The Marcinkiewicz multiplier condition for bilinear operators

Loukas Grafakos; Nigel J. Kalton

Displaying similar documents to “The Marcinkiewicz multiplier condition for bilinear operators”

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...

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.

Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet (2012)

Colloquium Mathematicae

Similarity:

Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T \otimes I d_{X}$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^{p} (G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient...

A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

Michał Wojciechowski (2000)

Studia Mathematica

Similarity:

It is proved that if $m : ℝ^{d} \to ℂ$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the $H^{1}$ space on the product domain $ℝ^{d_{1}} \times . . . \times ℝ^{d_{k}}$ . This implies an estimate of the norm $N (m, L^{p} (ℝ^{d})$ of the multiplier transformation of m on $L^{p} (ℝ^{d})$ as p→1. Precisely we get $N (m, L^{p} (ℝ^{d})) ≲ {(p - 1)}^{- k}$ . This bound is the best possible in general.

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...

The space of multipliers into $l_{1}$

P. Wojtaszczyk (1979)

Annales Polonici Mathematici

Similarity:

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.

Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators

G. Blower (1998)

Studia Mathematica

Similarity:

We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍ \mapsto Γ_{φ} ⨍^{'} (S)$ to be completely bounded on $H^{\infty} B (H)$ ; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1 - r) | | ⨍^{'} (r e^{i θ}) {| |}_{B (H)}^{2} r d r d θ$ and $(1 - r) | | ⨍ " (r e^{i θ}) {| |}_{B (H)} r d r d θ$ are Carleson measures, then ⨍ multiplies ${(H^{1} c^{1})}^{'}$ to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map $⨍ \mapsto Γ_{φ} ⨍^{'} (S)$ is bounded $A \to B (H^{2} (H), L^{2} (H) ⊖ H^{2} (H))$ . Thus we construct a functional calculus for operators of Foiaş-Williams-Peller...

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...

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

Tadeusz Pytlik (1984)

Colloquium Mathematicae

Similarity:

Multipliers of sequence spaces

Raymond Cheng, Javad Mashreghi, William T. Ross (2017)

Concrete Operators

Similarity:

This paper is selective survey on the space lAp and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality

Multipliers of the space $A^{p}$

I. Jovanović (1986)

Matematički Vesnik

Similarity:

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

Jolanta Dlugosz (1987)

Colloquium Mathematicae

Similarity:

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 $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

Transference and restriction of maximal multiplier operators on Hardy spaces

Zhixin Liu, Shanzhen Lu (1993)

Studia Mathematica

Similarity:

The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^{p} (ℝ^{n})$ and $H^{p} (^{n})$ , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty} (ℝ^{n})$ function m is a maximal multiplier on $H^{p} (ℝ^{n})$ if and only if it is a maximal multiplier on $H^{p} (^{n})$ . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

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...

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 $| | χ_{E} {| |}_{M_{c b} A ()}$ .

Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem

Anthony Carbery (1988)

Annales de l'institut Fourier

Similarity:

We show that the maximal operator associated to the family of rectangles in $R^{3}$ one of whose sides is parallel to $(1, 2^{j}, 2^{k})$ for some j,k $\in H Z$ is bounded on $L^{p}$ , $1 < p < \infty$ . We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.