Eigenvalue estimates for a class of operators related to self-similar measures

We obtain the sharp order of growth of the eigenvalue distribution function for the operator in the Sobolev space H^{X), generated by the quadratic form f^ H^, where X C R^ is a domain and p, is a probability self-similar fractal measure on X. Acknowledgement: Most of the results presented were obtained in cooperation with E. Verbitsky (1-dimensional case) and K. Naimark (d-dimensional case, d > 2); see [SV] and [NS]. 1. The notions of self-similar set and self-similar measure were introduced by Hutchinson [H]. Let S = {5i,..., Sm} be a set of contractive similitudes on E. , /ii,..., hm their coefficients of contraction. Also, let a system of positive numbers ("weights") p = {pi, ...,pm} be given, such that pi + + pm = 1 • Then there exists a unique non-empty compact set C = C{S} C R such that C = U^LI ^^A ^d a unique boundedly supported probability Borelian measure li = ^( 2, additional requirements on S and p are necessary. Denote Ak = h^pk , A(5, p) = maxAfc . Lemma. Let (2) be satisfied. Then the imbedding H^) C L^(^l,dp) is compact if and only if A ( < ? , p ) < l . (4) Necessity is quite straightforward and does not need (2). Sufficiency can be easily derived from [M], §8.8. Note that (2) and (4) imply that the Hausdorff dimension of ^ satisfies (3 :=dim^(5,p) > d2. xi-2 If (4) is satisfied, then Q^[u} generates in H^{X} a compact, self-adjoint and positive operator, say T^ or, in a more detailed notation, T^x . We are interested in the behaviour of its eigenvalues Afc(T^). As usual, we express this behaviour in terms of the corresponding distribution function n(̂ ) = #{fe : Afc(T^) > Q . If u € H^j (X) is an eigenfunction of T^ and A is the corresponding eigenvalue, then for any v e H^{X}, A / Vz& • Vvdx = / uvdp,. (5) Jx Jx For general self-similar p,, it is impossible to rewrite (5) in terms of any classical boundary value problem. Notice only that always u(x) = 0 outside C( 0 as the unique solution of m m E^-E^V^. (n k==l k=l Let X be an arbitrary open set in R^ such that X D fl.. Then there exists a number C=C(5,p,X) such that n(t,T^x) < Ct, any 00. (8) If, besides, (3) is satisfied, then there exist also c> 0 and to > 0, such that n{t,T^x) > ct~ , any < e ( 0 , < o ) . (9)