Cauchy–Leray–Fantappiè integral in linearly convex domains

Constructive description of Hardy–Sobolev spaces on strongly convex domains in Cn

“…To prove this we apply T 1-theorem for transformations with operator-valued kernels formulated by Hytönen and Weis in [9], taking in account that in our case concerned spaces are Hilbert. Some details of the proof are similar to the proof of the boundedness of operator Cauchy-Leray-Fantappiè K Ω for lineally convex domains introduced in [14]. Below we formulate the T 1-theorem, adapted to our context.…”

“…The function d(w, z) = | ∂ρ(w), w − z | defines on ∂Ω quasimetric, and if B(z, δ) = {w ∈ ∂Ω : d(w, z) < δ} is a quasiball with respect to d then σ(B(z, δ)) ≍ δ n , see for example [14]. Therefore {∂Ω, d, σ} is a space of homogeneous type.…”

External Area Integral Inequality for the Cauchy-Leray-Fantappiè Integral

“…The function d(w, z) = | ∂ρ(w), w − z | defines on ∂Ω quasimetric, and if B(z, δ) = {w ∈ ∂Ω : d(w, z) < δ} is a quasiball with respect to d then σ(B(z, δ)) ≍ δ n , see for example [13]. Therefore {∂Ω, d, σ} is a space of homogeneous type.…”

“…To prove this we apply T 1-theorem for transformations with operator-valued kernels formulated by Hytönen and Weis in [7], taking in account that in our case concerned spaces are Hilbert. Some details of the proof are similar to the proof of the boundedness of operator Cauchy-Leray-Fantappiè K Ω for lineally convex domains introduced in [13]. Below we formulate the T 1-theorem, adapted to our context.…”