Area integral estimates for higher order elliptic equations and systems

Elliptic equations in the plane satisfying a Carleson measure condition

“…We now consider the transformation Φ : R 2 + → Ω φ , used by Dahlberg, Kenig and Stein (see [3] and [5]), but also earlier by Nečas [13], defined as (1.2) Φ(x, t) = (x, c 0 t + (θ t * φ)(x)),…”

Commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains

“…In term of layer potential, we have the key observation that ∆q = 0 in R d \ ∂Ω, which leads two important facts. One is that the square function of q may be controlled by the boundary data (see Lemma 2.1), which is based on the equivalence between the square function and the nontangential maximal function (see [2,10]). The other is that |q(x)| 2 δ(x)dx could be a Carleson measure provided the velocity term u is bounded.…”

Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

“…p. 188 in [61]) and rediscovered by Maz'ya and Shaposhnikova [56] (see also [57], and later by Dahlberg et al (cf. [16] and the discussion in [19]), and Hofmann and Lewis [33].…”