## Integrability of Green potentials in fractal domains.(English)Zbl 0860.31002

Let $$Gf(x)$$ be the Green potential of a function $$f(x)$$ on an open connected and bounded set $$\Omega$$ in $$\mathbb{R}^n$$. The paper is devoted to the extension of the inequality $\Biggl( \int_\Omega |\nabla Gf|^q dx\Biggr)^{1/q}\leq c(\Omega,p) \Biggl( \int_\Omega |f|^p dx\Biggr)^{1/p} \tag{1}$ with $$1/q= 1/p-1/n$$, $$n/(n-1)< q< \infty$$, well known in case of a sufficiently smooth boundary $$\partial\Omega$$, to the case of a “highly non-rectifiable” boundary (the case of a Lipschitzian boundary was treated by B. Dahlberg [Math. Scand. 44, 149-170 (1979; Zbl 0418.31003)]). The restrictions on $$\Omega$$ are given in terms of the so called non-tangentially accessible (NTA) domains, this notion being developed by D. S. Jerison and C. E. Kenig [Adv. Math. 46, 80-147 (1982; Zbl 0514.31003)]. The restrictions on the exponent $$q$$ for (1) to be valid are given in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.
Reviewer: S.G.Samko (Faro)

### MSC:

 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 31B25 Boundary behavior of harmonic functions in higher dimensions

### Citations:

Zbl 0418.31003; Zbl 0514.31003
Full Text:

### References:

 [1] Adams, D. R. andHedberg, L. I.,Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg, 1996. [2] Ahlfors, L., Quasiconformal reflection.Acta Math. 109 (1963), 291–301. · Zbl 0121.06403 [3] Burkholder, D. L. andGundy, R., Distribution function inequalities for the area integral,Studia Math. 44 (1972), 527–544. · Zbl 0219.31009 [4] Coifman, R. R. andFefferman, C., Weighted norm inequalities for maximal functions and singular integrals,Studia Math. 51 (1974), 241–250. · Zbl 0291.44007 [5] Dahlberg, B. E. J., Estimates of harmonic measures,Arch. Rational Mech. Anal. 65 (1977), 149–179. · Zbl 0406.28009 [6] Dahlberg, B. E. J.,L q -estimates for Green potentials in Lipschitz domains,Math. Scand. 44 (1979), 149–170. · Zbl 0418.31003 [7] Dahlberg, B. E. J., On the Poisson integral for Lipschitz andC 1-domains,Studia Math. 66 (1979), 13–24. · Zbl 0422.31008 [8] Dahlberg, B. E. J., Weighted norm inequalities for the Lusin area integral and the non-tangential maximal function for harmonic functions in a Lipschitz domain,Studia Math. 67 (1980), 297–314. · Zbl 0449.31002 [9] Gehring, F. W.,Characteristic Properties of Quasidisks, Sém. Math. Sup.84, Univ. Montréal, Montréal, Que., 1982. · Zbl 0495.30018 [10] Gehring, F. W. andVäisälä, J., Hausdorff dimension and quasiconformal mappings,J. London Math. Soc. (2)6 (1973), 504–512. · Zbl 0258.30020 [11] de Guzman, M.,Real Variable Methods in Fourier Analysis, Notas Mat.75, North-Holland Math. Studies46, North-Holland, Amsterdam-New York, 1981. · Zbl 0449.42001 [12] Hedberg, L. I., Spectral synthesis in Sobolev spaces and uniqueness of solutions of the Dirichlet problem,Acta Math. 147 (1981), 237–264. · Zbl 0504.35018 [13] Helms, L. L.,Introduction to Potential Theory, John Wiley & Sons, New York, 1969. · Zbl 0188.17203 [14] Jerison, D. S. andKenig, C. E., Boundary behaviour of harmonic functions in non-tangentially accessible domains.Adv. in Math. 46 (1982), 80–147. · Zbl 0514.31003 [15] Jones, P. W., Extension theorems for BMO,Indiana J. Math. 29 (1980), 41–66. · Zbl 0432.42017 [16] Jones, P. W., Quasiconformal mappings and extendability of functions in Sobolev spaces,Acta Math. 47 (1981), 71–88. · Zbl 0489.30017 [17] Jones, P. W., A geometric localization theorem,Adv. in Math. 46 (1982), 71–79. · Zbl 0515.31004 [18] Maz’ya, V. G.,Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg, 1985. [19] Maz’ya, V. G. andHavin, V. P., Non-linear potential theory,Uspekhi Mat. Nauk 27:6 (1972), 67–138 (Russian). English transl.:Russian Math. Surveys 27:6 (1972), 71–148. [20] Muckenhoupt, B. andWheeden, R. L., Weighted norm inequalities for fractional integrals,Trans. Amer. Math. Soc. 192 (1974), 261–274. · Zbl 0289.26010 [21] Nyström, K.,Smoothness Properties of Dirichlet Problems in Domains with a Fractal Boundary, Ph. D. Dissertation, Umeå, 1994. [22] Stein, E. M.,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. · Zbl 0207.13501 [23] Väisälä, J.,Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin-Heidelberg, 1971. · Zbl 0221.30031 [24] Widman, K.-O., Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations,Math. Scand. 21 (1967), 17–37. · Zbl 0164.13101 [25] Wu, J. M., Content and harmonic measure; An extension of Hall’s lemma,Indiana Univ. Math. J. (2)36 (1987), 403–420. · Zbl 0639.31004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.