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

Let ࡭)࢘,ࡾ} = (ࢠ ∋ ℂ: ࢘| > ࢠ > |ࡾ {be an annular domain. We introduce a weighted polar Cauchy–Riemann operator ̅ࣔ ࢝, obtained by inserting a positive radial weight ࢝)࣋ (into the polar representation of the classical operator ࣔ/ࢠࣔ ̅, and we show that its kernel is carried, by an explicit radial diffeomorphism ࢝ࢶ built from ࢝, onto the space of holomorphic functions on a canonical annulus. Using this correspondence, we define weighted Hardy spaces ࢝ࡴ ࢖ ൫࡭)࢘,ࡾ(൯ attached to ࣔ ̅ ࢝, establish normequivalence and completeness results, and prove a boundary-regularity theorem extending the classical Hardy–Littlewood boundary theory to the weighted, doubly connected setting via a Laurent-series splitting argument. We equip ࢝ࡴ ૛ ൫࡭)࢘,ࡾ(൯ with an explicit, genuinely Hilbertian inner product equivalent to its defining norm, exhibit an orthogonal Laurent basis, and use it to compute the spectra of the composition operators induced by the full automorphism group of the annulus — rotations, the canonical inversion, and general reflection-type automorphisms — together with their adjoints and self-adjointness properties. We further study Toeplitz-type multiplication operators ࣐ࡹ on ࢝ࡴ ૛ ൫࡭)࢘,ࡾ(൯, obtaining sharp norm bounds and spectral inclusion results for the coordinate multiplier ࢠࡹ , and solve the inhomogeneous equation ࣔ ̅ ࢛࢝ = ࢎ explicitly via a weighted Cauchy–Pompeiu transform. Explicit norm estimates for power weights, a symmetric-weight class compatible with the inversion automorphism, and worked numerical examples illustrate the theory throughout.