Image of Lp(ℝn) under the Hermite Semigroup
It is shown that the Hermite (polynomial) semigroup {e−tℍ:t>0} maps Lp(ℝn,ρ) into the space of holomorphic functions in Lr(ℂn,Vt,p/2(r+ϵ)/2) for each ϵ>0, where ρ is the Gaussian measure, Vt,p/2(r+ϵ)/2 is a scaled version of Gaussian measure with r=p if 1<p<2 and r=p′ if 2<p<∞ with...
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2008/287218 |
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| author | R. Radha D. Venku Naidu |
| author_facet | R. Radha D. Venku Naidu |
| author_sort | R. Radha |
| collection | DOAJ |
| description | It is shown that the Hermite (polynomial) semigroup {e−tℍ:t>0} maps Lp(ℝn,ρ) into the space of holomorphic functions in Lr(ℂn,Vt,p/2(r+ϵ)/2) for each ϵ>0, where ρ is the Gaussian measure, Vt,p/2(r+ϵ)/2 is a scaled version of Gaussian measure with r=p if 1<p<2 and r=p′ if 2<p<∞ with 1/p+1/p′=1. Conversely if F is a holomorphic function which is in a “slightly” smaller space, namely Lr(ℂn,Vt,p/2r/2), then it is shown that there is a function f∈Lp(ℝn,ρ) such that e−tℍf=F. However, a single necessary and sufficient condition is obtained for the image of L2(ℝn,ρp/2) under e−tℍ, 1<p<∞. Further it is shown that if F is a holomorphic function such that F∈L1(ℂn,Vt,p/21/2) or F∈Lm1,p(ℝ2n), then there exists a function f∈Lp(ℝn,ρ) such that e−tℍf=F, where m(x,y)=e−x2/(p−1)e4t+1e−y2/e4t−1 and 1<p<∞. |
| format | Article |
| id | doaj-art-6ae1239e3fc843a18b1bc402f5cb09bb |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-6ae1239e3fc843a18b1bc402f5cb09bb2025-02-03T05:57:32ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252008-01-01200810.1155/2008/287218287218Image of Lp(ℝn) under the Hermite SemigroupR. Radha0D. Venku Naidu1Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, IndiaDepartment of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, IndiaIt is shown that the Hermite (polynomial) semigroup {e−tℍ:t>0} maps Lp(ℝn,ρ) into the space of holomorphic functions in Lr(ℂn,Vt,p/2(r+ϵ)/2) for each ϵ>0, where ρ is the Gaussian measure, Vt,p/2(r+ϵ)/2 is a scaled version of Gaussian measure with r=p if 1<p<2 and r=p′ if 2<p<∞ with 1/p+1/p′=1. Conversely if F is a holomorphic function which is in a “slightly” smaller space, namely Lr(ℂn,Vt,p/2r/2), then it is shown that there is a function f∈Lp(ℝn,ρ) such that e−tℍf=F. However, a single necessary and sufficient condition is obtained for the image of L2(ℝn,ρp/2) under e−tℍ, 1<p<∞. Further it is shown that if F is a holomorphic function such that F∈L1(ℂn,Vt,p/21/2) or F∈Lm1,p(ℝ2n), then there exists a function f∈Lp(ℝn,ρ) such that e−tℍf=F, where m(x,y)=e−x2/(p−1)e4t+1e−y2/e4t−1 and 1<p<∞.http://dx.doi.org/10.1155/2008/287218 |
| spellingShingle | R. Radha D. Venku Naidu Image of Lp(ℝn) under the Hermite Semigroup International Journal of Mathematics and Mathematical Sciences |
| title | Image of Lp(ℝn) under the Hermite Semigroup |
| title_full | Image of Lp(ℝn) under the Hermite Semigroup |
| title_fullStr | Image of Lp(ℝn) under the Hermite Semigroup |
| title_full_unstemmed | Image of Lp(ℝn) under the Hermite Semigroup |
| title_short | Image of Lp(ℝn) under the Hermite Semigroup |
| title_sort | image of lp rn under the hermite semigroup |
| url | http://dx.doi.org/10.1155/2008/287218 |
| work_keys_str_mv | AT rradha imageoflprnunderthehermitesemigroup AT dvenkunaidu imageoflprnunderthehermitesemigroup |