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,$$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Shiozawa, Yuichi [verfasserIn] |
|---|
| Format: |
Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
2021 |
|---|
| Schlagwörter: |
|---|
| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature B.V. 2021 |
|---|
| Übergeordnetes Werk: |
Enthalten in: Potential analysis - Springer Netherlands, 1992, 58(2021), 2 vom: 31. Juli, Seite 373-392 |
|---|---|
| Übergeordnetes Werk: |
volume:58 ; year:2021 ; number:2 ; day:31 ; month:07 ; pages:373-392 |
| Links: |
|---|
| DOI / URN: |
10.1007/s11118-021-09943-y |
|---|
| Katalog-ID: |
OLC2133723781 |
|---|
| LEADER | 01000naa a22002652 4500 | ||
|---|---|---|---|
| 001 | OLC2133723781 | ||
| 003 | DE-627 | ||
| 005 | 20230506153108.0 | ||
| 007 | tu | ||
| 008 | 230506s2021 xx ||||| 00| ||eng c | ||
| 024 | 7 | |a 10.1007/s11118-021-09943-y |2 doi | |
| 035 | |a (DE-627)OLC2133723781 | ||
| 035 | |a (DE-He213)s11118-021-09943-y-p | ||
| 040 | |a DE-627 |b ger |c DE-627 |e rakwb | ||
| 041 | |a eng | ||
| 082 | 0 | 4 | |a 510 |q VZ |
| 084 | |a 17,1 |2 ssgn | ||
| 100 | 1 | |a Shiozawa, Yuichi |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
| 264 | 1 | |c 2021 | |
| 336 | |a Text |b txt |2 rdacontent | ||
| 337 | |a ohne Hilfsmittel zu benutzen |b n |2 rdamedia | ||
| 338 | |a Band |b nc |2 rdacarrier | ||
| 500 | |a © The Author(s), under exclusive licence to Springer Nature B.V. 2021 | ||
| 520 | |a Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \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 | |
| 773 | 0 | 8 | |i Enthalten in |t Potential analysis |d Springer Netherlands, 1992 |g 58(2021), 2 vom: 31. Juli, Seite 373-392 |w (DE-627)165647787 |w (DE-600)33485-6 |w (DE-576)032989911 |x 0926-2601 |7 nnns |
| 773 | 1 | 8 | |g volume:58 |g year:2021 |g number:2 |g day:31 |g month:07 |g pages:373-392 |
| 856 | 4 | 1 | |u https://doi.org/10.1007/s11118-021-09943-y |z lizenzpflichtig |3 Volltext |
| 912 | |a GBV_USEFLAG_A | ||
| 912 | |a SYSFLAG_A | ||
| 912 | |a GBV_OLC | ||
| 912 | |a SSG-OLC-MAT | ||
| 912 | |a SSG-OPC-MAT | ||
| 951 | |a AR | ||
| 952 | |d 58 |j 2021 |e 2 |b 31 |c 07 |h 373-392 | ||
| author_variant |
y s ys j w jw |
|---|---|
| matchkey_str |
article:09262601:2021----::opcnsosmgopgnrtdyymtinnoadrcltomw |
| hierarchy_sort_str |
2021 |
| publishDate |
2021 |
| allfields |
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. $$ with $p\in [0,\infty )$ and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of $(\mathcal {E},\mathcal {F})$ heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for $(\mathcal {E},\mathcal {F})$. Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure. Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality Wang, Jian aut Enthalten in Potential analysis Springer Netherlands, 1992 58(2021), 2 vom: 31. Juli, Seite 373-392 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:58 year:2021 number:2 day:31 month:07 pages:373-392 https://doi.org/10.1007/s11118-021-09943-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 58 2021 2 31 07 373-392 |
| allfieldsGer |
10.1007/s11118-021-09943-y doi (DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Shiozawa, Yuichi verfasserin aut Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract Let $(\mathcal {E},\mathcal {F})$ be a symmetric non-local Dirichlet from with unbounded coefficient on $L^{2}(\mathbb {R}^{d};\mathrm {d} x)$ defined by E(f,g)=∬ℝd×ℝd(f(y)−f(x))(g(x)−g(y))W(x,y)J(x,dy)dx,f,g∈F,$$ \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 |
| language |
English |
| source |
Enthalten in Potential analysis 58(2021), 2 vom: 31. Juli, Seite 373-392 volume:58 year:2021 number:2 day:31 month:07 pages:373-392 |
| sourceStr |
Enthalten in Potential analysis 58(2021), 2 vom: 31. Juli, Seite 373-392 volume:58 year:2021 number:2 day:31 month:07 pages:373-392 |
| format_phy_str_mv |
Article |
| institution |
findex.gbv.de |
| topic_facet |
Non-local Dirichlet form Markov semigroup Compactness Essential super Poincaré inequality |
| dewey-raw |
510 |
| isfreeaccess_bool |
false |
| container_title |
Potential analysis |
| authorswithroles_txt_mv |
Shiozawa, Yuichi @@aut@@ Wang, Jian @@aut@@ |
| publishDateDaySort_date |
2021-07-31T00:00:00Z |
| hierarchy_top_id |
165647787 |
| dewey-sort |
3510 |
| id |
OLC2133723781 |
| language_de |
englisch |
| fullrecord |
<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">OLC2133723781</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230506153108.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">230506s2021 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s11118-021-09943-y</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2133723781</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s11118-021-09943-y-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Shiozawa, Yuichi</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2021</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Nature B.V. 2021</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="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.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Non-local Dirichlet form</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Markov semigroup</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Compactness</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Essential super Poincaré inequality</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Wang, Jian</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Potential analysis</subfield><subfield code="d">Springer Netherlands, 1992</subfield><subfield code="g">58(2021), 2 vom: 31. Juli, Seite 373-392</subfield><subfield code="w">(DE-627)165647787</subfield><subfield code="w">(DE-600)33485-6</subfield><subfield code="w">(DE-576)032989911</subfield><subfield code="x">0926-2601</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:58</subfield><subfield code="g">year:2021</subfield><subfield code="g">number:2</subfield><subfield code="g">day:31</subfield><subfield code="g">month:07</subfield><subfield code="g">pages:373-392</subfield></datafield><datafield tag="856" ind1="4" ind2="1"><subfield code="u">https://doi.org/10.1007/s11118-021-09943-y</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_OLC</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OPC-MAT</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">58</subfield><subfield code="j">2021</subfield><subfield code="e">2</subfield><subfield code="b">31</subfield><subfield code="c">07</subfield><subfield code="h">373-392</subfield></datafield></record></collection>
|
| author |
Shiozawa, Yuichi |
| spellingShingle |
Shiozawa, Yuichi ddc 510 ssgn 17,1 misc Non-local Dirichlet form misc Markov semigroup misc Compactness misc Essential super Poincaré inequality Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
| authorStr |
Shiozawa, Yuichi |
| ppnlink_with_tag_str_mv |
@@773@@(DE-627)165647787 |
| format |
Article |
| dewey-ones |
510 - Mathematics |
| delete_txt_mv |
keep |
| author_role |
aut aut |
| collection |
OLC |
| remote_str |
false |
| illustrated |
Not Illustrated |
| issn |
0926-2601 |
| topic_title |
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 |
| topic |
ddc 510 ssgn 17,1 misc Non-local Dirichlet form misc Markov semigroup misc Compactness misc Essential super Poincaré inequality |
| topic_unstemmed |
ddc 510 ssgn 17,1 misc Non-local Dirichlet form misc Markov semigroup misc Compactness misc Essential super Poincaré inequality |
| topic_browse |
ddc 510 ssgn 17,1 misc Non-local Dirichlet form misc Markov semigroup misc Compactness misc Essential super Poincaré inequality |
| format_facet |
Aufsätze Gedruckte Aufsätze |
| format_main_str_mv |
Text Zeitschrift/Artikel |
| carriertype_str_mv |
nc |
| hierarchy_parent_title |
Potential analysis |
| hierarchy_parent_id |
165647787 |
| dewey-tens |
510 - Mathematics |
| hierarchy_top_title |
Potential analysis |
| isfreeaccess_txt |
false |
| familylinks_str_mv |
(DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 |
| title |
Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
| ctrlnum |
(DE-627)OLC2133723781 (DE-He213)s11118-021-09943-y-p |
| title_full |
Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
| author_sort |
Shiozawa, Yuichi |
| journal |
Potential analysis |
| journalStr |
Potential analysis |
| lang_code |
eng |
| isOA_bool |
false |
| dewey-hundreds |
500 - Science |
| recordtype |
marc |
| publishDateSort |
2021 |
| contenttype_str_mv |
txt |
| container_start_page |
373 |
| author_browse |
Shiozawa, Yuichi Wang, Jian |
| container_volume |
58 |
| class |
510 VZ 17,1 ssgn |
| format_se |
Aufsätze |
| author-letter |
Shiozawa, Yuichi |
| doi_str_mv |
10.1007/s11118-021-09943-y |
| dewey-full |
510 |
| title_sort |
compactness of semigroups generated by symmetric non-local dirichlet forms with unbounded coefficients |
| title_auth |
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 |
| collection_details |
GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT |
| container_issue |
2 |
| title_short |
Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients |
| url |
https://doi.org/10.1007/s11118-021-09943-y |
| remote_bool |
false |
| author2 |
Wang, Jian |
| author2Str |
Wang, Jian |
| ppnlink |
165647787 |
| mediatype_str_mv |
n |
| isOA_txt |
false |
| hochschulschrift_bool |
false |
| doi_str |
10.1007/s11118-021-09943-y |
| up_date |
2024-07-03T21:11:45.382Z |
| _version_ |
1803593822120181760 |
| fullrecord_marcxml |
<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">OLC2133723781</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230506153108.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">230506s2021 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s11118-021-09943-y</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2133723781</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s11118-021-09943-y-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Shiozawa, Yuichi</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2021</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Nature B.V. 2021</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="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.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Non-local Dirichlet form</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Markov semigroup</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Compactness</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Essential super Poincaré inequality</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Wang, Jian</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Potential analysis</subfield><subfield code="d">Springer Netherlands, 1992</subfield><subfield code="g">58(2021), 2 vom: 31. Juli, Seite 373-392</subfield><subfield code="w">(DE-627)165647787</subfield><subfield code="w">(DE-600)33485-6</subfield><subfield code="w">(DE-576)032989911</subfield><subfield code="x">0926-2601</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:58</subfield><subfield code="g">year:2021</subfield><subfield code="g">number:2</subfield><subfield code="g">day:31</subfield><subfield code="g">month:07</subfield><subfield code="g">pages:373-392</subfield></datafield><datafield tag="856" ind1="4" ind2="1"><subfield code="u">https://doi.org/10.1007/s11118-021-09943-y</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_OLC</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OPC-MAT</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">58</subfield><subfield code="j">2021</subfield><subfield code="e">2</subfield><subfield code="b">31</subfield><subfield code="c">07</subfield><subfield code="h">373-392</subfield></datafield></record></collection>
|
| score |
7.400216 |