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Licensed Unlicensed Requires Authentication Published by De Gruyter October 26, 2022

Certain multiplier results on Bp spaces

  • Sunanda Naik EMAIL logo

Abstract

In this article, we use a particular case of convolution as an operator to discuss a number of problems concerning multiplier results between function spaces such as Hardy and B p -spaces. As a consequence, we extend certain well-known results on fractional derivatives and fractional integrals. Also, we find condition on the parameters b , c such that 𝒫 b , c in B p .

Acknowledgements

The author is extremely thankful to the anonymous referees whose reports have helped in improving the article.

References

[1] M. R. Agrawal, P. G. Howlett, S. K. Lucas, S. Naik and S. Ponnusamy, Boundedness of generalized Cesáro averaging operators on certain function spaces, J. Comput. Appl. Math. 180 (2005), no. 2, 333–344. 10.1016/j.cam.2004.11.004Search in Google Scholar

[2] G. D. Anderson, M. K. Vamanamurthy and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997. Search in Google Scholar

[3] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge University, Cambridge, 1999. 10.1017/CBO9781107325937Search in Google Scholar

[4] R. Balasubramanian and S. Ponnusamy, On Ramanujan asymptotic expansions and inequalities for hypergeometric functions, Proc. Indian Acad. Sci. Math. Sci. 108 (1998), no. 2, 95–108. 10.1007/BF02841543Search in Google Scholar

[5] R. Balasubramanian, S. Ponnusamy and M. Vuorinen, On hypergeometric functions and function spaces, J. Comput. Appl. Math. 139 (2002), no. 2, 299–322. 10.1016/S0377-0427(01)00417-4Search in Google Scholar

[6] D. Borgohain and S. Naik, An integral type operator on analytic function spaces, J. Anal. 27 (2019), no. 3, 829–836. 10.1007/s41478-018-0134-1Search in Google Scholar

[7] P. L. Duren, On the multipliers of H p spaces, Proc. Amer. Math. Soc. 22 (1969), 24–27. 10.1090/S0002-9939-1969-0241651-XSearch in Google Scholar

[8] P. L. Duren, Theory of H p Spaces, Academic Press, New York, 2000. Search in Google Scholar

[9] P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on H p spaces with 0 & l t ; p & l t ; 1 , J. Reine Angew. Math. 238 (1969), 32–60. 10.1515/crll.1969.238.32Search in Google Scholar

[10] P. L. Duren and A. L. Shields, Properties of H p ( 0 & l t ; p & l t ; 1 ) and its continuing Banach space, Trans. Amer. Math. Soc. 141 (1969), 255–262. 10.1090/S0002-9947-1969-0244751-8Search in Google Scholar

[11] P. L. Duren and A. L. Shields, Coefficient multipliers of H p and B p spaces, Pacific J. Math. 32 (1970), 69–78. 10.2140/pjm.1970.32.69Search in Google Scholar

[12] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403–439. 10.1007/BF01180596Search in Google Scholar

[13] I. R. Kayumov, D. M. Khammatova and S. Ponnusamy, On the Bohr inequality for the Cesáro operator, C. R. Math. Acad. Sci. Paris 358 (2020), no. 5, 615–620. 10.5802/crmath.80Search in Google Scholar

[14] J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series (II), Proc. Lond. Math. Soc. (2) 42 (1936), no. 1, 52–89. 10.1112/plms/s2-42.1.52Search in Google Scholar

[15] G. Liu and S. Ponnusamy, On harmonic ν-Bloch and ν-Bloch-type mappings, Results Math. 73 (2018), no. 3, Paper No. 90. 10.1007/s00025-018-0853-2Search in Google Scholar

[16] S. Naik, Cesáro type operators on spaces of analytic functions, Filomat 25 (2011), no. 4, 85–97. 10.2298/FIL1104085NSearch in Google Scholar

[17] S. Ponnusamy and F. Rø nning, Duality for Hadamard products applied to certain integral transforms, Complex Variables Theory Appl. 32 (1997), no. 3, 263–287. 10.1080/17476939708814995Search in Google Scholar

[18] S. Ponnusamy and F. Rø nning, Integral transforms of functions with the derivative in a halfplane, Israel J. Math. 114 (1999), 177–188. 10.1007/BF02785576Search in Google Scholar

[19] K. Stempak, Cesàro averaging operators, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 1, 121–126. 10.1017/S030821050002922XSearch in Google Scholar

[20] J. Xiao, Cesàro-type operators on Hardy, BMOA and Bloch spaces, Arch. Math. (Basel) 68 (1997), no. 5, 398–406. 10.1007/s000130050072Search in Google Scholar

Received: 2021-05-07
Revised: 2021-12-15
Accepted: 2022-02-04
Published Online: 2022-10-26
Published in Print: 2023-06-01

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