In Dragicevic et al. (2003, 2006) the Ahlfors-Beurling operator T and its powers Tn were represented as averages of two-dimensional martingale transforms. Our intention is to optimize the parameters of the averaging for the estimate of HT Hp from above. The second goal of ours is to give an estimate of HT nHp from below. (c) 2025 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We prove a variant of the so-called bilinear embedding theorem for operators in divergence form with complex coefficients and with nonnegative locally integrable potentials, subject to mixed boundary conditions, and acting on arbitrary open subsets of R d .

Let LA=−div(A∇) be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set Ω⊆Rd. We prove that the maximal operator MAf=supt>0⁡|exp⁡(−tLA)f| is bounded in Lp(Ω) whenever A is p-elliptic in the sense of [11]. The relevance of this result is that, in general, the semigroup generated by −LA is neither contractive in L∞ nor positive, therefore neither the Hopf–Dunford–Schwartz maximal ergodic theorem [16, Chap. VIII] nor Akcoglu's maximal ergodic theorem [2] can be used. We also show that if d⩾3 and the domain of the sesquilinear form associated with LA embeds into L2⁎(Ω) with 2⁎=2d/(d−2), then the range of Lp-boundedness of MA improves to (rd/((r−1)d+2),rd/(d−2)), where r⩾2 is such that A is r-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator sups,t>0⁡|TsA1TtA2f|.

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.

Let R_1,2 be scalar Riesz transforms on ℝ^2 . We prove that the L^p norms of k-th powers of the operator R_2+iR_1 behave exactly as | k| ^1-2/p^*(p^*-1) , uniformly in k∈ℤ\{0} and 1

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.

These notes are based on the ten lectures that the author held in September of 2013 at the University of Seville. The purpose of the course was to introduce basic concepts about the Ahlfors-Beurling operator and explain a few of the recent (already published or known) results and techniques focused around it.

We introduce a condition on accretive matrix functions, called p-ellipticity, and discuss its applications to the L-p theory of elliptic PDEs with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) L-p-contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindos and Pipher established close ties between p-ellipticity and (v) regularity theory of elliptic PDEs with complex coefficients. The p-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on the one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.

Prologue 2 Acknowledgements 2 1. Notation and main protagonists 3 1.1. Weak derivatives and Sobolev spaces 4 1.2. Riesz transforms 6 1.3. Ahlfors-Beurling transform 7 1.4. Muckenhoupt weights 8 2. Motivation 9 2.1. Some estimates concerning conformal mappings 9 2.2. Beyond conformality: quasiconformal maps and the Beltrami equation 14 2.3. Cauchy transform 15 2.4. Solving the Beltrami equation 19 2.5. The Iwaniec conjecture 21 2.6. Weak quasiregularity vs. quasiregularity: weighted estimates of T 22 2.7. Estimates of Tn on Lp 26 2.8. Weighted estimates of Tn 30 3. Weighted estimate for T : proof of Theorem 2.29. 31 3.1. Haar functions and martingale transforms 31 3.2. Main idea 32 3.3. The averaging 33 3.4. Estimates for S on L2(w) 42 3.5. Sharpness 47 3.6. Bellman function and Wittwer’s theorem 49 3.7. Unweighted estimates for Tn by means of averaging martingale transforms 55 4. Unweighted estimates for Tn 56 4.1. Lower estimates 56 4.2. Upper estimates 60 4.3. Candidates for ‖T‖p and some necessary conditions 61 5. Spectral theory for T 62 6. Open questions 66 6.1. Hypergeometric functions and de Branges’ theorem 66 References 74

Let $\Omega\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm div}(A\nabla)$ with mixed boundary conditions on $\Omega$. We extend the bilinear inequality that we proved in [16] in the special case when $\Omega=\mathbb{R}^{d}$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime}(t)+{\mathscr L}^{A}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^{p}(\Omega)$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that we introduced in [16]. This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on $\Omega$, in particular, we do not assume any regularity of $\partial\Omega$, nor the existence of a Sobolev embedding. The methods of [16] do not apply directly to the present case and a new argument is needed.

We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators ${\mathscr L}$. We prove that if $-{\mathscr L}$ generates an analytic semigroup on $L^{2}(\gamma_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(\gamma_{\infty})$, $1\theta^{*}_{r}$, where $\gamma_{\infty}$ is the associated invariant measure and $\theta^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(\gamma_{\infty})$. The angle $\theta^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1<p<\infty$, a H\"ormander-type holomorphic functional calculus on $L^p$ in the sector of angle $\phi^*_p=\arcsin|1-2/p|$. The obtained angle is optimal.

By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold (M,μφ) having the Bakry–Emery curvature bounded from below. The embedding, acting on the Cartesian product of Lp(M,μφ) and Lq(T⁎M,μφ), 1/p+1/q=1, involves estimates which are independent of the dimension of the manifold and linear in p. As a consequence we obtain linear dimension-free estimates of the Lp norms of the corresponding shifted Riesz transform. All our proofs are analytic.

We prove a bilinear embedding theorem for Schrödinger operators with nonnegative potentials. The embedding, acting on the cartesian product of Lp(ℝn) and its dual, involves estimates that are independent of the dimension n and linear in terms of p. This feature is achieved by means of a particular Bellman function which satisfies three crucial properties. Connections with known results on the Heisenberg group as well as with results for the Hilbert transform along the parabola are also explored. We believe our approach is quite universal in the sense that one could apply it to a whole range of Riesz transforms arising from various differential operators.

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The $p$-range of applicability of this estimate is $(1,\infty)$ for any real accretive (nonsymmetric) matrix $A$ of coefficients.

We provide a proof of sharp lower L p bounds for powers of the Ahlfors–Beurling operator T and improve the numerical constant in their asymptotic estimates. We also discuss the possibilities of applying de Branges’ theorem in the p − 1 problem.

We prove that for any n is an element of Z\{0}, p > 1 and any weight w from the Muckenhoupt A(p) class, the norm of the n-th power of the Ahlfors-Beurling operator T on the weighted Lebesgue space L-p(w) is majorized by C(p)vertical bar n vertical bar(3)[w](p)(max{1,1/(p-1)}), where [w](p) is the A(p) characteristic of w. We apply this estimate for a result concerning the spectrum of T on L-p(w).

We discuss bilinear embedding theorems for a certain class of Schrodinger operators on LP. The obtained estimates are dimension-free and linear in p. We outline a uniform proof of the theorem which relies on establishing three crucial properties of the concrete Bellman function we consider. To cite this article: O. Dragicevic, A. Volberg, C. R. Acad. Sci. Paris, Ser, 1347 (2009). (C) 2009 Acadamie des sciences. Published by Elsevier Masson SAS. All rights reserved.

We prove a self-improvement property regarding quadratic forms on arbitrary vector spaces. We discuss several consequences of this result, in particular those concerning dimension-free Lp estimates of certain singular integral operators (Riesz transforms).