Let Φ be a real valued function of one real variable, let L denote an elliptic second order formally self-adjoint differential operator with bounded measurable coefficients, and let P stand for the Poisson operator for L. A necessary and sufficient condition on Φ ensuring the equivalence of the Dirichlet integrals of Φ ◦ Ph and P(Φ ◦ h) is obtained. We illustrate this result by some sharp inequalities for harmonic functions. Bibliography: 1 title.
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V. Maz’ya, B. Plamenevskii, “On the asymptotics of the fundamental solutions of elliptic boundary value problems in regions with conical points” [in Russian], Probl. Mat. Anal. 7, 100–145 (1979); Engl. transl.: Sel. Math. Sov. 4, 363–397 (1985).
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Translated from Problems in Mathematical Analysis 43, November 2009, pp. 97–105.
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Cialdea, A., Maz’ya, V. A quasicommutativity property of the poisson and composition operators. J Math Sci 164, 415–426 (2010). https://doi.org/10.1007/s10958-009-9755-x
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DOI: https://doi.org/10.1007/s10958-009-9755-x