Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet

Displaying similar documents to “Unconditionality, Fourier multipliers and Schur multipliers”

Multipliers of the Hardy space H¹ and power bounded operators

Gilles Pisier (2001)

Colloquium Mathematicae

Similarity:

We study the space of functions φ: ℕ → ℂ such that there is a Hilbert space H, a power bounded operator T in B(H) and vectors ξ, η in H such that φ(n) = ⟨Tⁿξ,η⟩. This implies that the matrix ${(φ (i + j))}_{i, j \geq 0}$ is a Schur multiplier of B(ℓ₂) or equivalently is in the space (ℓ₁ ⊗̌ ℓ₁)*. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of H¹ which we call “shift-bounded”. We show that there is a φ which...

The space of multipliers into $l_{1}$

P. Wojtaszczyk (1979)

Annales Polonici Mathematici

Similarity:

Multilinear Fourier multipliers with minimal Sobolev regularity, I

Loukas Grafakos, Hanh Van Nguyen (2016)

Colloquium Mathematicae

Similarity:

We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces $H^{p_{k}}$ , $0 < p_{k} \leq 1$ , to Lebesgue spaces $L^{p}$ . These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral...

Multipliers of the space $A^{p}$

I. Jovanović (1986)

Matematički Vesnik

Similarity:

The Herz-Schur multiplier norm of sets satisfying the Leinert condition

Éric Ricard, Ana-Maria Stan (2011)

Colloquium Mathematicae

Similarity:

It is well known that in a free group , one has $| | χ_{E} {| |}_{M_{c b} A ()} \leq 2$ , where E is the set of all the generators. We show that the (completely) bounded multiplier norm of any set satisfying the Leinert condition depends only on its cardinality. Consequently, based on a result of Wysoczański, we obtain a formula for $| | χ_{E} {| |}_{M_{c b} A ()}$ .

A multiplier theorem for Fourier series in several variables

Nakhle Asmar, Florence Newberger, Saleem Watson (2006)

Colloquium Mathematicae

Similarity:

We define a new type of multiplier operators on $L^{p} (^{N})$ , where $^{N}$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on $L^{p} (^{N})$ , to which the theorem applies as a particular example.

The Marcinkiewicz multiplier condition for bilinear operators

Loukas Grafakos, Nigel J. Kalton (2001)

Studia Mathematica

Similarity:

This article is concerned with the question of whether Marcinkiewicz multipliers on $ℝ^{2 n}$ give rise to bilinear multipliers on ℝⁿ × ℝⁿ. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy...

Unitary multipliers on $L_{2} (G)$

Tadeusz Pytlik (1984)

Colloquium Mathematicae

Similarity:

Multipliers for the twisted Laplacian

E. K. Narayanan (2003)

Colloquium Mathematicae

Similarity:

We study $¹ - L^{p}$ boundedness of certain multiplier transforms associated to the special Hermite operator.

Symmetric Bessel multipliers

Khadija Houissa, Mohamed Sifi (2012)

Colloquium Mathematicae

Similarity:

We study the $L^{p}$ -boundedness of linear and bilinear multipliers for the symmetric Bessel transform.

(E,F)-Schur multipliers and applications

Fedor Sukochev, Anna Tomskova (2013)

Studia Mathematica

Similarity:

For two given symmetric sequence spaces E and F we study the (E,F)-multiplier space, that is, the space of all matrices M for which the Schur product M ∗ A maps E into F boundedly whenever A does. We obtain several results asserting continuous embedding of the (E,F)-multiplier space into the classical (p,q)-multiplier space (that is, when $E = l_{p}$ , $F = l_{q}$ ). Furthermore, we present many examples of symmetric sequence spaces E and F whose projective and injective tensor products are not isomorphic...

Endpoint bounds of square functions associated with Hankel multipliers

Jongchon Kim (2015)

Studia Mathematica

Similarity:

We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^{p}$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of...

$L^{p}$ -multipliers for the Laguerre expansions

Jolanta Dlugosz (1987)

Colloquium Mathematicae

Similarity:

Fourier coefficients of continuous functions and a class of multipliers

Serguei V. Kislyakov (1988)

Annales de l'institut Fourier

Similarity:

If $x$ is a bounded function on $Z$ , the multiplier with symbol $x$ (denoted by $M_{x})$ is defined by $(M_{x} f) \hat{} = x \hat{f}$ , $f \in L^{2} (T)$ . We give some conditions on $x$ ensuring the “interpolation inequality” $∥ M_{x} f ∥_{L^{p}} \leq C ∥ f ∥_{L^{1}}^{α} ∥ M_{x} f ∥_{L^{q}}^{1 - α}$ (here $1 < p < q$ and $α = α (p, q, x)$ is between 0 and 1). In most cases considered $M_{x}$ fails to have stronger $L^{1}$ -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E \subset Z$ every positive sequence in $ℓ^{2} (E)$ can be majorized by the sequence ${$ $| \hat{f} (n) |}_{n \in E}$ for some continuous funtion $f$ with spectrum...

Pointwise multipliers on martingale Campanato spaces

Eiichi Nakai, Gaku Sadasue (2014)

Studia Mathematica

Similarity:

We introduce generalized Campanato spaces $_{p, ϕ}$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then $_{p, ϕ} = B M O$ . We give a characterization of the set of all pointwise multipliers on $_{p, ϕ}$ .

Extensions of weak type multipliers

P. Mohanty, S. Madan (2003)

Studia Mathematica

Similarity:

We prove that if $Λ \in M_{p} (ℝ^{N})$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_{p} (ℝ^{N})$ is the space of multipliers of $L^{p} (ℝ^{N})$ .

Rearrangement of coefficients of Fourier series on $S U_{2}$

Giancarlo Travaglini (1987)

Colloquium Mathematicae

Similarity:

Fourier multipliers for Hölder continuous functions and maximal regularity

Wolfgang Arendt, Charles Batty, Shangquan Bu (2004)

Studia Mathematica

Similarity:

Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the $L^{p}$ -situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.

Spherical summation : a problem of E.M. Stein

Antonio Cordoba, B. Lopez-Melero (1981)

Annales de l'institut Fourier

Similarity:

Writing $(T_{R}^{λ} f) \hat{} (ξ) = (1 - | ξ |^{2} / R^{2})_{+}^{λ} \hat{f} (ξ)$ . E. Stein conjectured $∥ {(\sum_{j} | T_{R_{j}}^{λ} f_{i} |^{2})}^{1 / 2} ∥_{p} \leq C ∥ {(\sum_{j} | f_{j} |^{2})}^{1 / 2} ∥_{p}$ for $λ > 0$ , $\frac{4}{3} \leq p \leq 4$ and $C = C_{λ, p}$ . We prove this conjecture. We prove also $f (x) = {lim}_{j \to \infty} T_{2^{j}}^{λ} f (x)$ a.e. We only assume $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol (2006)

Studia Mathematica

Similarity:

We consider the multiplier $m_{μ}$ defined for ξ ∈ ℝ by $m_{μ} (ξ) ≐ {((1 - ξ ₁ ² - ξ ₂ ²) / (1 - ξ ₁))}^{μ} 1_{D} (ξ)$ , D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which $m_{μ}$ is a Fourier multiplier on $L^{p}$ is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show...

Two examples of subspaces in $L^{2 l}$ spanned by characters of finite order

Mats Erik Andersson (2001)

Colloquium Mathematicae

Similarity:

By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2 l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

Some results on $A_{p} (G)$ submodules of $P M_{p} (G)$

Edmond E. Granirer (1987)

Colloquium Mathematicae

Similarity:

$L^{p}$ spectral multipliers on the free group $N_{3, 2}$

Alessio Martini, Detlef Müller (2013)

Studia Mathematica

Similarity:

Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group $N_{3, 2}$ on three generators. We prove a theorem of Mikhlin-Hörmander type for the functional calculus of L, where the order of differentiability s > 6/2 is required on the multiplier.

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

Similarity:

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.