Various notions of dissipativity for partial differential operators and their applications are surveyed. We deal with functional dissipativity and its particular case Formula: see text-dissipativity. ...Most of the results are due to the authors.
We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for ...triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. 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.
We establish a new theory of regularity for elliptic complex valued second order equations of the form L=divA(∇⋅), when the coefficients of the matrix A satisfy a natural algebraic condition, a ...strengthened version of a condition known in the literature as Lp-dissipativity. Precisely, the regularity result is a reverse Hölder condition for Lp averages of solutions on interior balls, and serves as a replacement for the De Giorgi–Nash–Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for Lp-dissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragičević introduced a condition they termed p-ellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their p-ellipticity condition is exactly our strengthened version of Lp-dissipativity. The regularity results of the present paper are applied to solve Lp Dirichlet problems for L=divA(∇⋅)+B⋅∇ when A and B satisfy a Carleson measure condition, which previously was known only in the real valued case. We show solvability of the L2 Dirichlet problem, as well as solvability of the Lp Dirichlet boundary value problem for p in the range where A is p-elliptic.
Second-order divergence form operators are studied on an open set with various boundary conditions. It is shown that the
p
-ellipticity condition of Carbonaro–Dragičević and Dindoš–Pipher implies ...extrapolation to a holomorphic semigroup on Lebesgue spaces in a
p
-dependent range of exponents that extends the maximal range for general strictly elliptic coefficients. This has immediate consequences for the harmonic analysis of such operators, including
H
∞
-calculi and Riesz transforms.
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