Carleson Measure in Bergman-Orlicz Space of Polydisc
نویسندگان
چکیده
and Applied Analysis 3 2. Main Results and Proofs Lemma 2.1. For α α1, . . . , αn ∈ D, let uα z1, . . . , zn ∏n j 1 1 − |αj |2 / 1 − αjzj . Then uα z1, . . . , zn ∈ Lφa D , and ‖uα z ‖σn ≤ 1 φ−1 ∏n j 1 ( 1/δ2 j )) . 2.1 Proof. It is easy to see that ‖uα z ‖∞ ∏n j 1 1 |αj | / 1 − |αj | 2 ∏n j 1 2 − δj /δj . Since φ 0 0, the convexity of φ implies φ ax ≤ aφ x for 0 ≤ a ≤ 1. Hence, for every C > 0, we have ∫ Dn φ ⎛ ⎝ ∏n j 1 ( δj/ 2 − δj 2|uα z | C ⎞ ⎠dσn ≤ n ∏ j 1 ( δj 2 − δj )2 ∫ Dn |uα z |φ ( 1 C ) dσn n ∏ j 1 ( δj 2 − δj )2 ‖uα1 z ‖1φ ( 1 C ) , 2.2 but ∏n j 1 δj/ 2 − δj ‖uα1 z ‖1φ 1/C ≤ 1 if and only if C ≥ 1/φ−1 ∏n j 1 2 − δj/δj 2 1/‖uα z ‖1 , that is, ‖uα z ‖σn ≤ 1 φ−1 ∏n j 1 ( 2 − δj/δj )2 1/‖uα z ‖1 ) . 2.3
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