Estimates for singular integrals and extrapolation

Shuichi Sato

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Displaying similar documents to “Estimates for singular integrals and extrapolation”

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.

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.

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.

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.

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.

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.

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...

Hubert integrals, Singular integrals, and Radon transforms II.

E.M. Stein, D.H. Phong (1986)

Inventiones mathematicae

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On nonincreasing singular solutions to the Emden-Fowler equation.

Rabtsevich, V.A. (2000)

Memoirs on Differential Equations and Mathematical Physics

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On nondecreasing singular solutions of the Emden-Fowler equation.

Rabtsevich, V.A. (2000)

Memoirs on Differential Equations and Mathematical Physics

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Positive solutions for a singular second-order ordinary differential equation.

Zhou, W.S., Cai, S.F. (2006)

Lobachevskii Journal of Mathematics

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On complete solutions and complete singular solutions of second order ordinary differential equations

Masatomo Takahashi (2007)

Colloquium Mathematicae

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A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular...

Variational inequalities for singular integral operators

Albert Mas (2012)

Journées Équations aux dérivées partielles

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In these notes we survey some new results concerning the $ρ$ -variation for singular integral operators defined on Lipschitz graphs. Moreover, we investigate the relationship between variational inequalities for singular integrals on AD regular measures and geometric properties of these measures. An overview of the main results and applications, as well as some ideas of the proofs, are given.

On the numerical solution of the multidimensional singular integrals and integral equations, used in the theory of linear viscoelasticity.

Ladopoulos, E.G. (1988)

International Journal of Mathematics and Mathematical Sciences

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Singular solutions to systems of conservation laws and their algebraic aspects

V. M. Shelkovich* (2010)

Banach Center Publications

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We discuss the definitions of singular solutions (in the form of integral identities) to systems of conservation laws such as shocks, δ-, δ’-, and $δ^{(n)}$ -shocks (n = 2,3,...). Using these definitions, the Rankine-Hugoniot conditions for δ- and δ’-shocks are derived. The weak asymptotics method for the solution of the Cauchy problems admitting δ- and δ’-shocks is briefly described. The algebraic aspects of such singular solutions are studied. Namely, explicit formulas for flux-functions of...

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.

Analogies between differential equations: Singular solutions of difference equations of Clairaut's type.

J.D. Keckic (1977)

Publications de l'Institut Mathématique [Elektronische Ressource]

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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.

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...

Finite-part singular integral approximations in Hilbert spaces.

Ladopoulos, E.G., Tsamasphyros, G., Zisis, V.A. (2004)

International Journal of Mathematics and Mathematical Sciences

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On a Question of Pełczyński about Strictly Singular Operators

Jesús M. F. Castillo, Marilda Simoes, Jesús Suárez de la Fuente (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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We exhibit new examples of weakly compact strictly singular operators with dual not strictly cosingular and characterize the weakly compact strictly singular surjections with strictly cosingular adjoint as those having strictly singular bitranspose. We then obtain new examples of super-strictly singular quotient maps and show that the strictly singular quotient maps in Kalton-Peck sequences are not super-strictly singular.