Dirac operators, conformal transformations and aspects of classical harmonic analysis.
Directional operators and mixed norms.
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...
Discrete analogues of singular and maximal Radon transforms.
Distinctness of spaces of Lorentz-Zygmund multipliers
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
For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, , 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
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.
Eine axiomatische Charakterisierung der Hilbert-Transformation
Endpoint boundedness for multilinear integral operators of some sublinear operators on Herz and Herz type Hardy spaces.
Endpoint bounds for convolution operators with singular measures
Let be the graph of the function defined by with 1< and let the measure on 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 is - bounded.
Endpoint estimates and weighted norm inequalities for commutators of fractional integrals.
Endpoint estimates for multilinear commutator of Marcinkiewicz operator.
Entropy bump conditions for fractional maximal and integral 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.
Ergodic theory and connections with analysis and probability.
Errata of the paper “On the - boundedness of some fractional integral operators”
Estimates for differential operators of vector analysis involving -norm
Estimates for maximal singular integrals
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 -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
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 Convolution Operators on the Heisenberg Group.
Estimates for singular integral operators in terms of maximal functions
Estimates for singular integral operators with mixed type homogeneities in Hardy classes.