Abstract
In this paper, we establish certain embedding theorems and integral representations in weighted classes of functions harmonic or holomorphic in the unit disk of the complex plane. In our representations, we use the Lipschitz classes of O. V. Besov on the unit circle.
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References
M. M. Djrbashian, On canonical representation of functions meromorphic in the unit disk, Dokl. Akad. Nauk Arm. SSR, 3 No. 1 (1945), 3–9 (in Russian).
M. M. Djrbashian, On the representation problem of analytic functions, Soobshch. Inst. Matem. i Mekh. Akad. Nauk Arm. SSR, 2(1948), 3–40 (in Russian).
M. M. Djrbashian, Integral transforms and representations of functions in the complex domain, Nauka (Moscow, 1966) (in Russian).
P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on Hp spaces with 0 < p < 1, J. Reine Angew. Math., 238(1969), 32–60.
P. J. Enigenburg, The integral means of analytic functions, Quart. J. Math. Oxford, 32(1981), 313–322.
T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl., 38(1972), 746–765.
T. M. Flett, Lipschitz spaces of functions on the circle and the disc, J. Math. Anal. Appl., 39(1972), 125–158.
J. B. Garnett, Bounded analytic functions, Academic Press ( NewYork, 1981).
G. H. hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z., 34(1932), 403–439.
G. H. Hardy and J. E. Littlewood, Theorems concerning mean values of analytic or harmonic functions, Quart. J. Math. Oxford, 12(1941), 221–256.
F. Holland and J. B. Twomey, On Hardy classes and the area function, J. London Math. Soc. (2), 17(1978), 275–283.
J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. II, Proc. London Math. Soc. (2), 42(1936), 52–89.
F. A. Shamoyan, On a parametric representation of Nevanlinna-Djrbashian classes, Dokl. Akad. Nauk Arm. SSR, 90 No. 3(1990), 99–103 (in Russian).
F. A. Shamoyan, Some remarks on the parametric representation of Nevanlinna-Djrbashian classes, Mat. Zametki, 52 No. 1(1992), 128–140 (in Russian); Math. Notes 52(1993), 727–737.
E. M. Stein, Singular integrals and differentiability properties of functions, Univ. Press ( Princeton, NJ, 1970).
M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space. I. Principal properties, J. Math. Mech., 13(1964), 407–479.
A. Zabulionis, On the differential operator in the analytic function spaces, Litovsk. Mat. Sb., 24 No. 1 (1984), 53–59 (in Russian).
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Avetisian, K.L. Fractional Integration and Integral Representations in Weighted Classes of Harmonic Functions. Analysis Mathematica 26, 161–174 (2000). https://doi.org/10.1023/A:1005696901247
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DOI: https://doi.org/10.1023/A:1005696901247