Abstract
In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ℝn are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.H. Hardy and J.E. Littlewood, “Some properties of fractional integrals (II)”, Math. Z., 34, 403–439, 1932.
G.H. Hardy and J.E. Littlewood, “Theorems concerning mean values of analytic or harmonic functions”, Quart. J.Math. (Oxford), 12, 221–256, 1941.
P. Duren, Theory of H p Spaces (Academic Press, New York, London, 1970).
T.M. Flett, “Inequalities for the pth mean values of harmonic and subharmonic functions with p ≤ 1”, Proc. London Math. Soc., 20, 249–275, 1970.
T.M. Flett, “On the rate of growth of mean values of holomorphic and harmonic functions”, Proc. London Math. Soc., 20, 749–768, 1970.
M. Pavlović, “Decompositions of L p and Hardy spaces of polyharmonic functions”, J.Math. Anal. Appl., 216, 499–509, 1997.
M. Jevtić and M. Pavlović, “Harmonic Bergman functions on the unit ball in ρ n”, Acta Math. Hungar., 85, 81–96, 1999.
S. Stević, “On harmonic Hardy spaces and area integrals, J.Math. Soc. Japan, 56, 339–347, 2004.
S. Stević, “On harmonic function spaces. II”, J. Comput. Anal. Appl., 10, 205–228, 2008.
A.I. Petrosyan, “On weighted classes of harmonic functions in the unit ball of ρ n”, Complex Variables Theory Appl., 50, 953–966, 2005.
A.I. Petrosyan, “On weighted harmonic Bergman spaces”, Demonstratio Math., 41, 73–83, 2008.
K. Avetisyan, “Continuous inclusions and Bergman type operators in n-harmonic mixed norm spaces on the polydisc”, J.Math. Anal. Appl., 291, 727–740, 2004.
K. Avetisyan, “Weighted integrals and Bloch spaces of n-harmonic functions on the polydisc”, Potential Analysis, 29, 49–63, 2008.
A.B. Aleksandrov, “On boundary decay in the mean of harmonic functions”, St. Petersburg Math. J., 7, 507–542, 1996.
C. Fefferman and E.M. Stein, “H p spaces of several variables”, Acta Math., 129, 137–193, 1972.
C.W. Liu and J.H. Shi, “Invariant mean-value property and M-harmonicity in the unit ball of ρ n”, Acta Math. Sinica, 19, 187–200, 2003.
T. Holmstedt, “Interpolation of quasi-normed spaces”, Math. Scand., 26, 177–199, 1970.
E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, New Jersey 1970).
E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, New Jersey 1971).
A.P. Calderon and A. Zygmund, “On higher gradients of harmonic functions”, Studia Math., 24, 211–226, 1964.
N. Du Plessis, “Spherical Fejér-Riesz theorems”, J. London Math. Soc., 31, 386–391, 1956.
Y. Sagher, “On the Fejér-F. Riesz inequality in L p”, Studia Math., 61, 269–278, 1977.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © K. Avetisyan, Y. Tonoyan, 2012, published in Izvestiya NAN Armenii. Matematika, 2012, No. 5, pp. 3–20.
About this article
Cite this article
Avetisyan, K., Tonoyan, Y. Continuous embeddings in harmonic mixed norm spaces on the unit ball in ℝn . J. Contemp. Mathemat. Anal. 47, 209–220 (2012). https://doi.org/10.3103/S1068362312050019
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1068362312050019