14], the Riesz kernel is, therefore, perfect.
As usual, the infimum over the empty set is taken to be \(+\infty \). We also agree that \(1/(+\infty )=0\) and \(1/0 = +\infty \).
We also refer to the author’s recent work [26] providing a number of alternative characterizations of the inner capacity \(c_{\alpha }(A)\) and the inner equilibrium measure \(\gamma _A\), the results in [26] being actually valid even for quite a large class of general function kernels on locally compact spaces.
Concerning the orthogonal projection in a pre-Hilbert space, see e.g. [13, Theorem 1.12.3].
The outer \(\alpha \)-Riesz balayage was investigated by J. Bliedtner and W. Hansen [1] in the general framework of balayage spaces. See also N.S. Landkof [18, Section V.1.2], where, however, certain restrictions were imposed upon A and \(\mu \), e.g. that \(A\subset \mathbb {R}^n\) be Borel while \(\mu \in \mathfrak {M}^+\) be bounded.
For any \(\sigma \in \mathcal {E}^+_{\alpha }\) and any \(A\subset \mathbb {R}^n\), the measure \(\sigma ^A\in \mathcal {E}'_A\) having the property \(\kappa _{\alpha }\sigma ^A=\kappa _{\alpha }\sigma \) n.e. on A, exists and is unique. It is in fact the orthogonal projection of \(\sigma \) in the pre-Hilbert space \(\mathcal {E}_{\alpha }\) onto the convex, strongly complete cone \(\mathcal {E}'_A\); that is (compare with (3.1)),
$$\begin{aligned}\Vert \sigma -\sigma ^A\Vert =\min _{\nu \in \mathcal {E}'_A}\,\Vert \sigma -\nu \Vert .\end{aligned}$$Alternatively, \(\sigma ^A\) is uniquely characterized within \(\mathfrak {M}^+\) by the extremal property (3.3) with \(\mu :=\sigma \).
Such \(\mu _j\in \mathcal {E}^+_{\alpha }\), \(j\in {\mathbb {N}}\), do exist; they can be defined, for instance, by means of the formula
$$\begin{aligned}\kappa _{\alpha }\mu _j:=\min \,\bigl \{\kappa _{\alpha }\mu ,\,j\kappa _{\alpha }\lambda \bigr \},\end{aligned}$$\(\lambda \in \mathcal {E}^+_{\alpha }\) being fixed (see e.g. [18, p. 272] or [9, p. 257, footnote]). Here we have used the fact that for any \(\mu _1,\mu _2\in \mathfrak {M}^+\), there is \(\mu _0\in \mathfrak {M}^+\) such that \(\kappa _{\alpha }\mu _0:=\min \,\{\kappa _{\alpha }\mu _1,\,\kappa _{\alpha }\mu _2\}\) [18, Theorem 1.31].
This result has recently been extended to inner balayage on a locally compact space, see [27].
\(\varepsilon _y^A\) is said to be the (fractional) inner \(\alpha \)-harmonic measure of \(A\subset {\mathbb {R}}^n\) at \(y\in {\mathbb {R}}^n\). Being a natural generalization of the classical concept of (2-)harmonic measure [1,2,3, 18], \(\varepsilon _y^A\) serves as the main tool in solving the generalized Dirichlet problem for \(\alpha \)-harmonic functions. Besides, due to the integral representation formula \(\mu ^A=\int \varepsilon _y^A\,d\mu (y)\) [24, Theorem 5.1], the inner \(\alpha \)-harmonic measure \(\varepsilon _y^A\) is a powerful tool in the investigation of the inner balayage \(\mu ^A\) for arbitrary \(\mu \) (see [24]).
In view of (permanent) assumption (1.1), the inner \(\alpha \)-Riesz equilibrium measure \(\gamma _A\) does not exist if there is no \(\nu \in {\mathfrak {M}}^+\) with \(\kappa _{\alpha }\nu \geqslant 1\) n.e. on A. This implication can actually be reversed, and hence \(\gamma _A\) exists if and only if \(\Gamma _A\ne \varnothing \), see Theorem 4.2(ii).
References
Bliedtner, J., Hansen, W.: Potential Theory. An Analytic and Probabilistic Approach to Balayage. Springer, Berlin (1986)
Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Stud. Math. 123, 43–80 (1997)
Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of \(\alpha \)-harmonic functions. Probab. Theory Relat. Fields 140, 345–381 (2008)
Bourbaki, N.: Integration. Chapters 1–6. Springer, Berlin (2004)
Brelot, M.: On Topologies and Boundaries in Potential Theory. Lecture Notes in Math, vol. 175. Springer, Berlin (1971)
Brelot, M., Choquet, G.: Le théorème de convergence en théorie du potentiel. J. Madras Univ. Ser. B 27, 277–286 (1957)
Cartan, H.: Sur les fondements de la théorie du potentiel. Bull. Soc. Math. France 69, 71–96 (1941)
Cartan, H.: Théorie du potentiel newtonien: énergie, capacité, suites de potentiels. Bull. Soc. Math. France 73, 74–106 (1945)
Cartan, H.: Théorie générale du balayage en potentiel newtonien. Ann. Univ. Fourier Grenoble 22, 221–280 (1946)
Deny, J.: Les potentiels d’énergie finie. Acta Math. 82, 107–183 (1950)
Deny, J.: Méthodes Hilbertiennes en Théorie du Potentiel. In: Potential Theory. CIME Summer Schools 49, pp. 121–201. Springer, Berlin (2010)
Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin (1984)
Edwards, R.E.: Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York (1965)
Fuglede, B.: On the theory of potentials in locally compact spaces. Acta Math. 103, 139–215 (1960)
Fuglede, B., Zorii, N.: Green kernels associated with Riesz kernels. Ann. Acad. Sci. Fenn. Math. 43, 121–145 (2018)
Fuglede, B., Zorii, N.: An alternative concept of Riesz energy of measures with application to generalized condensers. Potential Anal. 51, 197–217 (2019)
Kurokawa, T., Mizuta, Y.: On the order at infinity of Riesz potentials. Hiroshima Math. J. 9, 533–545 (1979)
Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972)
Riesz, M.: Intégrales de Riemann-Liouville et potentiels. Acta Szeged 9, 1–42 (1938)
Zorii, N.: A problem of minimum energy for space condensers and Riesz kernels. Ukrainian Math. J. 41, 29–36 (1989)
Zorii, N.: Interior capacities of condensers in locally compact spaces. Potential Anal. 35, 103–143 (2011)
Zorii, N.: A concept of weak Riesz energy with application to condensers with touching plates. Anal. Math. Phys. 10, 43 (2020)
Zorii, N.: A theory of inner Riesz balayage and its applications. Bull. Pol. Acad. Sci. Math. 68, 41–67 (2020)
Zorii, N.: Harmonic measure, equilibrium measure, and thinness at infinity in the theory of Riesz potentials. Potential Anal. 57, 447–472 (2022)
Zorii, N.: Balayage of measures on a locally compact space. Anal. Math. 48, 249–277 (2022)
Zorii, N.: On the theory of capacities on locally compact spaces and its interaction with the theory of balayage. Potential Anal. (2022). https://doi.org/10.1007/s11118-022-10010-3
Zorii, N.: On the theory of balayage on locally compact spaces. Potential Anal. (2022). https://doi.org/10.1007/s11118-022-10024-x
Zorii, N.: Minimum Riesz energy problems with external fields. J. Math. Anal. Appl. (2023). https://doi.org/10.1016/j.jmaa.2023.127235
Acknowledgements
The author is deeply indebted to Douglas P. Hardin and Bent Fuglede for reading and commenting on the manuscript, as well as to the anonymous referee for his/her useful remarks, helping to improve the exposition of the paper.
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Zorii, N. On the role of the point at infinity in Deny’s principle of positivity of mass for Riesz potentials. Anal.Math.Phys. 13, 38 (2023). https://doi.org/10.1007/s13324-023-00793-y
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DOI: https://doi.org/10.1007/s13324-023-00793-y
Keywords
- Principle of positivity of mass for \(\alpha \)-Riesz
- Inner \(\alpha \)-thinness at infinity
- Inner \(\alpha \)-Riesz balayage
- A generalized concept of inner \(\alpha \)-Riesz equilibrium measure