arXiv subjects
Publications and source records attributed to Oliver Dragičević.
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 $\mathbb{R}^{d}$.
Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $Ω\subseteq\mathbb{R}^{d}$. We prove that the maximal operator ${\mathscr M}^{A} f=\sup_{t>0}|\exp(-tL_{A})f|$ is bounded in $L^{p}(Ω)$, whenever $A$ is $p$-elliptic in the sense of [10]. The relevance of this result is that, in general, the semigroup generated by $-L_{A}$ is neither contractive in $L^{\infty}$ nor positive, therefore neither the Hopf--Dunford--Schwartz maximal ergodic theorem [15, Chap.~VIII] nor Akcoglu's maximal ergodic theorem [1] can be used. We also show that if $d\geq 3$ and the domain of the sesquilinear form associated with $L_{A}$ embeds into $L^{2^{*}}(Ω)$ with $2^{*}=2d/(d-2)$, then the range of $L^{p}$-boundedness of ${\mathscr M}^{A}$ improves into the interval $(rd/((r-1)d+2),rd/(d-2))$, where $r\geq 2$ is such that $A$ is $r$-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator $\sup_{s,t>0}|T^{A_{1}}_{s}T^{A_{2}}_{t}f|$.
Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a complete asymptotic answer to a question suggested by Iwaniec and Martin in 1996. The main novelty are the lower estimates, of which we give three different proofs. We also conjecture the exact value of $\|(R_2+iR_1)^k\|_p$. Furthermore, we establish the sharp behaviour of weak $(1,1)$ constants of $(R_2+iR_1)^k$ and an $L^\infty$ to $BMO$ estimate that is sharp up to a logarithmic factor.
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 prove a dimension-free $L^p(Ω)\times L^q(Ω)\times L^r(Ω)\rightarrow L^1(Ω\times (0,\infty))$ 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 $p,q,r\in(1,\infty)$ mutually related by the identity $1/p+1/q+1/r=1$. Here $Ω$ 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.
Let $Ω\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $Ω$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm div}(A\nabla)$ with mixed boundary conditions on $Ω$. We extend the bilinear inequality that we proved in [16] in the special case when $Ω=\mathbb{R}^{d}$. 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 $L^{p}(Ω)$, 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 $Ω$, in particular, we do not assume any regularity of $\partialΩ$, 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 introduce a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE 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 Dindoš and Pipher (arXiv:1612.01568v3) established close ties between $p$-ellipticity and (v) regularity theory of elliptic PDE with complex coefficients. The $p$-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.
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 $L^{2}(γ_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(γ_{\infty})$, $1 θ^{*}_{r}$, where $γ_{\infty}$ is the associated invariant measure and $θ^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(γ_{\infty})$. The angle $θ^{*}_{r}$ 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 $σ$-finite measure space admits, for $1<p<\infty$, a Hörmander-type holomorphic functional calculus on $L^p$ in the sector of angle $ϕ^*_p=\arcsin|1-2/p|$. 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 $(M,μ_ϕ)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,μ_ϕ)$ and $L^q(T^*M,μ_ϕ)$, $1/p+1/q=1$, 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 $L^p$ norms of the corresponding shifted Riesz transform. All our proofs are analytic.
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 utilize the Bellman function technique to prove a bilinear dimension-free inequality for the Hermite operator. The Bellman technique is applied here to a non-local operator, which at first did not seem to be feasible. As a consequence of our bilinear inequality one proves dimension-free boundedness for the Riesz-Hermite transforms on L^p with linear growth in terms of p. A feature of the proof is a theorem establishing L^p(R^n) estimates for a class of spectral multipliers with bounds independent of n and p. 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. As a first step towards this goal we prove our dimension-free bilinear embedding theorem for quite a general family of Schroedinger semigroups.
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 L^p estimates of certain singular integral operators (Riesz transforms).