We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and α/n + 1/q = 1/p, the norm $\left|\right|[b,I_{\alpha }]:L^p\left({\mu }^p\right)\to L^q\left({\lambda }^q\right)\left|\right|$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda }^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

A sum of exponentials of the form $f\left(x\right)=exp\left(2\pi i{N}_{1}x\right)+exp\left(2\pi i{N}_{2}x\right)+\cdots +exp\left(2\pi i{N}_{m}x\right)$, where the $N_k$ are distinct integers is called an idempotent trigonometric polynomial (because the convolution of $f$ with itself is $f$) or, simply, an idempotent. We show that for every $p\>1,$ and every set $E$ of the torus $𝕋=ℝ/\mathbb{Z}$ with $\left|E\right|\>0,$ there are idempotents concentrated on $E$ in the $L^p$ sense. More precisely, for each $p\>1,$ there is an explicitly calculated constant $C_p\>0$ so that for each $E$ with $\left|E\right|\>0$ and $ϵ\>0$ one can find an idempotent $f$ such that the ratio ${\left({\int }_E{\left|f\right|}^p/{\int }_{𝕋}{\left|f\right|}^p\right)}^{1/p}$ is greater than $C_p-ϵ$. This is in fact...

We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $\alpha \le {\sum }_{j=1}^{n-1}1/(n²-j²)sin\left(jx\right)\le \beta$, with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0<{\sum }_{j=1}^{n-1}(n²-j²)sin\left(jx\right)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

We present a multidimensional analogue of an inequality by van der Corput-Visser concerning the coefficients of a real trigonometric polynomial. As an application, we obtain an improved estimate from below of the Bohr radius for the hypercone 𝓓₁ⁿ = {z ∈ ℂⁿ: |z₁|+. .. +|zₙ| < 1} when 3 ≤ n ≤ 10.