Bounded integral operators in some weighted spaces of analytic functions on product domains and related results

The expository article overviews new research results presented by the authors and their colleagues regarding to the existence problem of bounded projections of functions on weighted spaces in $n$-dimensional complex space $C^n.$ New extensions are discussed. Some new interesting problems in analytic spaces on product domains are also provided. In addition, we shortly review new results of other authors working in this direction.