Let ɸ be a given set of real-valued functions on the set T and let ß: ɸ → R given functional with values in the extended real line R = [−∞, ∞]. The objective of the paper is to construct contents or measures μ with good regularity and smoothness properties satisfying the integral inequalities ∫* φ dμ ≤ ß (φ), where ∫* denotes the upper integral.
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We provide a new division formula for holomorphic mappings. It is given in terms of residue currents and has the advantage of being more explicit and simpler to prove than the previously known formulas.
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Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br> $R_{m,n} = {Z ∈ M_{m,n}: I^{(m)} - ZZ* is positive definite}$. Here $I^{(m)} ∈ M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L^{p}_{α}(R_{m,n})$ is defined to be the space $L^p{R_{m,n}; [det(I^{(m)} - ZZ*)]^α dμ_{m,n}(Z)}$, where $μ_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H^p_α(R_{m,n}) ⊂ L^{p}_{α}(R_{m,n})$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Reβ > (α+1)/p -1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then $f(𝒵)= T^{β}_{m,n}(f)(𝒵), 𝒵 ∈ R_{m,n}, where $T^{β}_{m,n}$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p < ∞, we find conditions on α and β for $T^{β}_{m,n}$ to be a bounded projection of $L^p_α(R_{m,n})$ onto $H^p_α(R_{m,n})$. Some applications of this result are given.
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In 1945 the first author introduced the classes $H^p(α)$, 1 ≤ p<∞, α > -1, of holomorphic functions in the unit disk 𝔻 with finite integral (1) ∬_𝔻 |f(ζ)|^p (1-|ζ|²)^α dξ dη < ∞ (ζ=ξ+iη) and established the following integral formula for $f ∈ H^p(α)$: (2) f(z) = (α+1)/π ∬_𝔻 f(ζ) ((1-|ζ|²)^α)/((1-zζ̅)^{2+α}) dξdη, z∈ 𝔻 . We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(Ω;[K(w)]^α dm(w))$, where: 1) $Ω = {w = (w₁,...,w_n) ∈ ℂ^n: Im w₁ > ∑_{k=2}^n |w_k|²}$, $K(w) = Im w₁ - ∑_{k=2}^n |w_k|²$; 2) Ω is the matrix domain consisting of those complex m × n matrices W for which $I^{(m)} - W·W*$ is positive-definite, and $K(W) = det[I^{(m)} - W·W*]$; 3) Ω is the matrix domain consisting of those complex n × n matrices W for which $Im W = (2i)^{-1} (W - W*)$ is positive-definite, and K(W) = det[Im W]. Here dm is Lebesgue measure in the corresponding domain, $I^{(m)}$ denotes the unit m × m matrix and W* is the Hermitian conjugate of the matrix W.
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