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In this work we investigate semigroups of operators acting on noncommutative $L^p$-spaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and $H^\infty$ functional…

Let T_t = e^{-tA} be a bounded analytic semigroup on Lp, with 1<p<\infty. It is known that if A and its adjoint A^* both satisfy square function estimates \bignorm{\bigl(\int_{0}^{\infty}| A^{1/2} T_t(x)|^2\, dt\,\bigr)^{1/2}_{Lp} \lesssim…

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 $\phi^*_p=\arcsin|1-2/p|$.…

We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the…

For $p\in (1,\infty)$ and $\alpha\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g…

In this article we present a new proof of a sharp Reverse H\"older Inequality for $A_\infty$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes…

Let $\theta \in(0,1)$ and $(\mathcal{M},\tau)$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (denoted by $S_{d,\theta}$), showing that there exists a constant $d>0 $ depending on $p$, $0<p\le…

Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $\sigma$-finite metric measure space. When $\alpha \in (0,1)$, the subordinated semigroup $\{\exp(-tL^{\alpha}):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and…

We consider the fractional Laplacian operator $(-\Delta)^s$ (let $ s \in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\mathbb{R}^d) $ scalar product between a…

Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we…

We study the boundedness of the $H^{\infty}$ functional calculus for differential operators acting in (L^{p}(\mathbb{R}^{n};\mathbb{C}^{N})). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For…

We relate non integer powers ${\mathcal L}^{s}$, $s>0$ of a given (unbounded) positive self-adjoint operator $\mathcal L$ in a real separable Hilbert space $\mathcal H$ with a certain differential operator of order $2\lceil{s}\rceil$,…

Let $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha >0$ define the Parabolic Hermite-Zygmund spaces $$ \Lambda^\alpha_{\mathcal{L}}=\left\{f: \:f\in…

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $\mathcal{L}$ on a connected and simply connected nilpotent Lie group $G$ with the aid of the corresponding \emph{Rockland} operator…

In this paper, we obtain the desired noncommutative maximal inequalities of the truncated Calder\'on-Zygmund operators of non-convolution type acting on operator-valued $L_p$-functions for all $1<p<\infty$, answering a question left open in…

We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(\Omega)$ and satisfying (generalised) Gaussian estimates, where $\Omega$ is a metric measure space of homogeneous type of dimension $d$. The aim of the article is to show…

Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,\mu)$ with $1<p<\infty$. Let $E$ be a UMD Banach lattice of measurable functions on another measure space $(\Omega,\nu)$. For $f\in L_p(X; E)$ define…

We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^{p}(\mathbb{R}^{n},\exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional…

We begin by recalling the definition of nonnegative quasinearly subharmonic functions on locally uniformly homogeneous spaces. Recall that these spaces and this function class are rather general: among others subharmonic, quasisubharmonic…

Let $L$ be a one-to-one operator of type $\omega$ in $L^2(\mathbb{R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$…