{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:59:39Z","timestamp":1760144379894,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T00:00:00Z","timestamp":1713398400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NNSF of China","award":["12101219","2022JJ40141"],"award-info":[{"award-number":["12101219","2022JJ40141"]}]},{"name":"Hunan Provincial NSF","award":["12101219","2022JJ40141"],"award-info":[{"award-number":["12101219","2022JJ40141"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let \u03bc be a self-similar measure with compact support K. The Hausdorff dimension of K is \u03b1. The Cauchy transform of \u03bc is denoted by F(z). For 0&lt;\u03b2&lt;1, we define the function F[\u03b2], which compares with the fractional derivative of F of order \u03b2. Let \u03a6(z)=F(1\/z),|z|&lt;1. In this paper, we prove that \u03a6[\u03b2] belongs to Ap for 0&lt;p&lt;1\/(\u03b2+1), and (\u03a6\u2032)[\u03b2] belongs to Ap for 1\u2264p&lt;1\/\u03b2\u22641\/(2\u2212\u03b1), where Ap is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem.<\/jats:p>","DOI":"10.3390\/axioms13040268","type":"journal-article","created":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T10:30:52Z","timestamp":1713436252000},"page":"268","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bergman Space Properties of Fractional Derivatives of the Cauchy Transform of a Certain Self-Similar Measure"],"prefix":"10.3390","volume":"13","author":[{"given":"Songran","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shantou University, Shantou 515063, China"},{"name":"College of Computer Science and Mathematics, Central South University of Forestry and Technology, Changsha 410004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhimin","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Hunan University of Technology, Zhuzhou 412007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,18]]},"reference":[{"key":"ref_1","unstructured":"Mandelbrot, B.B. (1975). Les Objects Fractals: Forme, Hasard et Dimension, Flammarion."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/0022-1236(90)90009-A","article-title":"Fourier asymptotics of fractal measures","volume":"89","author":"Strichartz","year":"1990","journal-title":"J. Funct. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1512\/iumj.1990.39.39038","article-title":"Self-similar measures and their Fourier transforms I","volume":"39","author":"Strichartz","year":"1990","journal-title":"Indiana Univ. Math. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1090\/S0002-9947-1993-1081941-2","article-title":"Self-similar measures and their Fourier transforms II","volume":"336","author":"Strichartz","year":"1993","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1512\/iumj.1993.42.42018","article-title":"Self-similar measures and their Fourier transforms III","volume":"42","author":"Strichartz","year":"1993","journal-title":"Indiana Univ. Math. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00041-001-4001-z","article-title":"Self-similarity in harmonic analysis","volume":"1","author":"Strichartz","year":"1994","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Mattila, P. (1995). Geometry Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press.","DOI":"10.1017\/CBO9780511623813"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.4007\/annals.2005.162.1243","article-title":"Bilipschitz maps, analytic capacity, and the Cauchy integral","volume":"162","author":"Tolsa","year":"2005","journal-title":"Ann. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/s00039-007-0598-7","article-title":"Growth estimates for Cauchy integrals of measures and rectifiability","volume":"17","author":"Tolsa","year":"2007","journal-title":"Geom. Funct. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1080\/10586458.1998.10504368","article-title":"Cauchy transforms of self-similar measures","volume":"7","author":"Lund","year":"1998","journal-title":"Exp. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0022-1236(02)00069-1","article-title":"Cauchy transforms of self-similar measures: The Laurent coefficients","volume":"202","author":"Dong","year":"2003","journal-title":"J. Funct. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1080\/10586458.2004.10504549","article-title":"An integral related to the Cauchy transform on the Sierpinski gasket","volume":"13","author":"Dong","year":"2004","journal-title":"Exp. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.aim.2012.09.021","article-title":"Cantor boundary behavior of analytic functions","volume":"232","author":"Dong","year":"2013","journal-title":"Adv. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4817","DOI":"10.1090\/tran\/6819","article-title":"Cauchy transform of self-similar measures: Starlikeness and univalence","volume":"369","author":"Dong","year":"2017","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_15","unstructured":"Dong, X.H., and Lau, K.S. (2010). Recent Developments in Fractals and Related Fields, Birkh\u00e4user Boston, Inc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108653","DOI":"10.1016\/j.jfa.2020.108653","article-title":"Estimates for Taylor coefficients of Cauchy transforms of some Hausdorff measures I","volume":"280","author":"Li","year":"2021","journal-title":"J. Funct. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"108654","DOI":"10.1016\/j.jfa.2020.108654","article-title":"Estimates for Taylor coefficients of Cauchy transforms of some Hausdorff measures II","volume":"280","author":"Li","year":"2021","journal-title":"J. Funct. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Folconer, K.J. (1990). Fractal Geometry-Mathematical Foundations and Applications, John Wiley & Sons.","DOI":"10.2307\/2532125"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"6064","DOI":"10.3934\/math.2023306","article-title":"Function space properties of the Cauchy transform on the Sierpinski gasket","volume":"8","author":"Wang","year":"2023","journal-title":"AIMS Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1512\/iumj.1981.30.30055","article-title":"Fractals and self-similarity","volume":"30","author":"Hutchinson","year":"1981","journal-title":"Indiana Univ. Math. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/BF01180596","article-title":"Some properties of fractional integrals II","volume":"34","author":"Hardy","year":"1932","journal-title":"Math. Zeit."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1017\/S1446788716000409","article-title":"Limits of fractional derivatives and compositions of analytic functions","volume":"103","author":"MacGregor","year":"2017","journal-title":"J. Aust. Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1080\/17476933.2017.1385069","article-title":"Boundary limits of fractional derivatives of univalent functions","volume":"63","author":"TMacGregor","year":"2018","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/S0022-247X(03)00128-8","article-title":"Hadamard products with power functions and multipliers of Hardy spaces","volume":"282","author":"MacGregor","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","unstructured":"Duren, P. (1970). Theory of HP Spaces, Academic Press."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Duren, P., and Schuster, A. (2004). Bergman Spaces, Mathematical Surveys and Monographs, American Mathematical Society.","DOI":"10.1090\/surv\/100"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hedenmalm, H., Korenblum, B., and Zhu, K.H. (2000). Theory of Bergman Spaces, Springer.","DOI":"10.1007\/978-1-4612-0497-8"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1007\/s00209-015-1528-2","article-title":"Embedding Bergman spaces into tent spaces","volume":"281","author":"Rxaxttyxax","year":"2015","journal-title":"Math. Z."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1007\/s12220-017-9837-9","article-title":"Berezin transform and Toeplitz operators on Bergman spaces induced by regular weights","volume":"28","author":"Rxaxttyxax","year":"2018","journal-title":"J. Geom. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"109495","DOI":"10.1016\/j.aim.2024.109495","article-title":"Absolutely summing Carleson embeddings on Bergman spaces","volume":"439","author":"He","year":"2024","journal-title":"Adv. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00020-023-02742-7","article-title":"Some operator ideal properties of Volterra operators on Bergman and Bloch spaces","volume":"96","author":"Jreis","year":"2024","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1007\/s11401-021-0280-7","article-title":"Carleson measures on the weighted Bergman spaces with B\u00e9koll\u00e9 weights","volume":"42","author":"Tong","year":"2021","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1512\/iumj.1987.36.36006","article-title":"Analytic and univalent function with integral representations involving complex measures","volume":"86","author":"Macgregor","year":"1987","journal-title":"Indiana Univ. Math. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1090\/S0002-9939-1989-0938912-8","article-title":"Closure properties of families of Cauchy-Stieltjes transforms","volume":"105","author":"Hibschweiler","year":"1989","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1090\/S0002-9947-1992-1120775-6","article-title":"Multipliers of families of Cauchy- Stieltjes transforms","volume":"331","author":"Hibschweiler","year":"1992","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1112\/plms\/s3-72.1.157","article-title":"Fractional Cauchy transforms, inner functions and multipliers","volume":"72","author":"Hallenbeck","year":"1996","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1006\/jmaa.1993.1144","article-title":"On Cauchy integrals of logarithmic potentials and their multipliers","volume":"174","author":"Hallenbeck","year":"1993","journal-title":"J. Math. Anal. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1002\/sapm195938177","article-title":"Some elementary inequalities relating to the gamma and incomplete gamma function","volume":"38","author":"Gautschi","year":"1959","journal-title":"J. Math. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Cherry, W., and Ye, Z. (2001). Nevanlinna\u2019s Theory of Value Distribution, Springer Monographs in Mathematics; Springer.","DOI":"10.1007\/978-3-662-12590-8"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Vojta, P. (2011). Diophantine Approximation and Nevanlinna Theory, Springer.","DOI":"10.1007\/978-3-642-15945-9_3"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/268\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:30:06Z","timestamp":1760106606000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/268"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,18]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["axioms13040268"],"URL":"https:\/\/doi.org\/10.3390\/axioms13040268","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,4,18]]}}}