Let Γ be a compact d-set in ℝⁿ with 0 < d ≤ n, which includes various kinds of fractals. The author shows that the Besov spaces $B_{pq}^s\left(\Gamma \right)$ defined by two different and equivalent methods, namely, via traces and quarkonial decompositions in the sense of Triebel are the same spaces as those obtained by regarding Γ as a space of homogeneous type when 0 < s < 1, 1 < p < ∞ and 1 ≤ q ≤ ∞.

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:L_q\left(\mu \right)\to L_p\left(\nu \right)$, each n ∈ ℕ and functions $f_1,...,f_n\in L_q\left(\mu \right)$, $\left(ʃ\right({\sum }_{k=1}^n|T{f}_k{|}^r{)}^{p/r}{d\nu )}^{1/p}\le c\parallel T\parallel \left(ʃ\right({\sum }_{k=1}^n|f_k{|}^r{{)}^{q/r}d\mu )}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...

We determine the norm in $L^p\left(ℝ₊\right)$, 1 < p < ∞, of the operator $I-{ℱ}_s{ℱ}_c$, where ${ℱ}_c$ and ${ℱ}_s$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^p$-norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...

We consider complex-valued functions f ∈ L¹(ℝ), and prove sufficient conditions in terms of f to ensure that the Fourier transform f̂ belongs to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary in the case of real-valued functions f for which either xf(x) ≥ 0 or f(x) ≥ 0 almost everywhere.

Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies ${\sum }_{n\in \mathbb{Z}}{\left|f̂\left(n\right)\right|}^p\omega \left(n\right)<\infty$. Then there exists a weight ν on ℤ such that ${\sum }_{n\in \mathbb{Z}}{\left|\widehat{(1/f)}\left(n\right)\right|}^p\nu \left(n\right)<\infty$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An analogue of...