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Abstract
We study Hardy spaces of the conjugate Beltrami equation
over Dini-smooth finitely connected domains, for real contractive
with
, in the range
. We develop a theory of conjugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation
Acknowledgments
The research of the authors was partially supported by grant AHPI (ANR-07-BLAN-0247) and the “Region PACA”. They are grateful to S.B. Klimentov for bringing to their attention references [Citation6, Citation7, Citation19] while the present article was under review.
Notes
1. Indeed, any finitely connected domain is conformally equivalent to a domain whose boundary consists of circles or points [Citation49, Sec. V.6, Thm 2]; if the complement is infinite there is at least one circle whose interior can be mapped onto by a Möbius tranform.
2. i.e. the inductive topology of its subspaces comprised of functions whose support lies in a compact set
, each
being topologized by uniform convergence of a function and all its partial derivatives [Citation50, Sec. I.2]).
3. These are topologized by the family of semi-norms and
, respectively, with
a nested family of relatively compact open subset exhausting
.
4. The Hardy class is defined by the condition that has a harmonic majorant; the two classes coincide as soon as harmonic measure and arclength are comparable up to a multiplicative constant on
[Citation22, Ch. 10], [Citation21], which is the case for Dini-smooth domains thanks to Lemma A.1, Appendix A.
5. These works do not mention the case of unbounded domains, but it requires no change as we just stressed. The paper [Citation11] restricts to analytic boundaries, which is also unnecessary thanks to Lemma A.1.
6. There is no discrepancy in the notation since the nontangential limit coindes with the Sobolev trace whenever it exists [Citation10, Prop. 4.3.3].
7. is the unique harmonic function in
such that
is bounded in a neighborhood of
and
when
, see [Citation25, Sec. 4.4.].
8. Lemma 4 in that reference may have a problem, for the constant in the Sobolev embedding theorem for may blow up as
goes to 0; however, this has no consequence since Lemma 3 there easily implies boundedness of
for small
.
9. In this reference, Hardy spaces are defined through harmonic majorants, but we know this is equivalent to the definition based on (Equation10) for Dini-smooth domains.
10. The condition is that the sup on the Carleson domain of
, when viewed as a function of
, should be the density of a Carleson (or vanishing Carleson) measure. Now, for
the characteristic function of
,
, the radial function
lies in
for each
. If we put
, the corresponding density is not even integrable on Carleson domains.
11. This expresses that is comparable to harmonic measure on
.
12. From the point of view of regularity theory, and though we deal with two dimensions and scalar conductivity only, it is noteworthy that our assumptions are not covered by the Carleson condition set up in [Citation32, Citation33].
13. Indeed, since along any curve, its integral over a cycle
is zero by Green’s formula (Equation6) (applied with
on the domain bounded by
and all the
located inside
) because
for all
by construction, see [Citation47, Sec. 4.6].
14. Remember summability is understood with respect to area measure on the sphere.