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Directional operators and mixed norms.

Javier Duoandikoetxea (2002)

Publicacions Matemàtiques

We present a survey of mixed norm inequalities for several directional operators, namely, directional Hardy-Littlewood maximal functions and Hilbert transforms (both appearing in the method of rotations of Calderón and Zygmund), X-ray transforms, and directional fractional operators related to Riesz type potentials with variable kernel. In dimensions higher than two several interesting questions remain unanswered.[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential...

Distinctness of spaces of Lorentz-Zygmund multipliers

Kathryn E. Hare, Parasar Mohanty (2005)

Studia Mathematica

We study the spaces of Lorentz-Zygmund multipliers on compact abelian groups and show that many of these spaces are distinct. This generalizes earlier work on the non-equality of spaces of Lorentz multipliers.

Distributional fractional powers of the Laplacean. Riesz potentials

Celso Martínez, Miguel Sanzi, Francisco Periago (1999)

Studia Mathematica

For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, ( ( - Δ ) α u , ϕ ) = ( u , ( - Δ ) α ϕ ) , α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean...

Doubling properties and unique continuation at the boundary for elliptic operators with singular magnetic fields

Xiangxing Tao (2002)

Studia Mathematica

Let u be a solution to a second order elliptic equation with singular magnetic fields, vanishing continuously on an open subset Γ of the boundary of a Lipschitz domain. An elementary proof of the doubling property for u² over balls centered at some points near Γ is presented. Moreover, we get the unique continuation at the boundary of Dini domains for elliptic operators.

Endpoint bounds for convolution operators with singular measures

E. Ferreyra, T. Godoy, M. Urciuolo (1998)

Colloquium Mathematicae

Let S ⊂ ℝ n + 1 be the graph of the function ϕ : [ - 1 , 1 ] n → ℝ defined by ϕ ( x 1 , ⋯ , x n ) = ∑ j = 1 n | x j | β j , with 1< β 1 ≤ ⋯ ≤ β n , and let μ the measure on ℝ n + 1 induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with μ is L p - L q bounded.

Entropy bump conditions for fractional maximal and integral operators

Robert Rahm, Scott Spencer (2016)

Concrete Operators

We investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

Estimates for maximal singular integrals

Loukas Grafakos (2003)

Colloquium Mathematicae

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.

Estimates for oscillatory singular integrals on Hardy spaces

Hussain Al-Qassem, Leslie Cheng, Yibiao Pan (2014)

Studia Mathematica

For any n ∈ ℕ, we obtain a bound for oscillatory singular integral operators with polynomial phases on the Hardy space H¹(ℝⁿ). Our estimate, expressed in terms of the coefficients of the phase polynomial, establishes the H¹ boundedness of such operators in all dimensions when the degree of the phase polynomial is greater than one. It also subsumes a uniform boundedness result of Hu and Pan (1992) for phase polynomials which do not contain any linear terms. Furthermore, the bound is shown to be valid...

Estimates for singular integrals and extrapolation

Shuichi Sato (2009)

Studia Mathematica

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.

Currently displaying 141 – 160 of 544