Note on differentiation of integrals and the halo conjecture
We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces . This space has been introduced and the result for has been proved in Ding, Zhu (2017). In this paper, for , we establish its dual space .
Let and , where a(s) is a positive continuous function such that and b(s) is quasi-increasing and . Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants and such that for all ; (jj) there exist positive constants and such that for all .
We establish a decomposition of non-negative Radon measures on which extends that obtained by Strichartz [6] in the setting of -dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.
The Hardy inequality with holds for if is an open set with a sufficiently smooth boundary and if . P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for .
In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.
It is known that the weak type (1,1) for the Hardy-Littlewood maximal operator can be obtained from the weak type (1,1) over Dirac deltas. This theorem is due to M. de Guzmán. In this paper, we develop a technique that allows us to prove such a theorem for operators and measure spaces in which Guzmán's technique cannot be used.
In this paper we give a sufficient condition on the pair of weights for the boundedness of the Weyl fractional integral from into . Under some restrictions on and , this condition is also necessary. Besides, it allows us to show that for any there exist non-trivial weights such that is bounded from into itself, even in the case .
Let P(z,β) be the Poisson kernel in the unit disk , and let be the λ -Poisson integral of f, where . We let be the normalization . If λ >0, we know that the best (regular) regions where converges to f for a.a. points on ∂ are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of toward f in an weakly tangential region, if and p > 1. In the present paper we will extend the result to symmetric spaces X of...