engleski

Trilinear embedding for divergence-form operators

We give sufficient conditions that guarantee an $L^p(\Omega)\times L^q(\Omega)\times L^r(\Omega)\rightarrow L^1(\Omega\times (0, \infty))$ embedding for a triple of divergence-form elliptic operators with complex coefficients, and with Dirichlet, Neumann, or mixed boundary conditions on $\Omega$. Conditions on the exponents and coefficient matrices are expressed in terms of the so-called p-ellipticity, introduced previously by the first two authors. The proof relies on a particular Bellman function, found previously by the last two authors. We also present applications of the main result to paraproducts and conical square functions associated with the corresponding operator semigroups. This is joint work with Andrea Carbonaro, Oliver Dragičević, and Kristina Ana Škreb.

semigroup ; paraproduct

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