Trilinear embedding for divergence-form operators with complex coefficients
| Autor: | Carbonaro, Andrea, Dragičević, Oliver, Kovač, Vjekoslav, Škreb, Kristina |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Adv. Math. 431 (2023), article 109239 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.aim.2023.109239 |
| Popis: | We prove a dimension-free $L^p(\Omega)\times L^q(\Omega)\times L^r(\Omega)\rightarrow L^1(\Omega\times (0,\infty))$ embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on $\Omega$, and for triples of exponents $p,q,r\in(1,\infty)$ mutually related by the identity $1/p+1/q+1/r=1$. Here $\Omega$ is allowed to be an arbitrary open subset of $\mathbb{R}^d$. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as $p$-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato--Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new. Comment: A minor revision of v2; 51 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|