Displaying similar documents to “Weak-type (1,1) bounds for oscillatory singular integrals with rational phases”

On the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

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We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that g ' ( x ) = 2 x for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

Boundedness of certain oscillatory singular integrals

Dashan Fan, Yibiao Pan (1995)

Studia Mathematica

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We prove the L p and H 1 boundedness of oscillatory singular integral operators defined by Tf = p.v.Ω∗f, where Ω ( x ) = e i Φ ( x ) K ( x ) , K(x) is a Calderón-Zygmund kernel, and Φ satisfies certain growth conditions.

The Hardy-Lorentz spaces H p , q ( )

Wael Abu-Shammala, Alberto Torchinsky (2007)

Studia Mathematica

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We deal with the Hardy-Lorentz spaces H p , q ( ) where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

Estimates for maximal singular integrals

Loukas Grafakos (2003)

Colloquium Mathematicae

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It is shown that maximal truncations of nonconvolution L²-bounded singular integral operators with kernels satisfying Hörmander’s condition are weak type (1,1) and L p -bounded for 1 < p< ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.

Young's (in)equality for compact operators

Gabriel Larotonda (2016)

Studia Mathematica

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If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then λ k ( | a b * | ) λ k ( 1 / p | a | p + 1 / q | b | q ) for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if | a | p = | b | q .

L p ( ) bounds for commutators of convolution operators

Guoen Hu, Qiyu Sun, Xin Wang (2002)

Colloquium Mathematicae

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The L p ( ) 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 ( ) boundedness is obtained for commutators of homogeneous singular integral operators whose kernels satisfy the Grafakos-Stefanov condition.

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 A .

Generalized Hörmander conditions and weighted endpoint estimates

María Lorente, José María Martell, Carlos Pérez, María Silvina Riveros (2009)

Studia Mathematica

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We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded...

Boundedness of commutators of strongly singular convolution operators on Herz-type spaces

Zongguang Liu (2003)

Studia Mathematica

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The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, T b m , on Herz spaces K ̇ q α , p ( ) and K q α , p ( ) , and on a new class of Herz-type Hardy spaces H K ̇ q , b , m α , p , 0 ( ) and H K q , b , m α , p , 0 ( ) , where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

Ψ-pseudodifferential operators and estimates for maximal oscillatory integrals

Carlos E. Kenig, Wolfgang Staubach (2007)

Studia Mathematica

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We define a class of pseudodifferential operators with symbols a(x,ξ) without any regularity assumptions in the x variable and explore their L p boundedness properties. The results are applied to obtain estimates for certain maximal operators associated with oscillatory singular integrals.

An extension of a boundedness result for singular integral operators

Deniz Karlı (2016)

Colloquium Mathematicae

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We study some operators originating from classical Littlewood-Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one-dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are the G* and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on L p . Moreover, we generalize a classical multiplier theorem by weakening...

Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2

Michael Hitrik, Johannes Sjöstrand (2008)

Annales scientifiques de l'École Normale Supérieure

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We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint h -pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided...

Boundedness of vector-valuedB-singular integral operators in Lebesgue spaces

Seyda Keles, Mehriban N. Omarova (2017)

Open Mathematics

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We study the vector-valued B-singular integral operators associated with the Laplace-Bessel differential operator △B=∑k=1n−1∂ 2∂x k 2+(∂2∂x n 2+2vxn∂∂x n),v>0. B = k = 1 n - 1 2 x k 2 + ( 2 x n 2 + 2 v x n x n ) , v > 0 . We prove the boundedness of vector-valued B-singular integral operators A from [...] Lp,v(R+n,H1)toLp,v(R+n,H2), L p , v ( + n , H 1 ) to L p , v ( + n , H 2 ) , 1 < p < ∞, where H1 and H2 are separable Hilbert spaces.

Gropes and the rational lift of the Kontsevich integral

James Conant (2004)

Fundamenta Mathematicae

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We calculate the leading term of the rational lift of the Kontsevich integral, Z , introduced by Garoufalidis and Kricker, on the boundary of an embedded grope of class, 2n. We observe that it lies in the subspace spanned by connected diagrams of Euler degree 2n-2 and with a bead t-1 on a single edge. This places severe algebraic restrictions on the sort of knots that can bound gropes, and in particular implies the two main results of the author’s thesis [1], at least over the rationals. ...

Non-homogeneous strongly singular integrals

Bassam Shayya (2008)

Studia Mathematica

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We study the L p mapping properties of a family of strongly singular oscillatory integral operators on ℝⁿ which are non-homogeneous in the sense that their kernels have isotropic oscillations but non-isotropic singularities.

Boundedness of para-product operators on spaces of homogeneous type

Yayuan Xiao (2017)

Czechoslovak Mathematical Journal

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We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H p ( 𝒳 ) for 1 / ( 1 + ϵ ) < p 1 , where 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ϵ is the regularity exponent of the kernel of the singular integral operator T . Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast...

Boundedness of the Hausdorff operators in H p spaces, 0 < p < 1

Elijah Liflyand, Akihiko Miyachi (2009)

Studia Mathematica

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Sufficient conditions for the boundedness of the Hausdorff operators in the Hardy spaces H p , 0 < p < 1, on the real line are proved. Two related negative results are also given.

Commutators on ( q ) p

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on X = ( q ) p with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces

Sergio Antonio Tozoni (2004)

Studia Mathematica

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Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis ( e j ) j 1 . Given an operator T from L c ( X ) to L¹(X), we consider the vector-valued extension T̃ of T given by T ̃ ( j f j e j ) = j T ( f j ) e j . We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give...

Disjoint strict singularity of inclusions between rearrangement invariant spaces

Francisco L. Hernández, Víctor M. Sánchez, Evgueni M. Semenov (2001)

Studia Mathematica

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It is studied when inclusions between rearrangement invariant function spaces on the interval [0,∞) are disjointly strictly singular operators. In particular suitable criteria, in terms of the fundamental function, for the inclusions L ¹ L E and E L ¹ + L to be disjointly strictly singular are shown. Applications to the classes of Lorentz and Marcinkiewicz spaces are given.