Let $E \subset \mathbb R^{n+1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $$\partial_tu- \Delta u=0, \quad \text{in} R^{n+1}\setminus E$$ satisfies Carleson measure estimate condition. An important technical novelty of our work is we develop corona domain approximation scheme for $E$ in terms regular Lip(1/2,1) graph domains. This has an analogous ell...

and Applied Analysis 3 2. Main Results and Proofs Lemma 2.1. For α α1, . . . , αn ∈ D, let uα z1, . . . , zn ∏n j 1 1 − |αj |2 / 1 − αjzj . Then uα z1, . . . , zn ∈ Lφa D , and ‖uα z ‖σn ≤ 1 φ−1 ∏n j 1 ( 1/δ2 j )) . 2.1 Proof. It is easy to see that ‖uα z ‖∞ ∏n j 1 1 |αj | / 1 − |αj | 2 ∏n j 1 2 − δj /δj . Since φ 0 0, the convexity of φ implies φ ax ≤ aφ x for 0 ≤ a ≤ 1. Hence, for every C > 0...

In Ho and Russell (SIAM J Control Optim 21(4):614–640, 1983), and Weiss (Syst Control Lett 10(1): 79–82, 1988), a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given.We extend their results from the Hilbert space situation (L2-admissibility on the state space 2) to the more general situation of L p-admissibility on the stat...

We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the ¯ ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the ¯ ∂-equation, which satisfies simultaneously a good L ∞ estimate and a good L 1 estimate. Th...