We show that for every there exists a weight such that the Lorentz Gamma space is reflexive, its lower Boyd and Zippin indices are equal to zero and its upper Boyd and Zippin indices are equal to one. As a consequence, the Hardy-Littlewood maximal operator is unbounded on the constructed reflexive space and on its associate space .
We show that there is a measure y defined on the hyperbolic plane and with polynomial growth, such that the centered maximal operator associated to y does not satisfy weak type (1, 1) bounds.
The existence of a continuous right inverse of the divergence operator in , 1 < p < ∞, is a well known result which is basic in the analysis of the Stokes equations. The object of this paper is to show that the continuity also holds for some weighted norms. Our results are valid for Ω ⊂ ℝⁿ a bounded domain which is star-shaped with respect to a ball B ⊂ Ω. The continuity results are obtained by using an explicit solution of the divergence equation and the classical theory of singular integrals...
We establish an exponential estimate for the relationship between the ergodic maximal function and the maximal operator associated with convolution powers of a probability measure.
Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.
The purpose of this paper is to improve the known results (specifically [1]) concerning Lp boundedness of maximal functions formed using 1 x δ x ... x δ tubes.
The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from to L² for any s > 3/8. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of α = 2. The extension from α = 2 to general α > 1 faces three essential obstacles: the lack of Lee’s reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable...
2000 Mathematics Subject Classification: 42B10, 43A32.In this paper we take the strip KL = [0, +∞[×[−Lπ, Lπ], where L is a positive integer. We consider, for a nonnegative real number α, two partial differential operators D and Dα on ]0, +∞[×] − Lπ, Lπ[. We associate a generalized Fourier transform Fα to the operators D and Dα. For this transform Fα, we establish an Lp − Lq − version of the Morgan's theorem under the assumption 1 ≤ p, q ≤ +∞.
We prove the continuity of an oscillatory singular integral operator T with polynomial phase P(x,y) on an atomic space related to the phase P. Moreover, we show that the cancellation condition to be imposed on T holds under more general conditions. To that purpose, we obtain a van der Corput type lemma with integrability at infinity.
We prove an x-ray estimate in general dimension which is a stronger version of Wolff's Kakeya estimate [12]. This generalizes the estimate in [13], which dealt with the n = 3 case.
We study the problem of -boundedness () of operators of the form for a commuting system of self-adjoint left-invariant differential operators on a Lie group of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when is a homogeneous group and are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.