| Title: | Trilinear embedding for divergence-form operators with complex coefficients |
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| Authors: | ID Carbonaro, Andrea (Author)
ID Dragičević, Oliver (Author)
ID Kovač, Vjekoslav (Author)
ID Škreb, Kristina Ana (Author) |
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| Files: |  URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0001870823003821
 PDF - Presentation file, download (1,10 MB)
MD5: 8B1845458CDDCA277D057F58DBE69B2E
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | elliptic differential operator, p-ellipticity, operator semigroup, multilinear estimate |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.10.2023 |
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| Year of publishing: | 2023 |
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| Number of pages: | 72 str. |
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| Numbering: | Vol. 431, article no. 109239 |
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| PID: | 20.500.12556/DiRROS-18411  |
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| UDC: | 517.9 |
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| ISSN on article: | 0001-8708 |
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| DOI: | 10.1016/j.aim.2023.109239 |
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| COBISS.SI-ID: | 177492995 |
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| Note: |
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| Publication date in DiRROS: | 15.03.2024 |
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| Views: | 964 |
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| Downloads: | 539 |
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