Perturbation of harmonic structures and an index-zero theorem
Annales de l'Institut Fourier, Volume 20 (1970) no. 1, p. 317-359
In the framework of an axiomatic theory of sheaves of “harmonic” functions, a notion of perturbation of these sheaves is introduced which corresponds to the replacement of the operator Δ by an operator Δ+f, in the classical situation. The “harmonic” functions with which the paper is concerned are assumed to satisfy certain hypotheses (weaker than the axioms of Bauer); it is shown that the perturbed harmonic functions also satisfy these hypotheses. Moreover, the results obtained imply that the dimensions of the spaces H 0 (W,H) and H 1 (W,H) are (finite and) equal whenever the base space W of a sheaf H satisfying these hypotheses is compact. That fact generalizes the classical theorem that the index of any second order elliptic operator on a (trivial bundle over a) compact manifold is zero. Further, it implies that whenever H satisfies the Brelot axioms and its adjoint sheaf H * exists, the spaces H W and H W * (where W is again compact) have the same (finite) dimension.
Dans le cadre d’une théorie des faisceaux de fonctions “harmoniques”, on introduit une notion de perturbation de ces faisceaux, qui correspond au remplacement de l’opérateur Δ par Δ+f . dans la théorie classique. Les faisceaux pris au point de départ satisfont à certaines hypothèses, plus faibles que les axiomes de Bauer, et on trouve que les faisceaux perturbés satisfont encore à ces mêmes hypothèses. Les résultats entraînent la finitude et l’égalité des dimensions des espaces H 0 (W,H) et H 1 (W,H), dans le cas où la base W du faisceau H est compacte. Ceci est une généralisation du théorème classique qui dit que l’indice d’un opérateur elliptique du second ordre sur une variété compacte est nul. Comme conséquence, on trouve que les espaces linéaires H W et H W * (où W est encore compacte) ont la même dimension finie, si le faisceau H satisfait aux axiomes de Brelot et si son adjoint existe.
@article{AIF_1970__20_1_317_0,
     author = {Walsh, Bertram},
     title = {Perturbation of harmonic structures and an index-zero theorem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {20},
     number = {1},
     year = {1970},
     pages = {317-359},
     doi = {10.5802/aif.344},
     mrnumber = {43 \#554},
     zbl = {0187.04303},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1970__20_1_317_0}
}
Perturbation of harmonic structures and an index-zero theorem. Annales de l'Institut Fourier, Volume 20 (1970) no. 1, pp. 317-359. doi : 10.5802/aif.344. https://aif.centre-mersenne.org/item/AIF_1970__20_1_317_0/

[1] H. Bauer, Harmonische Räume und ihre Potentialtheorie, Springer Lecture Notes in Mathematics 22 (1966). | Zbl 0142.38402

[2] N. Boboc, C. Constantinescu and A. Cornea, Axiomatic theory of harmonic functions. Nonnegative superharmonic functions, Ann. Inst. Fourier (Grenoble), 15 (1965), 283-312. | Numdam | MR 33 #1476 | Zbl 0139.06604

[3] M. Brelot, Lectures on Potential Theory, Tata Institute, Bombay, 1960. | MR 22 #9749 | Zbl 0098.06903

[4] C.H. Dowker, Lectures on Sheaf Theory, Tata Institute, Bombay, 1957.

[5] N. Dunford and J.T. Schwartz, Linear Operators I, Interscience, New York, 1958. | MR 22 #8302 | Zbl 0084.10402

[6] R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR 31 #4927 | Zbl 0141.08601

[7] R.M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble), 12 (1962), 415-571. | Numdam | MR 25 #3186 | Zbl 0101.08103

[8] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Göttingen-Heidelberg, 1966. | MR 34 #3324 | Zbl 0148.12601

[9] P.A. Meyer, Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Ann. Inst. Fourier (Grenoble), 13 (1963), 357-372. | Numdam | MR 29 #260 | Zbl 0116.30404

[10] H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. | MR 33 #1689 | Zbl 0141.30503

[11] B. Walsh, Flux in axiomatic potential theory. I : Cohomology, Inventiones Math. 8 (1969), 175-221. | MR 42 #532 | Zbl 0179.15203

[12] B. Walsh, Flux in axiomatic potential theory. II : Duality, Ann. Inst. Fourier, (Grenoble), 19 (1969). | Numdam | MR 42 #2023 | Zbl 0181.11703