Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866917827465707520 |
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| author | Zorii, Natalia |
| author_facet | Zorii, Natalia |
| contents | In the theory of $g_α$-potentials on a domain $D\subset\mathbb R^n$, $n\geqslant2$, $g_α$ being the $α$-Green kernel associated with the $α$-Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,n)$, $α\leqslant2$, we establish the existence and uniqueness of the $g_α$-balayage $μ^F$ of a positive Radon measure $μ$ onto a relatively closed set $F\subset D$, we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for $μ^F(D)=μ(D)$ to hold, given in terms of the $α$-harmonic measure of suitable Borel subsets of $\overline{\mathbb R^n}$, the one-point compactification of $\mathbb R^n$. As a by-product, we find necessary and/or sufficient conditions for the existence of the $g_α$-equilibrium measure $γ_F$, $γ_F$ being understood in an extended sense where $γ_F(D)$ might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for $g_α$-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials Zorii, Natalia Classical Analysis and ODEs Complex Variables 31C15 In the theory of $g_α$-potentials on a domain $D\subset\mathbb R^n$, $n\geqslant2$, $g_α$ being the $α$-Green kernel associated with the $α$-Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,n)$, $α\leqslant2$, we establish the existence and uniqueness of the $g_α$-balayage $μ^F$ of a positive Radon measure $μ$ onto a relatively closed set $F\subset D$, we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for $μ^F(D)=μ(D)$ to hold, given in terms of the $α$-harmonic measure of suitable Borel subsets of $\overline{\mathbb R^n}$, the one-point compactification of $\mathbb R^n$. As a by-product, we find necessary and/or sufficient conditions for the existence of the $g_α$-equilibrium measure $γ_F$, $γ_F$ being understood in an extended sense where $γ_F(D)$ might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for $g_α$-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed. |
| title | Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials |
| topic | Classical Analysis and ODEs Complex Variables 31C15 |
| url | https://arxiv.org/abs/2411.01221 |