Two problems of Calderón-Zygmund theory on product-spaces

Jean-Lin Journé

Displaying similar documents to “Two problems of Calderón-Zygmund theory on product-spaces”

Calderón-Zygmund operators on product spaces.

Jean-Lin Journé (1985)

Revista Matemática Iberoamericana

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Regularity properties of singular integral operators

Abdellah Youssfi (1996)

Studia Mathematica

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For s>0, we consider bounded linear operators from $D (ℝ^{n})$ into $D^{'} (ℝ^{n})$ whose kernels K satisfy the conditions $| \partial_{x}^{γ} K (x, y) | \leq C_{γ} {| x - y |}^{- n + s - | γ |}$ for x≠y, |γ|≤ [s]+1, $| \nabla_{y} \partial_{x}^{γ} K (x, y) | \leq C_{γ} {| x - y |}^{- n + s - | γ | - 1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^{2} (ℝ^{n})$ into the homogeneous Sobolev space $Ḣ^{s} (ℝ^{n})$ . This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some...

Singular integrals on product H spaces.

Robert Fefferman (1985)

Revista Matemática Iberoamericana

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

G. Blower (1998)

Studia Mathematica

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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...

On multilinear singular integrals of Calderón-Zygmund type.

Loukas Grafakos, Rodolfo H. Torres (2002)

Publicacions Matemàtiques

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A variety of results regarding multilinear singular Calderón-Zygmund integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators....

$L^{p}$ estimates for Schrödinger operators with certain potentials

Zhongwei Shen (1995)

Annales de l'institut Fourier

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We consider the Schrödinger operators $- Δ + V (x)$ in $ℝ^{n}$ where the nonnegative potential $V (x)$ belongs to the reverse Hölder class $B_{q}$ for some $q \geq n / 2$ . We obtain the optimal $L^{p}$ estimates for the operators $(- Δ + V)^{i γ}, \nabla^{2} (- Δ + V)^{- 1}, \nabla (- Δ + V)^{- 1 / 2}$ and $\nabla (- Δ + V)^{- 1}$ where $γ \in ℝ$ . In particular we show that $(- Δ + V)^{i γ}$ is a Calderón-Zygmund operator if $V \in B_{n / 2}$ and $\nabla (- Δ + V)^{- 1 / 2}, \nabla (- Δ + V)^{- 1} \nabla$ are Calderón-Zygmund operators if $V \in B_{n}$ .

The boundedness of Calderón-Zygmund operators on the spaces F .

Michel Frazier, Rodolfo Torres, Guido Weiss (1988)

Revista Matemática Iberoamericana

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Calderón-Zygmund operators are generalizations of the singular integral operators introduced by Calderón and Zygmund in the fifties [CZ]. These singular integrals are principal value convolutions of the form Tf(x) = lím_ε→0 ∫_|x-y|>ε K(x-y) f(y) dy = p.v.K * f(x), where f belongs to some class of test functions.

$L^{p} (ℝ ⁿ)$ bounds for commutators of convolution operators

Guoen Hu, Qiyu Sun, Xin Wang (2002)

Colloquium Mathematicae

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The $L^{p} (ℝ ⁿ)$ boundedness is established for commutators generated by BMO(ℝⁿ) functions and convolution operators whose kernels satisfy certain Fourier transform estimates. As an application, a new result about the $L^{p} (ℝ ⁿ)$ boundedness is obtained for commutators of homogeneous singular integral operators whose kernels satisfy the Grafakos-Stefanov condition.

Weighted estimates for commutators of linear operators

Josefina Alvarez, Richard Bagby, Douglas Kurtz, Carlos Pérez (1993)

Studia Mathematica

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We study boundedness properties of commutators of general linear operators with real-valued BMO functions on weighted $L^{p}$ spaces. We then derive applications to particular important operators, such as Calderón-Zygmund type operators, pseudo-differential operators, multipliers, rough singular integrals and maximal type operators.

The trace inequality and eigenvalue estimates for Schrödinger operators

R. Kerman, Eric T. Sawyer (1986)

Annales de l'institut Fourier

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Suppose $Φ$ is a nonnegative, locally integrable, radial function on $R^{n}$ , which is nonincreasing in $| x |$ . Set $(T f) (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ when $f \geq 0$ and $x \in R^{n}$ . Given $1 < p < \infty$ and $v \geq 0$ , we show there exists $C > 0$ so that $\int_{R^{n}} (T f) (x)^{p} v (x) d x \leq C \int_{R^{n}} f (x)^{p} d x$ for all $f \geq 0$ , if and only if $C^{'} > 0$ exists with $\int_{Q} T (x_{Q} v) (x)^{p^{'}} d x \leq C^{'} \int_{Q} v (x) d x < \infty$ for all dyadic cubes Q, where $p^{'} = p / (p - 1)$ . This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

ω-Calderón-Zygmund operators

Sijue Wu (1995)

Studia Mathematica

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We prove a T1 theorem and develop a version of Calderón-Zygmund theory for ω-CZO when $ω \in A_{\infty}$ .