Generalization of Schur’s Test and its Application to a Class of Integral Operators on the Unit Ball of $${\mathbb C^n}$$

Abstract

We generalize the classical Schur's test to the boundedness of integral operators from L p to L q spaces equipped with possibly different measures, for 1 ≤ p ≤ q < ∞. As an application, we determine exactly when a class of integral operators are bounded from L p (Bn, dvα) to L q (Bn, dv β ), where 1 ≤ p ≤ q < ∞, Bn is the unit ball of n-dimensional complex Euclidean space C n , and dvα and dv β are weighted area measures on C n . The result generalizes a result by Kures and Zhu.

Bergman–Toeplitz operators on weakly pseudoconvex domains

“…The following corollary follows immediately from Theorem 1.3, that is a refined result by McNeal [15], Phong and Stein [19] for the L p estimates of the Bergman projection P and also generalises the works by Zhu [24] and Zhao [23] for the upper bound of the norm }P } L p pΩqÑL p pΩq . Let Ω be a bounded, pseudoconvex domain in C n with smooth boundary.…”

“…Proof. The proof of this theorem follows from a standard argument, as in [9,23]. Using Hölder's inequality and (5.1), it follows…”

On the Boundedness of the Fractional Bergman Operators

“…Unlike the case of the unit ball (see [5]), the above result can not be deduced from the boundedness of the families of Bergman-type operators considered in [2,6]. …”

Boundedness of a family of Hilbert-type operators and its Bergman-type analogue

“…When p = r, take t = 1 q to obtain the classical Schur's test. The following limit case of Okikiolu result is proved in [11]. Lemma 3.4.…”