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 introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagonal estimates as a substitute for Calderón-Zygmund kernel estimates. As an application, we study differentiability properties of paraproducts in terms of fractional powers of the operator $L$. The results of this paper are fundamental for the proof of a T(1)-Theorem for operators beyond Calderón-Zygmund theory, which will be the subject of a forthcoming paper.
We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.
We consider the dyadic paraproducts $π_\f$ on $\T$ associated with an $\M$-valued function $\f.$ Here $\T$ is the unit circle and $\M$ is a tracial von Neumann algebra. We prove that their boundedness on $L^p(\T,L^p(\M))$ for some $1<p<\infty $ implies their boundedness on $L^p(\T,L^p(\M))$ for all $1<p<\infty$ provided $\f$ is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M)).
Denote by $M_n$ the algebra of $n\times n$ matrices. We consider the dyadic paraproducts $π_b$ associated with $M_n$ valued functions $b$, and show that the $L^\infty (M_n)$ norm of $b$ does not dominate $||π_b||_{L^2(\ell _n^2)\to L^2(\ell_n^2)}$ uniformly over $n$. We also consider paraproducts associated with noncommutative martingales and prove that their boundedness on bounded noncommutative $L^p-$% martingale spaces implies their boundedness on bounded noncommutative $L^q-$% martingale spaces for all $1<p<q<\infty $.
We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calderón--Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. We get two weight $A_2$-$A_\infty$ estimates, that in the one weight case give us the estimate $$ \|T\|_{L^2(W)\to L^2 (W)} \le C [W]_{A_2}^{1/2} [W]_{A_\infty} \le C[W]_{A_2}^{3/2} $$ where $T$ is either a Calderon--Zygmund operator (with…
Let $(T_t)_{t \geq 0}$ be a markovian (resp. submarkovian) semigroup on some $σ$-finite measure space $(Ω,μ)$. We prove that its negative generator $A$ has a bounded $H^\infty(Σ_θ)$ calculus on the weighted space $L^2(Ω,wdμ)$ as long as the weight $w : Ω\to (0,\infty)$ has finite characteristic defined by $Q^A_2(w) = \sup_{t > 0} \left\| T_t(w) T_t \left(w^{-1} \right) \right\|_{L^\infty(Ω)}$ (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle $θ> \fracπ{2}$ is admissib…
Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$ type estimates. In conjunction with $L_\infty \to BMO$, we get certain row/column …
We prove spectral multiplier theorems for Hörmander classes $\mathcal{H}^α\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(Σ\_σ)$ calculus for some $σ\in (0,π)$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}\_+$, the wave operators $e^{isA}$ for $s \in \mathbb{R}$, the resolvent $(λ-- A)^{-1}$ on $\mathbb{C} \backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t \in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u)^α\_+$ for $u > 0.$ In contrast to the existing literature we neither assume t…
This note deals with the boundedness of the $H^\infty$ functional calculus of Ritt operators $T$ and associated square function estimates. The purpose is to give a shorter, concise and slightly more general approach towards Le~Merdy's results [19] on this subject.
Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,μ)$ with $1<p<\infty$. Let $E$ be a UMD Banach lattice of measurable functions on another measure space $(Ω,ν)$. For $f\in L_p(X; E)$ define $$\mathcal M(f)(x, ω)=\sup_{t>0}\frac1t\Big|\int_0^tT_s(f(\cdot,ω))(x)ds\Big|,\quad (x,ω)\in X\timesΩ.$$ Then the following maximal ergodic inequality holds $$\big\|\mathcal M(f)\big\|_{L_p(X; E)}\lesssim \big\|f\big\|_{L_p(X; E)},\quad f\in L_p(X; E).$$ If the semigroup $\{T_t\}_{t>0}$ is additionally assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analyt…
In this paper we extend the $H^\infty$ functional calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called $S$-functional calculus. The $S$-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties. The $S$-functional calculus is based on the notion of $S$-spectru…
We develop a very general operator-valued functional calculus for operators with an $H^{\infty}-$calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an $H^{\infty}$calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of $L_p-$maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structu…
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 calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
By using an $H^{\infty}$ joint functional calculus for strongly commuting operators, we derive a scheme to deduce the $L^p$ boundedness of certain $d$-dimensional Riesz transforms from the $L^p$ boundedness of appropriate one-dimensional Riesz transforms. Moreover, the $L^p$ bounds we obtain are independent of the dimension. The scheme is applied to Riesz transforms connected with orthogonal expansions and discrete Riesz transforms on products of groups with polynomial growth.
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 \norm{x} and \bignorm{\bigl(\int_{0}^{\infty}|A^{*}^{1/2} T_t^*(y)|^2\, dt\,\bigr)^{1/2}_{Lp'} \lesssim \norm{y} for x in Lp and y in Lp', then A admits a bounded H^{\infty}(Σ_θ) functional calculus for any θ>\fracπ{2}. We show that this actually holds true for some θ<\fracπ{2}.
This paper is devoted to the multivariable $H^\infty$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $(A_1,\ldots, A_d)$ is such a family, if $A_k$ is $R$-sectorial of $R$-type $ω_k\in(0,π)$, $k=1,\ldots,d$, and if $(A_1,\ldots, A_d)$ admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for some $θ_k\in (ω_k,π)$, then it admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for all $θ_k\in (ω_k,π)$, $k=1,\ldots,d$. Second we introduce square funct…
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
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 $L^{r}(γ_{\infty})$, $1<r<\infty$, in any sector of angle $θ>θ^{*}_{r}$, 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 dimensio…
We show that a bounded analytic semigroup on an $L_p$-space has a bounded $H^{\infty}(Σ_φ)$-calculus for some $φ< \fracπ{2}$ if and only if the semigroup can be obtained, after restricting to invariant subspaces, factorizing through invariant subspaces and similarity transforms, from a bounded analytic semigroup on some bigger $L_p$-space which is positive and contractive on the real line.
We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.
We introduce and investigate $H^\infty$-functional calculus for commuting finite families of Ritt operators on Banach space $X$. We show that if either $X$ is a Banach lattice or $X$ or $X^*$ has property $(α)$, then a commuting $d$-tuple $(T_1,\ldots, T_d)$ of Ritt operators on $X$ has an $H^\infty$ joint functional calculus if and only if each $T_k$ admits an $H^\infty$ functional calculus. Next for $p\in(1,\infty)$, we characterize commuting $d$-tuple of Ritt operators on $L^p(Ω)$ which admit an $H^\infty$ joint functional calculus, by a joint dilation property. We also obtain a similar cha…
We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty}$ functional calculus with respect to some angle $< π/2$ can be dilated into a bounded analytic semigroup $(R_t)_{t\geq 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t$. We also establish a discrete analogue for Ritt operators and consider the case when $L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations o…
We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^\infty$ has a bounded $H^\infty(S_η^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $η>π/2$, provided the semigroup satisfies Bakry-Emry's $Γ_2 $ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,α}=e^{-tL^α},0<α<1$, that is proved in the first half of the article.
Given a Hilbert space and the generator $A$ of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in {\mathcal H}_{\infty}$ we show that there exists an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the "normal" definition of $g(A)$. In particular, when $g(s)=1/(s + α)$, $ α\in {\mathbb C}_0^+$, then this admissible output operator equals $(αI - A)^{-1}$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every (s…
Using notions from the geometry of Banach spaces we introduce square functions $γ(Ω,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(Ω,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $γ(Ω,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a…
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.
In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,\infty)^2$. Our construction has special features that functions on $[0,\infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions wit…
The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical ``$T(1)\in BMO$'' assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical $T(1)$ criterion still guarantees their $L^2$-boundedness on Hilbert space …
We show that, given a Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction. Finally, it is shown that the calculus coincides with one f…
$H^\infty$ Functional Calculus for a Commuting Pair of $\text{Ritt}_{\text{E}}$ Operators
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
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 calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that $L$ has a bounded $H^{\infty}$ functional calculus with this optimal angle.
We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(Ω)$ and satisfying (generalised) Gaussian estimates, where $Ω$ is a metric measure space of homogeneous type of dimension $d$. The aim of the article is to show that $A \otimes \mathrm{Id}_Y$ admits a Hörmander type $\mathcal{H}^β_2$ functional calculus on $L^p(Ω;Y)$ where $Y$ is a UMD lattice, thus extending the well-known Hörmander calculus of $A$ on $L^p(Ω)$. We show that if $T_t$ is lattice positive (or merely admits an $H^\infty$ calculus on $L^p(Ω;Y)$) then this is indeed the case. Here the derivation exponent has to sa…
On a doubling metric measure space $(M,d,μ)$ endowed with a "carré du champ", let $\mathcal{L}$ be the associated Markov generator and $\dot L^{p}_α(M,\mathcal{L},μ)$ the corresponding homogeneous Sobolev space of order $0<α<1$ in $L^p$, $1<p<+\infty$, with norm $\left\|\mathcal{L}^{α/2}f\right\|_p$. We give sufficient conditions on the heat semigroup $(e^{-t\mathcal{L}})_{t>0}$ for the spaces $\dot L^{p}_α(M,\mathcal{L},μ) \cap L^\infty(M,μ)$ to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and…
We derive conditions that ensure the existence of a bounded $H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions $\underline{A}_T$ of a differential operator $A$ on a conic manifold with boundary, subject to differential boundary conditions $T$. In general, these conditions ask for a particular pseudodifferential structure of the resolvent $(λ-\underline{A}_T)^{-1}$ in a sector $Λ\subset\mathbf{C}$. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem $(A,T)$. Examples concern the Dirichlet and Neumann Laplacians.
We discuss the notion of $γ$-$H^{\infty}$-bounded calculus, strong $γ$-$m$-$H^{\infty}$-bounded calculus on half-plane and weak-$γ$-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of $γ$-bounded $C_0$-semigroup in $K$-convex space, which leads to a version of Gearhart-Prüss on $K$-convex space.
On a manifold $X$ with boundary and bounded geometry we consider a strongly elliptic second order operator $A$ together with a degenerate boundary operator $T$ of the form $T=φ_0γ_0 + φ_1γ_1$. Here $γ_0$ and $γ_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $φ_0,φ_1\in C^{\infty}_b(\partial X)$, $φ_0,φ_1\ge 0$, and $φ_0+φ_1\geq c$, for some $c>0$. We also assume that the highest order coefficients of $A$ belong to $C^τ(X)$ for some $τ>0$ and the lower order coefficients are in $L_\infty(X)$. We show that the $L_p(X)$-rea…
A Ritt operator T : X --> X on Banach space is a power bounded operator such that the sequence of all n(T^{n} -T^{n-1}) is bounded. When X=Lp for some 1<p<\infty, we study the validity of square functions estimates Norm{(\sum_k k |T^{k}(x) - T^{k-1}(x)|^2)^{1/2}}_{Lp} \leq K Norm{x}_{Lp} for such operators. We show that T and its adjoint T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analog of the famous Cowling-Doust-McIntosh-Yagi characterization of bounded H^\infty-calculus on $Lp$-spaces by the…
We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T : X --> X on some Banach space X with the following property: T has a bounded $\H^\infty$ functional calculus with respect to the unit disc $\D$ (that is, T is polynomially bounded) but T does not have any bounded $\H^\infty$ functional calculus with respect to a Stolz domain of $\D$ with vertex at 1. Also we show that for an R-Ritt operator, the unconditional Ritt condition of Kalton-Portal is equivalent to the existence of a bounded $\H^\infty$ functional calculus with respect to s…
This paper generalizes the classical Sz.-Nagy--Foias $H^{\infty}(\mathbb{D})$ functional calculus for Hilbert space contractions. In particular, we replace the single contraction $T$ with a tuple $T=(T_1, \dots, T_d)$ of commuting bounded operators on a Hilbert space and replace $H^{\infty}(\mathbb{D})$ with a large class of multiplier algebras of Hilbert function spaces on the unit ball in $\mathbb C^d$.
We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas $b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analy…
The main purpose of this paper is to introduce the geometric difference sequence space $l_\infty^{G} (Δ_G)$ and prove that $l_\infty^{G} (Δ_{G})$ is a Banach space with respect to the norm $\left\|.\right\|^G_{Δ_G}.$ Also we compute the $α$-dual, $β$-dual and $γ$-dual spaces. Finally we obtain the Geometric Newton-Gregory interpolation formulae.
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 $\mathcal{L}_0$ on the `local' contraction $G_0$ of $G$, as well as of the corresponding Rockland operator $\mathcal{L}_\infty$ on the `global' contraction $G_\infty$ of $G$. We provide asymptotic estimates of the Riesz potentials associated with $\mathcal{L}$ at $0$ and at $\infty$, as well as of the kernels associated with functions of $\mathcal{L}$ satisfying M…
Let $A$ be a generator of an analytic semigroup having a H{ö}rmander functional calculus on $X = L^p(Ω,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{ö}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(Ω,Y)} \lesssim \|f\|_{L^p(Ω,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12}…
In this paper, we present a new class of operators, which we name to be $n$-Ritt operators. This produces a discrete analogue of $n$-sectorial operators and generalizes the notion of Ritt operators. We develop a $H^\infty$-functional calculus for $n$-Ritt operators and prove an useful transference result. We also generalize the notions of quadratic functional calculus and $n$-$R$-Ritt operator and discuss some examples.
