Non-homogeneous strongly singular integrals

Bassam Shayya

Displaying similar documents to “Non-homogeneous strongly singular integrals”

Boundedness of higher order commutators of oscillatory singular integrals with rough kernels

Huoxiong Wu (2005)

Studia Mathematica

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The author studies the commutators generated by a suitable function a(x) on ℝⁿ and the oscillatory singular integral with rough kernel Ω(x)|x|ⁿ and polynomial phase, where Ω is homogeneous of degree zero on ℝⁿ, and a(x) is a BMO function or a Lipschitz function. Some mapping properties of these higher order commutators on $L^{p} (ℝ ⁿ)$ , which are essential improvements of some well known results, are given.

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.

Maximal singular integrals on product homogeneous groups

Yong Ding, Shuichi Sato (2014)

Studia Mathematica

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We prove $L^{p}$ boundedness for p ∈ (1,∞) of maximal singular integral operators with rough kernels on product homogeneous groups under a sharp integrability condition of the kernels.

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.

Singular integrals with highly oscillating kernels on product spaces

Elena Prestini (2000)

Colloquium Mathematicae

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We prove the $L^{2} (^{2})$ boundedness of the oscillatory singular integrals $P_{0} f (x, y) = \int_{D_{x}} e^{i (M_{2} (x) y^{'} + M_{1} (x) x^{'})} ο v e r x^{'} y^{'} f (x - x^{'}, y - y^{'}) d x^{'} d y^{'}$ for arbitrary real-valued $L^{\infty}$ functions $M_{1} (x), M_{2} (x)$ and for rather general domains $D_{x} \subseteq^{2}$ whose dependence upon x satisfies no regularity assumptions.

$L^{p} (ℝ ⁿ)$ boundedness for the commutator of a homogeneous singular integral operator

Guoen Hu (2003)

Studia Mathematica

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The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $Ω \in L {(l o g L)}^{k + 1} (S^{n - 1})$ is a sufficient condition for the kth order commutator to be bounded on $L^{p} (ℝ ⁿ)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.

Estimates for singular integrals and extrapolation

Shuichi Sato (2009)

Studia Mathematica

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We study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove $L^{p}$ boundedness of the singular integrals under a certain sharp size condition on their kernels.

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...

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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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.

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.

Boundedness of commutators of an oscillatory integral operator

Xia Xia, Shanzhen Lu (2008)

Studia Mathematica

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We obtain a necessary and sufficient condition for $L^{p}$ boundedness of commutators of certain oscillatory integral operators and Lipschitz functions.

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 $λ_{k} (| a b * |) \leq λ_{k} {(1 / p | a |}^{p} + 1 / {q | b |}^{q})$ 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 ${| a |}^{p} = {| b |}^{q}$ .

Convergence of singular integrals with general measures

Pertti Mattila, Joan Verdera (2009)

Journal of the European Mathematical Society

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We show that $L^{2}$ -bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.

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 $L ¹ \cap L^{\infty} ↪ E$ and $E ↪ L ¹ + L^{\infty}$ to be disjointly strictly singular are shown. Applications to the classes of Lorentz and Marcinkiewicz spaces are given.

Singular $φ$ -Laplacian third-order BVPs with derivative dependance

Smaïl Djebali, Ouiza Saifi (2016)

Archivum Mathematicum

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This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $φ$ -Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition...

On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE

Martin Rohleder (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The paper investigates the singular initial problem[4pt] ${(p (t) u^{'} (t))}^{'} + q (t) f (u (t)) = 0, u (0) = u_{0}, u^{'} (0) = 0$ [4pt] on the half-line $[0, \infty)$ . Here $u_{0} \in [L_{0}, L]$ , where $L_{0}$ , $0$ and $L$ are zeros of $f$ , which is locally Lipschitz continuous on $ℝ$ . Function $p$ is continuous on $[0, \infty)$ , has a positive continuous derivative on $(0, \infty)$ and $p (0) = 0$ . Function $q$ is continuous on $[0, \infty)$ and positive on $(0, \infty)$ . For specific values $u_{0}$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$ , $p$ and $q$ it is shown that the problem has for each specified...

$L^{p}$ type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson (2006)

Studia Mathematica

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We show in two dimensions that if $K f = \int_{ℝ ₊ ²} k (x, y) f (y) d y$ , $k (x, y) = (e^{i x^{a} \cdot y^{b}}) {/ (| x - y |}^{η})$ , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_{p} (y) = y^{(p / p^{'}) (1 ̅ - b / a)}$ , then ${| | K f | |}_{p} {\leq C | | f | |}_{p, v_{p}}$ if η + α₁ + α₂ < 2, $α_{j} = 1 - b_{j} / a_{j}$ , j = 1,2. Our methods apply in all dimensions and also for more general kernels.

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 $1 / (1 + ϵ) < p \leq 1$ , 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...

A counter-example in singular integral theory

Lawrence B. Difiore, Victor L. Shapiro (2012)

Studia Mathematica

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An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let $f \in C ¹ (ℝ^{N} ∖ 0)$ and suppose f vanishes outside of a compact subset of $ℝ^{N}$ , N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the $L^{p}$ -sense. Set $F (x) = \int_{ℝ^{N}} k (x - y) f (y) d y \forall x \in ℝ^{N} ∖ 0$ . Then F(x) = O(log²r) as r → 0 in the $L^{p}$ -sense, 1 <...

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 ${(e_{j})}_{j \geq 1}$ . Given an operator T from $L_{c}^{\infty} (X)$ to L¹(X), we consider the vector-valued extension T̃ of T given by $T ̃ (\sum_{j} f_{j} e_{j}) = \sum_{j} T (f_{j}) e_{j}$ . 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...

Commutators on ${(\sum ℓ_{q})}_{p}$

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

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ 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.

Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Sergei V. Astashkin, Francisco L. Hernández, Evgeni M. Semenov (2009)

Studia Mathematica

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If G is the closure of $L_{\infty}$ in exp L₂, it is proved that the inclusion between rearrangement invariant spaces E ⊂ F is strictly singular if and only if it is disjointly strictly singular and E ⊊ G. For any Marcinkiewicz space M(φ) ⊂ G such that M(φ) is not an interpolation space between $L_{\infty}$ and G it is proved that there exists another Marcinkiewicz space M(ψ) ⊊ M(φ) with the property that the M(ψ) and M(φ) norms are equivalent on the Rademacher subspace. Applications are given and a question...

Common zero sets of equivalent singular inner functions

Keiji Izuchi (2004)

Studia Mathematica

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Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $| ψ_{μ ₀} | < 1 \cap | ψ_{λ ₀} | < 1 = \emptyset$ , where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν; ν \sim μ} Z (ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$ . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty}$ which relates naturally to the set of measures equivalent...

Carleson's theorem with quadratic phase functions

Michael T. Lacey (2002)

Studia Mathematica

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It is shown that the operator below maps $L^{p}$ into itself for 1 < p < ∞. $C f (x) : = s u p_{a, b} | p . v . \int f (x - y) e^{i (a y ² + b y)} d y / y |$ . The supremum over b alone gives the famous theorem of L. Carleson [2] on the pointwise convergence of Fourier series. The supremum over a alone is an observation of E. M. Stein [12]. The method of proof builds upon Stein’s observation and an approach to Carleson’s theorem jointly developed by the author and C. M. Thiele [7].

Anisotropic classes of homogeneous pseudodifferential symbols

Árpád Bényi, Marcin Bownik (2010)

Studia Mathematica

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We define homogeneous classes of x-dependent anisotropic symbols $Ṡ_{γ, δ}^{m} (A)$ in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent...

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. $▵_{B} = \sum_{k = 1}^{n - 1} \frac{\partial^{2}}{\partial x_{k}^{2}} + (\frac{\partial^{2}}{\partial x_{n}^{2}} + \frac{2 v}{x_{n}} \frac{\partial}{\partial x_{n}}), v > 0 .$ We prove the boundedness of vector-valued B-singular integral operators A from [...] Lp,v(R+n,H1)toLp,v(R+n,H2), $L_{p, v} (ℝ_{+}^{n}, H_{1}) to L_{p, v} (ℝ_{+}^{n}, H_{2}),$ 1 < p < ∞, where H1 and H2 are separable Hilbert spaces.