Carleson type measures for harmonic mixed norm spaces with application to Toeplitz operators

Abstract

Let Ω be a bounded domain in ℝ ⁿ with a smooth boundary, and let h _p,q be the space of all harmonic functions with a finite mixed norm. The authors first obtain an equivalent norm on h _p,q, with which the definition of Carleson type measures for h _p,q is obtained. And also, the authors obtain the boundedness of the Bergman projection on h _p,q which turns out the dual space of h _p,q. As an application, the authors characterize the boundedness (and compactness) of Toeplitz operators T _µ on h _p,q for those positive finite Borel measures µ.

Keywords

Carleson type measure / Harmonic mixed norm space / Toeplitz operator / Bergman projection

References

[1] Stein E M. Boundary Behavior of Holomorphic Functions of Several Complex Variables, 1972, Princeton: Princeton Univ. Press

[2] Ahern P, Jevtić M. Duality and multipliers for mixed norm spaces. Michigan Math. J., 1983, 30(1): 53-64

[3] Gadbois S. Mixed norm generalizations of Bergman spaces and duality. Proc. Amer. Math. Soc., 1988, 104: 1171-1180

[4] Shi J H. Duality and multipliers for mixed norm spaces in the ball I, II. Complex Variables Theory Appl., 1994, 25(2): 119-157

[5] Hu Z J. Extended Cesaro operators on mixed norm spaces. Proc. Amer. Math. Soc., 2003, 131: 2171-2179

[6] Hu Z J. Estimates for the integral means of harmonic functions on bounded domains in R ⁿ. Sci. China Ser. A, 1995, 38(1): 36-46

[7] Kang H, Koo H. Estimates of the harmonic Bergman kernel on smooth domains. J. Funct. Anal., 2001, 185: 220-239

[8] Choe B R, Koo H, Yi H. Positive Toeplitz operators between the harmonic Bergman spaces. Potential Anal., 2002, 17(4): 307-335

[9] Luecking D H. Trace ideal criteria for Toeplitz operators. J. Funct. Anal., 1987, 73(2): 345-368

[10] Zhu K H. Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Operator Th., 1988, 20: 329-357

[11] Miao J. Toeplitz operators on harmonic Bergman spaces. Integ. Equ. Oper. Th., 1997, 27(4): 426-438

[12] Choe B R, Lee Y J, Na K. Toeplitz operators on harmonic Bergman spaces. Nagoya Math. J., 2004, 174: 165-186

[13] Choe B R, Lee Y J, Na K. Positive Toeplitz operators from a harmonic Bergman space into another. Tohoku Math. J., 2004, 56(2): 255-270

[14] Choe B R, Koo H, Lee Y. Positive Schatten class Toeplitz operators on the ball. Studia Math., 2008, 189(1): 65-90

[15] Choe B R, Koo H, Na K. Positive Toeplitz operators of Schatten-Herz type. Nagoya Math. J., 2007, 185: 31-62

[16] Choe B R, Nam K. Toeplitz operators and Herz spaces on the half-space. Integ. Equ. Oper. Th., 2007, 59(4): 501-521

[17] Choe B R, Nam K. Berezin transform and Toeplitz operators on harmonic Bergman spaces. J. Funct. Anal., 2009, 257(10): 3135-3166

[18] Zhu K H. Spaces of Holomorphic Functions in the Unit Ball, 2005, New York: Springer-Verlag

[19] Oleinik O L. Embedding theorems for weighted classes of harmonic and analytic functions. J. Soviet Math., 1978, 9: 228-243

[20] Tchoundja E. Carleson measures for the generalizaed Bergman spaces via a T (1)-type theorem. Ark. Mat., 2008, 46: 377-406

[21] Taylor M E. Partial Defferential Equations III, 1996, New York: Springer-Verlag

[22] Rudin W. Real and Complex Analysis, 1987 3rd ed. New York: McGraw-Hill