Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients
Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)...
Ausführliche Beschreibung
Autor*in: |
Shiozawa, Yuichi [verfasserIn] |
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Sprache: |
Englisch |
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2021 |
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Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
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Übergeordnetes Werk: |
Enthalten in: Potential analysis - Springer Netherlands, 1992, 58(2021), 2 vom: 31. Juli, Seite 373-392 |
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Übergeordnetes Werk: |
volume:58 ; year:2021 ; number:2 ; day:31 ; month:07 ; pages:373-392 |
Links: |
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DOI / URN: |
10.1007/s11118-021-09943-y |
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Katalog-ID: |
OLC2133723781 |
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520 | |a Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. | ||
650 | 4 | |a Non-local Dirichlet form | |
650 | 4 | |a Markov semigroup | |
650 | 4 | |a Compactness | |
650 | 4 | |a Essential super Poincaré inequality | |
700 | 1 | |a Wang, Jian |4 aut | |
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10.1007/s11118-021-09943-y doi (DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Shiozawa, Yuichi verfasserin aut Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality Wang, Jian aut Enthalten in Potential analysis Springer Netherlands, 1992 58(2021), 2 vom: 31. Juli, Seite 373-392 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:58 year:2021 number:2 day:31 month:07 pages:373-392 https://doi.org/10.1007/s11118-021-09943-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 58 2021 2 31 07 373-392 |
spelling |
10.1007/s11118-021-09943-y doi (DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Shiozawa, Yuichi verfasserin aut Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality Wang, Jian aut Enthalten in Potential analysis Springer Netherlands, 1992 58(2021), 2 vom: 31. Juli, Seite 373-392 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:58 year:2021 number:2 day:31 month:07 pages:373-392 https://doi.org/10.1007/s11118-021-09943-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 58 2021 2 31 07 373-392 |
allfields_unstemmed |
10.1007/s11118-021-09943-y doi (DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Shiozawa, Yuichi verfasserin aut Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality Wang, Jian aut Enthalten in Potential analysis Springer Netherlands, 1992 58(2021), 2 vom: 31. Juli, Seite 373-392 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:58 year:2021 number:2 day:31 month:07 pages:373-392 https://doi.org/10.1007/s11118-021-09943-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 58 2021 2 31 07 373-392 |
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10.1007/s11118-021-09943-y doi (DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Shiozawa, Yuichi verfasserin aut Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality Wang, Jian aut Enthalten in Potential analysis Springer Netherlands, 1992 58(2021), 2 vom: 31. Juli, Seite 373-392 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:58 year:2021 number:2 day:31 month:07 pages:373-392 https://doi.org/10.1007/s11118-021-09943-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 58 2021 2 31 07 373-392 |
allfieldsSound |
10.1007/s11118-021-09943-y doi (DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Shiozawa, Yuichi verfasserin aut Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality Wang, Jian aut Enthalten in Potential analysis Springer Netherlands, 1992 58(2021), 2 vom: 31. Juli, Seite 373-392 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:58 year:2021 number:2 day:31 month:07 pages:373-392 https://doi.org/10.1007/s11118-021-09943-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 58 2021 2 31 07 373-392 |
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We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. 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Shiozawa, Yuichi |
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510 VZ 17,1 ssgn Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality |
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Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
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compactness of semigroups generated by symmetric non-local dirichlet forms with unbounded coefficients |
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Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
abstract |
Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. © The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
abstractGer |
Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. © The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
abstract_unstemmed |
Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$. In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and W(x,y)=(1+|x|)p+(1+|y|)p,|x−y|<1(1+|x|)q+(1+|y|)q,|x−y|≥1$$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. © The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
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Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
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