The ρ-variation as an operator between maximal operators and singular integrals

Abstract

The ρ-variation and the oscillation of the heat and Poisson semigroups of the Laplacian and Hermite operators (i.e. and − + |x| 2 ) are proved to be bounded fromin the case p = 1) for 1 ≤ p < ∞ and w being a weight in the Muckenhoupt's A p class. In the case p = ∞ it is proved that these operators do not map L ∞ into itself. Even more, they map L ∞ into B M O but the range of the image is strictly smaller that the range of a general singular integral operator.

Weighted variation inequalities for differential operators and singular integrals in higher dimensions

“…Poisson) semigroup relative to the Laplacian of R d , up to a multiple constant. For this two examples, the above corollary goes back to [4] Both Theorems 1 and 2 can be extended to the vector-valued case. The following result for the differential operators improves Fefferman-Stein's celebrated vector-valued.…”

L p -boundedness properties of variation operators in the Schrödinger setting

“…This result applies to the symmetric diffusion semigroup {W t } t>0 generated by the Euclidean Laplacian ∆. Recently, Crescimbeni, Macías, Menárguez, Torrea and Viviani ( [9]), by using vector valued Calderón-Zygmund theory, have proved that the operators V ρ (W t ) map L 1 (R n ) into L 1,∞ (R n ), for each ρ > 2. These results are contained in the following.…”

“…According to [20,Theorem 3.3] T is bounded from L 2 (R n ) into L 2 Eρ (R n ). Moreover, T is a Calderón-Zygmund operator associated with the E ρ -valued kernel K(x, y; t) = W t (x, y), x, y ∈ R n , t > 0, that satisfies the following properties (see [9]):…”

Weighted Lépingle inequality

“…Lépingle's inequality also found applications in ergodic theory [Bou89] and harmonic analysis [NOT10], see [MSZ18] and [DOP17; DDU18] and references therein, respectively, for recent developments in these directions. Weighted inequalities in harmonic analysis go back to [Muc72], and weighted variational inequalities have been studied since [Cre+09]. A major motivation of the weighted theory is the Rubio de Francia extrapolation theorem that allows to obtain vector-valued L p inequalities for all 1 < p < ∞ from scalar-valued weighted L p inequalities for a single p, see [Duo11, Section 3] for the most basic version of that result and [DPW17, Theorem 8.1] for a version applicable to martingales.…”