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In this paper, we give a survey of some recent progress in terms of verifying Carleson measures; this includes the difference between two definitions of a Carleson measure, the Bergman tree condition, the T1 condition for Besov-Sobolev spaces on a complex ball, vector-valued Carleson measures, Carleson measures in strongly pseudoconvex domains and reverse Carleson measures.

We study Toeplitz operators from Hardy spaces to weighted Bergman spaces in the unit ball of Cn. Toeplitz operators are closely related to many classical mappings, such as composition operators, the Volterra type integration operators and Carleson embeddings. We characterize the boundedness and compactness of Toeplitz operators from Hardy spaces Hp to weighted Bergman spaces A for the different values of p and q in the unit ball.

This article is devoted to characterizing the boundedness and compactness of multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball of ℂn.

This paper is devoted to give the connections between Carleson measures for Besov-Sobolev spaces B-p(sigma)(B) and p-Carleson measure in the unit ball of Cm. As applications, we characterize the Riemann-Stieltjes operators and multipliers acting on B-p(sigma)(B) spaces by means of Carleson measures for B-p(sigma)(B).