In this article, we study Toeplitz operators with nonnegative symbols on the -weighted harmonic Bergman space. We characterize the boundedness, compactness, and invertibility of Toeplitz operators with nonnegative symbols on this space.
References
[1] T. C. Anderson and A. Vagharshakyan, A simple proof of the sharp weighted estimate for Calderón–Zygmund operators on homogeneous spaces, J. Geom. Anal. 24 (2014), no. 3, 1276–1297. 1302.42018 10.1007/s12220-012-9372-7[1] T. C. Anderson and A. Vagharshakyan, A simple proof of the sharp weighted estimate for Calderón–Zygmund operators on homogeneous spaces, J. Geom. Anal. 24 (2014), no. 3, 1276–1297. 1302.42018 10.1007/s12220-012-9372-7
[3] B. R. Choe and Y. J. Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J. 46 (1999), no. 1, 163–174. 0969.47023 10.1307/mmj/1030132367 euclid.mmj/1030132367[3] B. R. Choe and Y. J. Lee, Commuting Toeplitz operators on the harmonic Bergman space, Michigan Math. J. 46 (1999), no. 1, 163–174. 0969.47023 10.1307/mmj/1030132367 euclid.mmj/1030132367
[4] E. S. Choi, Positive Toeplitz operators on pluriharmonic Bergman spaces, J. Math. Kyoto Univ. 47 (2007), no. 2, 247–267. 1158.32001 10.1215/kjm/1250281046 euclid.kjm/1250281046[4] E. S. Choi, Positive Toeplitz operators on pluriharmonic Bergman spaces, J. Math. Kyoto Univ. 47 (2007), no. 2, 247–267. 1158.32001 10.1215/kjm/1250281046 euclid.kjm/1250281046
[5] O. Constantin, Discretizations of integral operators and atomic decompositions in vector-valued weighted Bergman spaces, Integral Equations Operator Theory 59 (2007), no. 4, 523–554. MR2370047 1148.32005 10.1007/s00020-007-1536-7[5] O. Constantin, Discretizations of integral operators and atomic decompositions in vector-valued weighted Bergman spaces, Integral Equations Operator Theory 59 (2007), no. 4, 523–554. MR2370047 1148.32005 10.1007/s00020-007-1536-7
[6] O. Constantin, Carleson embeddings and some classes of operators on weighted Bergman spaces, J. Math. Anal. Appl. 365 (2010), no. 2, 668–682. 1189.46018 10.1016/j.jmaa.2009.11.035[6] O. Constantin, Carleson embeddings and some classes of operators on weighted Bergman spaces, J. Math. Anal. Appl. 365 (2010), no. 2, 668–682. 1189.46018 10.1016/j.jmaa.2009.11.035
[8] R. G. Douglas and K. Wang, A harmonic analysis approach to essential normality of principal submodules, J. Funct. Anal. 261 (2011), no. 11, 3155–3180. 1254.47004 10.1016/j.jfa.2011.07.021[8] R. G. Douglas and K. Wang, A harmonic analysis approach to essential normality of principal submodules, J. Funct. Anal. 261 (2011), no. 11, 3155–3180. 1254.47004 10.1016/j.jfa.2011.07.021
[9] K. Guo and D. Zheng, Toeplitz algebra and Hankel algebra on the harmonic Bergman space, J. Math. Anal. Appl. 276 (2002), no. 1, 213–230. MR1944347 1030.47020 10.1016/S0022-247X(02)00429-8[9] K. Guo and D. Zheng, Toeplitz algebra and Hankel algebra on the harmonic Bergman space, J. Math. Anal. Appl. 276 (2002), no. 1, 213–230. MR1944347 1030.47020 10.1016/S0022-247X(02)00429-8
[12] D. H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), no. 1, 1–11. 0437.30025 euclid.ijm/1256047358[12] D. H. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), no. 1, 1–11. 0437.30025 euclid.ijm/1256047358
[14] D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), no. 1, 85–111. 0584.46042 10.2307/2374458[14] D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), no. 1, 85–111. 0584.46042 10.2307/2374458
[15] D. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), no. 2, 319–336. 0538.32004 10.1512/iumj.1985.34.34019[15] D. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), no. 2, 319–336. 0538.32004 10.1512/iumj.1985.34.34019
[16] J. Miao, Toeplitz operators on harmonic Bergman spaces, Integral Equations Operator Theory 27 (1997), no. 4, 426–438. 0902.47026 10.1007/BF01192123[16] J. Miao, Toeplitz operators on harmonic Bergman spaces, Integral Equations Operator Theory 27 (1997), no. 4, 426–438. 0902.47026 10.1007/BF01192123
[17] J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), no. 1, 25–35. 0907.46020 10.1007/BF01489456[17] J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), no. 1, 25–35. 0907.46020 10.1007/BF01489456
[18] M. Mitkovski and B. D. Wick, A reproducing kernel thesis for operators on Bergman-type function spaces, J. Funct. Anal. 267 (2014), no. 7, 2028–2055. 1302.32008 10.1016/j.jfa.2014.07.020[18] M. Mitkovski and B. D. Wick, A reproducing kernel thesis for operators on Bergman-type function spaces, J. Funct. Anal. 267 (2014), no. 7, 2028–2055. 1302.32008 10.1016/j.jfa.2014.07.020
[19] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192, no. 1 (1974), 261–274. 0289.26010 10.1090/S0002-9947-1974-0340523-6[19] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192, no. 1 (1974), 261–274. 0289.26010 10.1090/S0002-9947-1974-0340523-6
[21] J. Á. Peláez and J. Rättyä, Trace class criteria for Toeplitz and composition operators on small Bergman spaces, Adv. Math. 293 (2016), no. 1, 606–643.[21] J. Á. Peláez and J. Rättyä, Trace class criteria for Toeplitz and composition operators on small Bergman spaces, Adv. Math. 293 (2016), no. 1, 606–643.
[22] Y. L. Shu and X. F. Zhao, Positivity of Toeplitz operators on harmonic Bergman space, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 2, 175–186. 1338.47027 10.1007/s10114-016-5138-7[22] Y. L. Shu and X. F. Zhao, Positivity of Toeplitz operators on harmonic Bergman space, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 2, 175–186. 1338.47027 10.1007/s10114-016-5138-7
[24] X. Zhao and D. Zheng, Invertibility of Toeplitz operators via Berezin transforms, J. Operator Theory 75 (2016), no. 2, 475–495. 1389.47087 10.7900/jot.2015jul07.2082[24] X. Zhao and D. Zheng, Invertibility of Toeplitz operators via Berezin transforms, J. Operator Theory 75 (2016), no. 2, 475–495. 1389.47087 10.7900/jot.2015jul07.2082
[25] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Math. Surveys Monogr. 138, Amer. Math. Soc., Providence, 2007. 1123.47001[25] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Math. Surveys Monogr. 138, Amer. Math. Soc., Providence, 2007. 1123.47001