Weighted Polar Cauchy-Riemann Operators and Hardy Space Theory on Annular Domains

Abstract

Let A(r,R)={z?C:r<|z|

Introduction

The text introduces a new approach to studying weighted Hardy spaces on an annulus, a doubly connected domain in complex analysis. Unlike the unit disk, an annulus has two boundary circles and a more complex automorphism group. The main objective is to investigate how a positive radial weight affects the Cauchy–Riemann equation itself, rather than merely modifying the norm or integration measure used to define a function space.

The paper defines a weighted Cauchy–Riemann operator, ∂‾w∂w?, in which the radial derivative is multiplied by a positive continuous weight w(ρ)w(ρ). Through a radial change of variables, the study demonstrates that the solutions of this weighted equation correspond to ordinary holomorphic functions on a transformed annulus. This result provides the foundation for developing weighted Hardy spaces and extending classical boundary-value theory to the weighted setting.

The study further investigates the mathematical properties of these spaces, including integral means, convexity, boundary behavior, and completeness. It also examines composition operators induced by annulus automorphisms, multiplication operators of Toeplitz type, and solutions to the inhomogeneous weighted Cauchy–Riemann equation. The annulus's compact automorphism group allows explicit descriptions of composition-operator spectra, which differ from the more complicated behavior often found in the unit disk.

The literature review distinguishes this approach from classical Hardy-space theory and existing weighted Hardy and Bergman spaces. Earlier research, including work by Sarason, established Hardy-space theory on annuli, while other researchers investigated weighted spaces in which the weight modifies the integration measure. In contrast, the present study incorporates the weight directly into the differential operator, effectively transforming the underlying notion of holomorphy.

The preliminary section establishes essential definitions and mathematical results, including the polar form of the Wirtinger operator, integral means, their logarithmic convexity, and sub-mean-value estimates. These results support the later development of the weighted function spaces and associated operator theory.

Conclusion

The starting point of this paper was a small deformation — let the weight in the Cauchy–Riemann operator depend on the radius — and it is a little surprising how much structure comes along with it once one works on an annulus rather than a disk. The operator ? ?_w turned out to be, in disguise, the ordinary Cauchy–Riemann operator on a rescaled annulus (Theorem 3.3), and almost every other result in the paper is really that fact being unpacked in a different direction: norm equivalence and completeness for the resulting Hardy spaces (Section IV), a boundary theory that reduces to the classical Hardy–Littlewood theorem through an elementary Laurent splitting (Section V), and, once H_w^2 (A(r,R)) is given a genuine inner product (Proposition 6.1), closed-form spectra for every composition operator coming from an automorphism of the annulus — a finite cyclic group or the whole unit circle for rotations, and always {1,-1} for the reflection-type automorphisms, with a concrete symmetric-weight class (Example 7.4) showing the latter is not vacuous. Section VIII then carried the same operator-theoretic questions over to Toeplitz-type multiplication operators, and Section X solved the inhomogeneous equation outright. A few things did not get finished, and it seems more honest to say so than to leave them out. Proposition 8.4 only proves one half of the expected spectral identity for the coordinate multiplier — the inclusion ?(M_z )?¯A (r,R) is easy, but the reverse inclusion, which one would expect by analogy with the classical unilateral shift, seems to need an index-theoretic or approximate-eigenvector argument I have not worked out for the bilateral, weighted setting. The continuity hypothesis on w in Theorem 5.2 is almost certainly stronger than necessary; a Muckenhoupt-type A_p condition ought to suffice, at the cost of replacing Proposition 4.2 with a genuine maximal-function argument, but I have not checked the details. The Toeplitz operators T_? of Definition 8.1 were only really considered here for holomorphic symbols; a general continuous symbol ? would bring in essentially all of the machinery of Douglas [13] and Zhu [11], which felt like a separate paper rather than a short addition to this one. And it would be worth knowing how much of Theorem 3.3 survives once the annulus is replaced by a finitely connected domain with more than two boundary circles, where a single weight and a single diffeomorphism ?_w would presumably have to be replaced by a family of local uniformizing data glued together in some way. I hope to come back to at least the first of these.

References

L. V. Ahlfors, Complex Analysis, 3rd ed. New York: McGraw-Hill, 1979. P. L. Duren, Theory of H^p Spaces. New York: Academic Press, 1970. W. Rudin, Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, 1987. J. B. Conway, Functions of One Complex Variable I, 2nd ed. New York: Springer-Verlag, 1978. J. B. Garnett, Bounded Analytic Functions, revised 1st ed. New York: Springer, 2007. B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Trans. Amer. Math. Soc., vol. 165, pp. 207–226, 1972. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions. Boca Raton: CRC Press, 1995. N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Vol. I. Providence: Amer. Math. Soc., 2002. E. Hille, Analytic Function Theory, Vol. II. Boston: Ginn and Company, 1962. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable. Providence: Amer. Math. Soc. Translations, 1969. K. Zhu, Operator Theory in Function Spaces, 2nd ed. Providence: Amer. Math. Soc., 2007. D. Sarason, “The H^p spaces of an annulus,” Mem. Amer. Math. Soc., No. 56, 1965. R. G. Douglas, Banach Algebra Techniques in Operator Theory, 2nd ed. New York: Springer-Verlag, 1998. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces. New York: Springer-Verlag, 2000. I. N. Vekua, Generalized Analytic Functions. Oxford: Pergamon Press, 1962. R. R. Coifman and C. Fefferman, “Weighted norm inequalities for maximal functions and singular integrals,” Studia Math., vol. 51, pp. 241–250, 1974.

Copyright

Copyright © 2026 Sachin R. Wadje, Avinash D. Rajegore. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.