A Mean Value Property of Poly-Temperatures on a Strip Domain |
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Authors: | Nishio Masaharu; Shimomura Katsunori; Suzuki Noriaki |
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Institution: | Department of Mathematics, Osaka City University Sugimoto, Sumiyoshi, Osaka 558, Japan
Department of Mathematical Sciences, Ibaraki University Mito, Ibaraki 310, Japan
Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya 464-8602, Japan |
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Abstract: | We consider the iterates of the heat operator
on Rn+1={(X, t); X=(x1, x2, ..., xn) Rn, t R}. Let ![{Omega}](http://jlms.oxfordjournals.org/math/Omega.gif) Rn+1 be a domain,and let m 1 be an integer. A lower semi-continuous and locallyintegrable function u on is called a poly-supertemperatureof degree m if (H)mu 0 on (in the sense of distribution). If u and u are both poly-supertemperatures of degreem, then u is called a poly-temperature of degree m. Since His hypoelliptic, every poly-temperature belongs to C ( ), andhence (H)m u(X, t)=0 (X, t)![isin](http://jlms.oxfordjournals.org/math/isin.gif) . For the case m=1, we simply call the functions the supertemperatureand the temperature. In this paper, we characterise a poly-temperature and a poly-supertemperatureon a strip D={(X, t);X Rn, 0<t<T} by an integral mean on a hyperplane. To state our result precisely,we define a mean A·, ·]. This plays an essentialrole in our argument. |
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