Variational characterization of Hp

In this paper, we obtain the variational characterization of Hardy space Hp for $p\in (((n)/({n+1})),1]$, and get estimates for the oscillation operator and the λ-jump operator associated with approximate identities acting on Hp for $p\in (((n)/({n+1})),1]$. Moreover, we give counterexamples to show that the oscillation and λ-jump associated with some approximate identity cannot be used to characterize Hp for $p\in (((n)/({n+1})),1]$.

Oscillation and variation for the Riesz transform associated with Bessel operators

Let λ > 0 and letbe the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.

NON-PERIODIC Γ-CONVERGENCE: APPLICATION TO A MAIN EXAMPLE OF A NON-FINER OSCILLATION OPERATOR IN HIGHER DIMENSION

We generalize, to a higher dimension, a main example of an operator that enjoys the NFO property introduced recently in [13]. It turns out that this issue is intimately connected to Γ-convergence of functionals in a non-periodic, general setting. Under a main structural assumption on the sequence of functionals, the Γ-convergence limit is computable. As a consequence of our analysis, we obtain the generalization just mentioned, opening thus the way to treat the existence for some nonlinear PDEs in divergence form.