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Green’s formula and a characterization of the harmonic functions withBMO traces

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Riassunto

Abbiamo recentemente caratterizzato (si veda [1]) lo spazio di tutte le funzioni armonicheu(x,t) suR n×(0, ∞) che hanno traccia (come limite verticale, o non tangenziale) nello spazio delle funzioni con oscillazione media limitata. Diamo qui una nuova e più esplicita dimostrazione di quel risultato. Essa, essendo più libera da certi procedimenti speciali di analisi armonica, si può adattare a corrispondenti caratterizzazioni per le soluzioni di problemi ben posti di varie equazioni a derivate parziali. In [4] per esempio abbiamo dato un risultato analogo per il problema di valori iniziali per l’equazione del calore con dati iniziali inBMO(R n).

Summary

We have recently characterized (see ref. [1]) the space of all harmonic functionsu(x,t) onR n×(0, ∞) having traces (as vertical, or non-tangential limits)f(x) inBMO(R n), the space of functions with bounded mean oscillation. We give here a new and more explicit proof of that result which, being freer of certain «special features» of harmonic analysis, is adaptable to corresponding characterizations for the solutions of well-posed problems in various partial differential equations. In [4], for example, we have given an analogous result for the initial value problem for the heat equation, with initial data inBMO(R n).

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References

  1. E. Fabes—R. Johnson—U. Neri,Spaces of harmonic functions representable by Poisson integrals of functions in BMO and ℑ p, δ , Indiana Univ. Math. Journal (to appear).

  2. C. Fefferman—E. Stein,H p spaces of several variables, Acta Math.,129 (1972), pp. 137–193.

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  3. E. Stein—G. Weiss,Introduction to Fourier analysis on Euclidean spaces, Princeton Press, New Jersey, 1971.

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  4. E. Fabes—U. Neri,Characterization of temperatures with initial data in BMO, in corso di stampa sul Duke Math. Jr., 1975.

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Author notes

  1. U. Neri

    Present address: Department of Mathematics, University of Maryland, 20742, College Park, Maryland, USA

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    1. E. B. Fabes
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    2. R. L. Johnson
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    3. U. Neri
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    Fabes, E.B., Johnson, R.L. & Neri, U. Green’s formula and a characterization of the harmonic functions withBMO traces. Ann. Univ. Ferrara 21, 147–157 (1975). https://doi.org/10.1007/BF02826787

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    • DOI: https://doi.org/10.1007/BF02826787

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