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It is well known that, in the classical potential theory, a polar set is removable for Dirichlet finite harmonic functions (see, e.g., [1]). This result was extended by the author to the case of self-adjoint harmonic spaces in [5]. But, as remarked in Remark 3 in [5], there seems to be no known results of this type in the non-elliptic case, even for solutions of the heat equation. In the present note, we prove that, on harmonic spaces with adjoint structure, polar sets are removable for energy finite harmonic functions. In case the constant functions are harmonic, the energy coincides with the Dirichlet integral, so that our result implies the removability of heat polar sets for Dirichlet-finite solutions of the heat equation. Also, our proof provides a new proof to the classical result, which is quite different from the known proofs (cf. e.g., [1] and [5]). A preliminary abridged version of the present paper is given in the APPENDIX of [7].

Removability of polar sets for energy finite harmonic functions on harmonic spaces with adjoint structure