INTEGRABILITY OF SUPERHARMONIC FUNCTIONS IN A OHN OMAIN Hiroaki Aikawa Abstract. The integrability of positive erharmonic functions on a bounded fat ohn domain is established. No exterior conditions are assumed. For a general bounded ohn domain the L p -integrability is proved with the estimate of p in terms of the ohn constant.. Introduction Let be a bounded domain in R n with n 2. By S + () we denote the family of all positive erharmonic functions in. Armitage [5, 6] proved that S + () L p () for 0 < p < n/(n ), provided is smooth. This result was extended by Maeda-Suzuki [] to a Lipschitz domain. They gave an estimate of p in terms of Lipschitz constant. Their estimate has the correct asymptotic behavior: p n/(n ) as the Lipschitz constant tends to 0. As a result they showed that S + () L p () for 0 < p < n/(n ), provided is a C domain. Masumoto [2, 3] succeeded in obtaining the sharp value of p for planar domains bounded by finitely many ordan curves. For the higher dimensional case Aikawa [] gave the sharp value of p for Lipschitz domains with the aid of the coarea formula and the boundary Harnack principle. On the other hand, Stegenga-Ullrich [6] treated very non-smoot domains, such as ohn domains and domains satisfying the quasihayperbolic boundary condition [7, 3.6], which are called Hölder domains by Smith-Stegenga [5]. Let δ (x) = dist(x, ) and x 0. We say that is a ohn domain with ohn constant c > 0 if each x can be joined to x 0 by a rectifiable curve γ such that (.) δ (ξ) c l(γ(x, ξ)) for all ξ γ, where γ(x, ξ) is the subarc of γ from x to ξ and l(γ(x, ξ)) is the length of γ(x, ξ). A ohn domain may be visualized as a domain satisfying a twisted cone condition. The quasi-hyperbolic metric k (x, x 2 ) is defined by k (x, x 2 ) = inf γ γ ds δ (x), 99 Mathematics Subject Classification. 3B05. This work was ported in part by Grant-in-Aid for Scientific Research (B) (No. 09440062), apanese Ministry of Education, Science and Culture. Typeset by AMS-TEX
2 HIROAKI AIKAWA where the infimum is taken over all rectifiable arcs γ joining x to x 2 in. We say that satisfies a quasi-hyperbolic boundary condition if there are positive constants A and A 2 such that ( ) k (x, x 0 ) A log + A 2 for all x. δ (x) Smith-Stegenga [5] called a domain satisfying the quasihayperbolic boundary condition a Hölder domain. It is easy to see that a ohn domain satisfies the quasihyperbolic boundary condition (see [7, Lemma 3.]). Stegenga-Ullrich [6] proved that S + () L p () with small p > 0 for a domain satisfying the quasihayperbolic boundary condition. Lindqvist [0] extends the result to positive ersolutions of certain nonlinear elliptic equations, such as the p-laplace equation. Gotoh [8] also studies L p -integrability. Unfortunately, their p > 0 is very small and it does not seem that p is obtained by their methods. The main aim of the present paper is to show that S + () L () for a fat ohn domain. Theorem. Let be a bounded ohn domain with ohn constant c 2 n. Then S + () L (). The above bound 2 n is not sharp. For more specific ohn domains we obtain the sharp bound. We say that satisfies the interior cone condition with aperture ψ, 0 < ψ < π/2, if for each point x there is a truncated cone with vertex at x, aperture ψ and a fixed radius lying in. Obviously, a domain satisfying the interior cone condition with aperture ψ is a ohn domain with ohn constant sin ψ. Theorem 2. Let be a bounded domain satisfying the interior cone condition with aperture ψ with cos ψ > / n. Then S + () L (). For a slim ohn domain we will show S + () L p () for some 0 < p < with the estimate of p. This will give a larger p than that in Stegenga-Ullrich [6]. For details see Section 3. In the previous paper [2], the above theorems are obtained with additional assumption: the capacity density condition (CC). See [3] for more illustrations. For the 2-dimensional case CC is equivalent to the uniform perfectness of the boundary; planar domains bounded by finitely many ordan curves satisfy CC. Thus all the known results for S + () L p () with p required CC or some other stronger exterior condition. The above Theorems and 2 first establish S + () L () for a domain satisfying only an interior condition. Recently, Gustafsson, Sakai and Shapiro [9] considered the L -integrability in connection with quadrature domains. They showed that if is a quadrature domain and the Green functions does not decay so fast near the boundary, then S + () L () ([9, Corollary 5.4]). 2. Proof of Theorems For an open set U we denote by G U the Green function for U. Throughout this section is a bounded ohn domain or a domain satisfying an interior condition. For
INTEGRABILITY OF SUPERHARMONIC FUNCTIONS IN A OHN OMAIN 3 simplicity we press the subscript and write G for the Green function for. Moreover, x 0 is a fixed point and let g(x) = G(x, x 0 ). By the symbol A we denote an absolute positive constant whose value is unimportant and may change from line to line. If necessary, we use A, A 2,..., to specify them. We shall say that two positive functions f and f 2 are comparable, written f f 2, if and only if there exists a constant A such that A f f 2 Af. The constant A will be called the constant of comparison. The proof of Theorems uses the following lower estimate of the Green function. For 0 < c < we let [ (2.) α = log c ( + c ) n ]/ log( c 2 ). We observe that lim α = as c. Let c n be the solution of the equation ( + t) n+ ( t) = for 0 < t <. Then α = 2 for c = c n and < α < 2 for c n < c <. We see that n/(n + 2) < c n < 2 n. Lemma. (see [2, Lemma 2]) (i) If is a ohn domain with ohn constant c, then g(x) Aδ (x) α. (ii) If satisfies the interior cone condition with aperture ψ with cos ψ > / n, then there is < α(ψ) < 2 such that g(x) Aδ (x) α(ψ). Theorems and 2 readily follows from Lemma and the following. Theorem 3. Let be a ohn domain and pose g(x) Aδ (x) α for α > 0. For ε > 0 let V = (min{g, }) ε 2/α. Then u(x)v (x)g(x)dx Au(x 0 ) for any u S + (), where A is independent of u S + (). Moreover, if 0 < α < 2, then S + () L (). We need one of the main results in [2]. efine the Green capacity Cap U (E) for E U by Cap U (E) = {µ(e) : G U µ on U, µ is a Borel measure ported on E}. By B(x, r) we denote the open ball with center at x and radius r. Lemma 2. ( [2, Theorem ]) Let 0 < η <. Then for an open set U with Green function G U G U (x, y)dy Aw η (U) 2, x U U where { w η (U) = inf ρ > 0 : Cap } B(x,2ρ)(B(x, ρ) \ U) η for all x U. Cap B(x,2ρ) (B(x, ρ)) The above quantity w η (U) is called the capacitary width of U. The definition of ohn domain readily implies the following.
4 HIROAKI AIKAWA Lemma 3. Let be a ohn domain. Then w η ({x : δ (x) r}) Ar. The following estimate of the Green potential is called the basic estimate [4, Theorem 3]. Lemma 4. Let u be a positive continuous erharmonic function on. For an integer j we put j = {x : 2 j < u(x) < 2 j+2 } and let G j be the Green function for j. If f is a nonnegative measurable function on, then x u(x) G(x, y)f(y)dy 4 x j u(x) j G j (x, y)f(y)dy. Proof of Theorem 3. Apply Lemma 4 to u = g and f = V g to obtain x g(x) G(x, y)v (y)g(y)dy 32 x j j G j (x, y)v (y)dy, where j = {x : 2 j < g(x) < 2 j+2 }. Since g(x) Aδ (x) α, it follows that j {x : δ (x) A2 j/α } and hence from Lemmas 2 and 3 that 0 x j j G j (x, y)v (y)dy A 0 (2 j ) ε 2/α (2 j/α ) 2 A 0 2 εj <. On the other hand, if j, then j B(x 0, A2 j/(2 n) ) if n 3 and j B(x 0, exp( A2 j )) if n = 2. Hence Lemma 2 implies Thus x j G j (x, y)v (y)dy = G j (x, y)dy j x j j A 2 2j/(2 n) < if n 3, A exp( A2 j ) < if n = 2. G(x, y)v (y)g(y)dy Ag(x) = AG(x, x 0 ). Integrate the above inequality with respect to dµ(x) and use Fubini s theorem. Then we have u(y)v (y)g(y)dy Au(x 0 )
INTEGRABILITY OF SUPERHARMONIC FUNCTIONS IN A OHN OMAIN 5 with u = Gµ. Every u S + () can be approximated from below by a Green potential, so that the monotone convergence theorem proves the first assertion. Finally, pose 0 < α < 2. Let ε = + 2/α > 0 and observe that V g and udx Au(x 0) for u S + (). If u(x 0 ) <, then u L () obviously. If u(x 0 ) =, then replace u by its Poisson integral over a small ball with center at x 0. The replaced function belongs to S + () and its value at x 0 is finite, so that it belongs to L () by the previous observation. This, together with the local integrability of u, proves u L (). 3. L p -integrability For a bounded ohn domain with ohn constant smaller than that in Theorem, we shall obtain L p -integrability of positive erharmonic functions with 0 < p <. The exponent p will be estimated in terms of ohn constant. To this end we show the following lemma, which is inspired by [4, Theorem 4]. Lemma 5. Let be a bounded ohn domain with ohn constant c. Then there is a positive constant τ depending only on c and the dimension n such that δ (x) τ dx < for 0 < τ < τ. Here τ can be estimated as τ log( + (c /20) n ). log 2 Proof. Let j = {x : 2 j δ (x) < 2 j }. Then j=j 0 j is a disjoint decomposition of with some j 0. Observe that j=j 0 j = <, where j denotes the volume of j. Without loss of generality we may assume that j 0 = 0 and x 0 0. Suppose x i=j+ i with j, i.e., δ (x) < 2 j. By definition there is a rectifiable curve γ connecting x and x 0 with (.). We find a point ξ γ such that δ (ξ) = 2 j. By (.) 2 j = δ (ξ) c l(γ(x, ξ)) c x ξ, so that x ξ c 2 j. Hence i=j+ i δ (ξ)=2 j C(ξ, c 2 j ), where C(ξ, c 2 j ) is the closed ball with center at ξ and radius c 2 j. Suppose for a moment δ (ξ) = 2 j. Then, by definition, there is a point x ξ such that x ξ ξ = 2 j. Let ξ be the point on the line segment x ξ ξ with ξ ξ = 2 j. Then an elementary geometrical observation shows that δ (ξ ) = 2 (2 j + 2 j ) and B(ξ, 2 j 2 ) j, so that (3.) j C(ξ, c 2 j ) A 0 (2 j 2 ) n = ( c ) n C(ξ, 5c 20 2 j ),
6 HIROAKI AIKAWA where A 0 is the volume of a unit ball. By the covering lemma (see e.g. [7, Theorem.3.]) we can find ξ k such that δ (ξ k ) = 2 j, {C(ξ k, c 2 j )} k is disjoint and i C(ξ k, 5c 2 j ). i=j+ k In view of (3.) we have i ( ) n 20 ( ) n C(ξ k, 5c 2 j ) c 20 j C(ξ k, c 2 j ) i=j+ c j. k k Multiply the above inequalities by r j and take the summation for j =,..., N, where r > is a constant to be determined. Then ( ) n 20 r j min{n,i} j r j i = r j i so that c Letting N, we obtain > i=j+ i=2 r r i r r = r r i r j i = = r ( r i Let < r < + (c /20) n. Then ( r and the above inequality implies We observe that whence This proves the lemma. r i i ( 20 c i=2 i=2 ( r ( ) n ) 20 > 0 c r i i <. r i r r i r r i, ) n ) r i i. ( ) n ) 20 r i i. c r i δ (x) log r/ log 2 for x i, δ (x) log r/ log 2 dx <.
INTEGRABILITY OF SUPERHARMONIC FUNCTIONS IN A OHN OMAIN 7 Theorem 4. Let be a bounded ohn domain with ohn constant c. Suppose g(x) Aδ (x) α for x with α 2 and (3.2) δ (x) τ dx < with τ > 0. Then S + () L p () for 0 < p < τ/(α 2 + τ). Remark. Let α be as in (2.) and let τ be as in Lemma 5. If α 2, then Lemmas and 5 show that S + () L p () with 0 < p < p = τ /(α 2 + τ ). Observe that p c n+ as c 0; p as c c n. Proof. Let 0 < p < τ/(α 2 + τ). Put ε = α ( ( p)τ p ) α + 2. Then ε > 0 and α(ε 2/α + )p/( p) = τ. Let = {x : g(x) }. Take u S + (). Then Hölder s inequality and Theorem 3 yield ( ) p ( u p dx ug ε 2/α+ dx g (ε 2/α+)p/( p dx ( ) p Au(x 0 ) p δ τ dx Au(x 0 ) p, ) p where (3.2) is used in the last inequality. By the same reasoning as in the proof of Theorem 3, we have u p dx <. This, together with the local integrability of a erharmonic function, proves the theorem. References [] H. Aikawa, Integrability of erharmonic functions and subharmonic functions, Proc. Amer. Math. Soc. 20 (994), 09 7. [2] H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of erharmonic functions (to appear). [3] H. Aikawa, Norm estimate for the Green operator with applications, Proceedings of Essén Conference at Uppsala in une 997 (to appear). [4] H. Aikawa and M. Murata, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains,. Analyse Math. 69 (996), 37 52. [5]. H. Armitage, On the global integrability of erharmonic functions in balls,. London Math. Soc. (2) 4 (97), 365 373. [6]. H. Armitage, Further result on the global integrability of erharmonic functions,. London Math. Soc. (2) 6 (972), 09 2. [7] F. W. Gehring and O. Martio, Lipschitz classes and quasiconformally homogeneous domains, Ann. Acad. Sci. Fenn. 0 (985), 203 29. [8] Y. Gotoh, Integrability of erharmonic functions, uniform domains, and Hölder domains, Proc. Amer. Math. Soc. (to appear).
8 HIROAKI AIKAWA [9] B. Gustafsson, M. Sakai and H. S. Shapiro, On domains in which harmonic functions satisfy generalized mean value properties, Potential Analysis 7 (997), 467 484. [0] P. Lindqvist, Global integrability and degenerate quasilinear elliptic equations,. Analyse Math. 6 (993), 283 292. [] F.-Y. Maeda and N. Suzuki, The integrability of erharmonic functions on Lipschitz domains, Bull. London Math. Soc. 2 (989), 270 278. [2] M. Masumoto, A distortion theorem for conformal mappings with an application to subharmonic functions, Hiroshima Math.. 20 (990), 34 350. [3] M. Masumoto, Integrability of erharmonic functions on plane domains,. London Math. Soc. (2) 45 (992), 62 78. [4] W. Smith and. A. Stegenga, Hölder domains and Poincaré domains, Trans. Amer. Math. Soc. 39 (990), 67 00. [5] W. Smith and. A. Stegenga, Exponential integrability of the quasi-hyperbolic metric on Hölder domains, Ann. Acad. Sci. Fenn. 6 (99), 345 360. [6]. A. Stegenga and. C. Ullrich, Superharmonic functions in Hölder domains, Rocky Mountain. Math. 25 (995), 539 556. [7] W. P. Ziemer, Weakly ifferentiable Functions, Springer-Verlag, 989. epartment of Mathematics, Shimane University, Matsue 690, apan E-mail address: haikawa@riko.shimane-u.ac.jp