Calculus of multilinear differential operators, operator L?-algebras and IBL?-algebras

We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series such operators. The is built upon the notion a multifiltration linear operad generalizing concept filtration associative algebra. describe particular way constructing and analyzing multifiltrations based on presentation in terms generators relations. In particular, that allows us to observe special role played this context by Lie, Lie-admissible L-infinity structures. As main application, original motivation present work, we show how certain generalization well-known big bracket construction Lecomte\textendash Roger Kosmann-Schwarzbach encompassing case homotopy involutive Lie bialgebras can be obtained.