Weighted composition operators on weighted Bergman spaces induced by doubling weights

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi $ on Bergman type spaces $A_\omega ^p $ induced by a doubling weight ω. Let $X=\{u\in H(\mathbb{D} ): uC_\varphi \colon A_\omega ^p\to A_\omega ^p\ \text {is bounded}\}$. For some regular weights ω, we obtain that $X=H^\infty $ if and only if ϕ is a finite Blaschke product.

Trace inequalities for weighted Hardy—Sobolev spaces in the non-diagonal case

We give characterizations of the positive Borel measures μ on $\mathbb{S}^n$ so that the weighted Hardy–Sobolev spaces $H_s^p(w)$ are imbedded in Lq(dμ), for a range of s > 0, 0 < p, q < + ∞, q ≠ p, where w is a doubling weight in the unit sphere of ℂn.

Extension Theorems on Weighted Sobolev Spaces and Some Applications

We extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

On Weighted Sobolev Spaces

We study density and extension problems for weighted Sobolev spaces on bounded (ε, δ) domains𝓓when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of 𝓓 and when it is in the MuckenhouptApclass locally in 𝓓. Moreover, when the weightswi(x) are of the form dist(x,Mi)αi,αi∈ ℝ,Mi⊂ 𝓓that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.