arXiv subjects

Emiel Lorist

Publications and source records attributed to Emiel Lorist.

At least 19 recordsLinked to original sources

arXiv2026

The bounds $\hbar_{p,X}\lesssim(\beta_{p,X})^2$ and $\beta_{p,X}\lesssim(\hbar_{p,X})^2$ are sharp

It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\R;X)$, and that the UMD constant $\beta_{p,X}$ and the Hilbert transform constant $\hbar_{p,X}$ are related by the quadratic bounds \begin{equation*} \hbar_{p,X}\lesssim(\beta_{p,X})^2, \qquad \beta_{p,X}\lesssim(\hbar_{p,X})^2. \end{equation*} In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit $2^n$-dimensional Banach spaces for which the Hilbert transform constant grows like $n$ and the UMD constant like $\sqrt n$, and a second family with the reverse behaviour.

math.FA↗

arXiv2026

Sparse domination implies convex body domination

We prove that sparse domination of a bilinear form implies convex body domination. More precisely, if a bilinear form admits an $(r,s)$-sparse bound, then its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound. The proof relies on a randomization argument. We establish the result both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application, we deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.

math.CA↗

arXiv2026

Necessary conditions for deterministic and stochastic maximal regularity

We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.

math.FA↗

arXiv2026

A solution to the inverse generator problem and related questions

We give a negative solution to the inverse generator problem on Hilbert spaces. More precisely, we construct a bounded operator $A$ with dense range on a Hilbert space $H$ that generates a bounded, strongly stable $C_0$-semigroup, while $A^{-1}$ does not generate a $C_0$-semigroup. We also construct an exponentially stable generator $A$ with $0 \in \rho(A)$ such that the inverse semigroup is unbounded and grows at least double logarithmically. For the latter generator, every Cayley transform satisfies the ordinary Kreiss resolvent condition but is neither strongly Kreiss bounded nor power bounded. Moreover, its powers satisfy a doubly logarithmic lower bound. Therefore, the Crank--Nicolson scheme is unstable in operator norm both for every fixed step size over long times and under mesh refinement at any fixed final time. Our counterexamples are deduced from a common finite-dimensional construction. For $\alpha\in(0,1)$, we use explicit bases of $\mathbb C^{2n}$ whose partial-sum projections are uniformly bounded and whose unconditionality constants are comparable to $n^\alpha$. The matrices underlying the counterexamples are then obtained as Schauder multipliers with respect to these bases, using a sequence of eigenvalues whose moduli decay doubly exponentially.

math.FA↗

arXiv2026

A solution to Crouzeix's conjecture

We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.

math.CA↗

arXiv2026

The two-weight fractional Poincar\'e-Sobolev sandwich

We establish a two-weight fractional Poincar\'e-Sobolev sandwich, consisting of a two-weight fractional Poincar\'e-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.

math.CA↗

arXiv2026

Cancellative sparse domination

We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the $H^1$ norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calder\'on-Zygmund operators acting on $H^p$ spaces.

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arXiv2026

Operator-valued Fourier multipliers of bounded s-variation

In this paper, we establish an operator-valued Fourier multiplier theorem in weighted Lebesgue spaces, Besov and Triebel--Lizorkin spaces, assuming the multiplier has $\mathcal{R}$-bounded range and satisfies an $\ell^r$-summability condition on its bounded $s$-variation seminorms over dyadic intervals. The exponents $r$ and $s$ reflect the relationship between the geometric properties of the underlying Banach spaces (type and cotype) and the boundedness of Fourier multiplier operators. As our main tool we prove a weighted vector-valued variational Carleson inequality and deduce an estimate of Littlewood--Paley--Rubio de Francia type.

math.FA↗

arXiv2025

$H^\infty$-calculus for the Dirichlet Laplacian on conical domains

We establish boundedness of the $H^\infty$-calculus for the Dirichlet Laplacian on conical domains in $\mathbb{R}^d$ and corresponding wedges on $L^p$-spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip/edge of the cone/wedge. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal $L^p$-regularity for the Poisson equation on conical domains in appropriate weighted Sobolev spaces.

math.FA↗

arXiv2025

Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains

We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded $C^{1,\lambda}$-domains with $\lambda\in[0,1]$, revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable.

math.AP↗

arXiv2025

Optimization algorithms for Carleson and sparse collections of sets

Carleson and sparse collections of sets play a central role in dyadic harmonic analysis. We employ methods from optimization theory to study such collections. First, we present a strongly polynomial algorithm to compute the Carleson constant of a collection of sets, improving on the recent approximation algorithm of Rey. Our algorithm is based on submodular function minimization. Second, we provide an algorithm showing that any Carleson collection is sparse, achieving optimal dependence of the respective constants and thus providing a constructive proof of a result of H\"anninen. Our key insight is a reformulation of the duality between the Carleson condition and sparseness in terms of the duality between the maximum flow and the minimum cut in a weighted directed graph.

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arXiv2024

Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. We prove that this operator admits a bounded $H^\infty$-calculus on Sobolev spaces with power weights measuring the distance to the boundary. These weights do not necessarily belong to the class of Muckenhoupt $A_p$ weights. We additionally study the corresponding Dirichlet and Neumann heat semigroup. It is shown that these semigroups, in contrast to the $L^p$-case, have polynomial growth. Moreover, maximal regularity results for the heat equation are derived on inhomogeneous and homogeneous weighted Sobolev spaces.

math.FA↗

arXiv2024

Extrapolation of compactness on Banach function spaces

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator $T$ in the weighted Lebesgue scale and the compactness of $T$ in the unweighted Lebesgue scale yields compactness of $T$ on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.

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arXiv2023

Banach function spaces done right

In this survey, we discuss the definition of a (quasi-)Banach function space. We advertise the original definition by Zaanen and Luxemburg, which does not have various issues introduced by other, subsequent definitions. Moreover, we prove versions of well-known basic properties of Banach function spaces in the setting of quasi-Banach function spaces.

math.FA↗

arXiv2023

Strongly Kreiss Bounded Operators in UMD Banach Spaces

In this paper we give growth estimates for $\|T^n\|$ for $n\to \infty$ in the case $T$ is a strongly Kreiss bounded operator on a UMD Banach space $X$. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and $L^p$-spaces, but also intermediate UMD spaces such as non-commutative $L^p$-spaces and variable Lebesgue spaces.

math.FA↗

arXiv2023

Euclidean structures and operator theory in Banach spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators $Γ$ on a Hilbert space. Our assumption on $Γ$ is expressed in terms of $α$-boundedness for a Euclidean structure $α$ on the underlying Banach space $X$. This notion is originally motivated by $\mathcal{R}$- or $γ$-boundedness of sets of operators, but, for example, any operator ideal from the Euclidean space $\ell^2_n$ to $X$ defines such a structure. Therefore, our method is quite flexible. Conversely we show that $Γ$ has to be $α$-bounded for some Euclidean structure $α$ to be representable on a Hilbert space. By choosing the Euclidean structure $α$ accordingly, we get a unified and more general approach to classical factorization and extension theorems. Furthermore we use these Euclidean structures to build vector-valued function spaces and define an interpolation method based on these spaces, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We define generalizations of the classical square function estimates in $L^p$-spaces and establish, via the $H^\infty$-calculus, a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our results for sectorial operators lead to some sophisticated counterexamples.

math.FA↗

arXiv2023

Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner-Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón--Zygmund operators, the previously known quantitative Bloom weighted estimates are not sharp for the second and higher order commutators.

math.CA↗

arXiv2023

Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting

As our main result, we supply the missing characterization of the $L^p(\mu)\to L^q(\lambda)$ boundedness of the commutator of a non-degenerate Calder\'on--Zygmund operator $T$ and pointwise multiplication by $b$ for exponents $1<q<p<\infty$ and Muckenhoupt weights $\mu\in A_p$ and $\lambda\in A_q$. Namely, the commutator $[b,T]\colon L^p(\mu)\to L^q(\lambda)$ is bounded if and only if $b$ satisfies the following new, cancellative condition: $$M^\#_\nu b\in L^{pq/(p-q)}(\nu),$$ where $M^\#_\nu b$ is the weighted sharp maximal function defined by $$ M^\#_\nu b:=\sup_{Q} \frac{\mathbf{1}_Q}{\nu(Q)} \int_{Q} |b-\langle b\rangle_Q |\,\mathrm{d}x$$ and $\nu$ is the Bloom weight defined by $\nu^{1/p+1/q'}:= \mu^{1/p} \lambda^{-1/q}$. In the unweighted case $\mu=\lambda=1$, by a result of Hyt\"onen the boundedness of the commutator $[b,T]$ is, after factoring out constants, characterized by the boundedness of pointwise multiplication by $b$, which amounts to the non-cancellative condition $b\in L^{pq/(p-q)}$. We provide a counterexample showing that this characterization breaks down in the weighted case $\mu\in A_p$ and $\lambda\in A_q$. Therefore, the introduction of our new, cancellative condition is necessary. In parallel to commutators, we also characterize the weighted boundedness of dyadic paraproducts $\Pi_b$ in the missing exponent range $p\neq q$. Combined with previous results in the complementary exponent ranges, our results complete the characterisation of the weighted boundedness of both commutators and of paraproducts for all exponents $p,q \in (1,\infty)$.

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