A characterization of Hilbert spaces and the vector-valued Littlewood–Paley theorem

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H ∞ -calculus on weighted L p -spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

show abstract

“…Moreover, in the X-valued setting this is even wrong for w = 1 unless X is isomorphic to a Hilbert space (see [35]…”

Section: Preliminariesmentioning

confidence: 99%

The heat equation with rough boundary conditions and holomorphic functional calculus

Lindemulder

Veraar

2020

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…Note that, by Kwapien's theorem [52], unless Y is isomorphic to a Hilbert space, its type and cotype do not coincide, and therefore by combining Theorem 36 with (130) one does not obtain an equivalence between the norms · H s,p (R n ,Y ) and · W s,p (R n ,Y ) for non-Hilbertian targets Y . See [38] for a related characterization of Hilbert space.…”

Section: Quantitative Affine Approximation For Umd Targetsmentioning

confidence: 99%

Quantitative affine approximation for UMD targets

Hytönen¹,

Li²

2016

Discrete Analysis

View full text Add to dashboard Cite

It is shown here that if (Y, · Y ) is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists c = c(Y ) ∈ (0, ∞) with the following property. For every n ∈ N and ε ∈ (0, 1/2], if (X, · X ) is an n-dimensional normed space with unit ball B X and f : B X → Y is a 1-Lipschitz function then there exists an affine mapping Λ : X → Y and a sub-ball B * = y + ρB X ⊆ B X of radius ρ exp(−(1/ε) cn ) such that f (x) − Λ(x) Y ερ for all x ∈ B * . This estimate on the macroscopic scale of affine approximability of vector-valued Lipschitz functions is an asymptotic improvement (as n → ∞) over the best previously known bound even when X is R n equipped with the Euclidean norm and Y is a Hilbert space.

show abstract

“…if and only if X has the UMD property (see [34,62]), and L p (R d ; X) = F 0 p,2 (R d ; X) if and only if X can be renormed as a Hilbert space (see [21] and [44,Remark 7]).…”

Section: 2mentioning

confidence: 99%

“…Strichartz' proof of the multiplier assertion for the characteristic function on H s,p (R d ) is based on a difference norm for these spaces, see [52,Section 2]. It generalizes to H-spaces with values in a Hilbert space, see [60, [21], and Proposition 5.8 for a refinement of this assertion in terms of type and cotype of X). As a substitute, a randomized Littlewood-Paley decomposition is available if X has UMD.…”

Section: Introductionmentioning

confidence: 99%

Pointwise Multiplication on Vector-Valued Function Spaces with Power Weights

Meyries

Veraar

2014

J Fourier Anal Appl

View full text Add to dashboard Cite

Abstract. We investigate pointwise multipliers on vector-valued function spaces over R d , equipped with Muckenhoupt weights. The main result is that in the natural parameter range, the characteristic function of the half-space is a pointwise multiplier on Bessel-potential spaces with values in a UMD Banach space. This is proved for a class of power weights, including the unweighted case, and extends the classical result of Shamir and Strichartz. The multiplication estimate is based on the paraproduct technique and a randomized Littlewood-Paley decomposition. An analogous result is obtained for Besov and Triebel-Lizorkin spaces.

show abstract

A characterization of Hilbert spaces and the vector-valued Littlewood–Paley theorem

Abstract: ABSTRACT. In this note we prove that the existence of the Banach space-valued Littlewood-Paley theorem implies that a Banach space is isomorphic to a Hilbert space.

Cited by 10 publications

References 1 publication

The heat equation with rough boundary conditions and holomorphic functional calculus

The heat equation with rough boundary conditions and holomorphic functional calculus

Quantitative affine approximation for UMD targets

Pointwise Multiplication on Vector-Valued Function Spaces with Power Weights

Contact Info

Product

Resources

About