Some properties of solutions of the heat equation
Abstract
In this paper we investigate some properties of solutions of the heat equation. Their basic properties are established in [3]. Our object is to prove some partial distribution function inequalities for the area integral which can be used to study the local and the global behavior of solutions of the heat equation. Theorem 3 shows that the area integral A and the nontangential maximal function N are remarkably closely related. The method used in this paper is based on the treatment of analogous problems for harmonic functions in [1].
References
2. A. Friedman, Partial differential equations of parabolic type, Prentice - Hall Englewood Cliffs, 1964.
3. J.R. Hattemer, Boundary behavior of temperatures I, Studia Math. 25 (1964), 111-155.
4. И.Г. Пeтpoвcкий, Лeкции oб ypaвнeниях c чacтными пpoизвoдными, Mocквa, 1953.
5. E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N. J., 1970.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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