Sandwich Bernstein-Sato Polynomials and Bernstein's Inequality

Bernstein's inequality is a central result in the theory of $D$-modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of \`{A}lvarez Montaner, Hern\'andez, Jeffries, N\'u\~nez-Betancourt, Teixeira, and Witt establishes Bernstein's inequality for invariants of finite groups in characteristic zero and certain other mild singularities in positive characteristic. Motivated by extending this result to new classes of singular rings, we introduce a ``two-sided'' analogue of the Bernstein-Sato polynomial which we call the sandwich Bernstein-Sato polynomial. We apply this notion to give an effective criterion to verify Bernstein's inequality, and apply this to show that Bernstein's inequality holds for the coordinate ring of $\mathbb{P}^a \times \mathbb{P}^b$ via the Segre embedding. We also establish a number of examples and basic results on sandwich Bernstein-Sato polynomials.