Expansion of certain parabolic means (Q793904)

The author considers analogues, for the heat equation, of the classical formulae of Pizetti for the surface and volume means of a harmonic function. The means he discusses are the weighted volume means, featured in his earlier papers [Dokl. Akad. Nauk SSSR 242, 529-532 (1978; Zbl 0436.35040); ibid. 252, 296-301 (1980; Zbl 0484.35051); Mat. Zametki 29, 211-223 (1981; Zbl 0465.35042)] along with related surface means. However, he seems unaware that the principal one of the surface means \((m=0)\) was known to \textit{B. Pini} [Ann. Mat. Pura Appl., IV. Ser. 37, 249-264 (1954; Zbl 0057.327)] in the case of one space variable, and to \textit{W. Fulks} [Proc. Am. Math. Soc. 17, 6-11 (1966; Zbl 0152.105)] in the general case. The principal volume mean was derived by the reviewer [Proc. Lond. Math. Soc., III. Ser. 26, 385-417 (1973; Zbl 0253.35045)], who also obtained results mentioned in the paper under review, with weaker smoothness hypotheses [loc. cit.; J. Lond. Math. Soc., II. Ser. 7, 195-198 (1973; Zbl 0271.35033)]. Both means have since been used or studied by various authors, including \textit{L. C. Evans} and \textit{R. F. Gariepy} [Arch. Ration. Mech. Anal. 78, 293-314 (1982; Zbl 0508.35038)] and \textit{H. Bauer} [C. R. Acad. Sci. Paris, Sér. I. 298, 165-167 (1984); Ann. Acad. Sci. Fenn., Sér. AI (to appear)].