A theory of inner Riesz balayage and its applications

• Autorzy: Zorii Natalia

Identyfikatory

DOI

10.4064/ba191104-31-1

• Języki publikacji: EN

Abstrakty

EN

We establish a theory of balayage for the Riesz kernel │x - y│^α-n, α∈(0; 2], on Rⁿ, n ≥ 3, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan’s ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner equilibrium measure γA for A ⊂ Rⁿμ arbitrary, in particular given in terms of the total mass of the inner swept measure μ^A with μ suitably chosen. For example, ϒ^A exists if and only if ε^A* ≠ε, where ε is the Dirac measure at x = 0 and A* the inverse of A relative to the sphere │x│= 1, which leads to a Wiener type criterion of inner α-irregularity. The results obtained are illustrated by examples.

Słowa kluczowe

EN

inner Riesz balayage; inner Riesz equilibrium measure; Wiener type criterion of inner α-irregularity; μ-adequate family of Radon measures

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2020
• Tom: Vol. 68, no. 1
• Strony: 41--67
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Zorii Natalia

• Institute of Mathematics, Academy of Sciences of Ukraine, Tereshchenkivska 3, 01601 Kyiv, Ukraine

Bibliografia

• [1] J. Bliedtner and W. Hansen, Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer, Berlin, 1986.
• [2] N. Bourbaki, General Topology, Chapters 1-4, Springer, Berlin, 1989.
• [3] N. Bourbaki, Integration, Chapters 1-6, Springer, Berlin, 2004.
• [4] M. Brelot, Éléments de la Théorie Classique du Potentiel, Les Cours Sorbonne, Paris, 1961.
• [5] M. Brelot, On Topologies and Boundaries in Potential Theory, Lecture Notes in Math. 175, Springer, Berlin, 1971.
• [6] H. Cartan, Théorie générale du balayage en potentiel newtonien, Ann. Univ. Grenoble 22 (1946), 221-280.
• [7] J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer, Berlin, 1984.
• [8] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965.
• [9] B. Fuglede, On the theory of potentials in locally compact spaces, Acta Math. 103 (1960), 139-215.
• [10] B. Fuglede and N. Zorii, Green kernels associated with Riesz kernels, Ann. Acad. Sci. Fenn. Math. 43 (2018), 121-145.
• [11] N. S. Landkof, Foundations of Modern Potential Theory, Springer, Berlin, 1972.
• [12] M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Univ. Szeged 9 (1938), 1-42.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).