In [8], we studied the problem of local solvability of complex coefficient second order left-invariant differential operators on the Heisenberg group ℍₙ, whose principal parts are "positive combinations of generalized and degenerate generalized sub-Laplacians", and which are homogeneous under the Heisenberg dilations. In this note, we shall consider the same class of operators, but in the presence of left invariant lower order terms, and shall discuss local solvability for these operators in a complete...

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O\left(\right|x{|}^m{e}^{-\alpha x²})$ and $O\left(\right|x\left|ⁿe^{-x²/\left(4\alpha \right)}\right)$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P\left(x\right)e^{-\alpha x²}$ and $P^{\text{'}}\left(x\right)e^{-x²/\left(4\alpha \right)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o\left(e^{-x²/\left(4\alpha \right)}\right)$, then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

Let G be a connected locally compact group with a left invariant Haar measure μ. We prove that the function ξ(x) = inf μ̅(AB): μ(A) = x is concave for any fixed bounded set B ⊂ G. This is used to give a new proof of Kemperman’s inequality $\mu ̲\left(AB\right)\ge min\left(\mu ̲\right(A)+\mu ̲(B),\mu (G\left)\right)$ for unimodular G.

For some time it has been known that there exist continuous Helson curves in ${\mathbf{R}}^2$. This result, which is related to Lusin’s rearrangement problem, had been proved first by Kahane in 1968 with the aid of Baire category arguments. Later McGehee and Woodward extended this result, giving a concrete construction of a Helson $k$-manifold in ${\mathbf{R}}^{nk}$ for $n\ge k+1$. We present a construction of a Helson 2-manifold in ${\mathbf{R}}^3$. With modification, our method should even suffice to prove that there are Helson hypersurfaces in any ${\mathbf{R}}^n$.

We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^M={[0,1)}^M$, with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^M$; (3) the measures have the convolution property that $\mu \ast L^p\subseteq L^{p+\epsilon }$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $\mu \ast L^p\subseteq L^q$ for any measure μ in our class.

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^p\left(\right)$ to $L^q\left(\right)$. We also give a condition on p which is necessary if this operator maps $L^p\left(\right)$ into L²().

In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if $G$ and $H$ are countable amenable non-type I groups, then the unitary duals of $G$ and $H$ are Borel isomorphic.

Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu$. The $\delta$-ball $B_{\delta }$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $\mu \left(B_{2{\delta }^{\prime }}\right)\le 2^d\mu \left(B_{{\delta }^{\prime }}\right)$ for all ${\delta }^{\prime }\subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu \left(nA\right)\le n^d\mu \left(A\right)$ for all $n\ge dlogd$, then $A$ is contained in an $O\left(d{log}^{3}d\right)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu \left(B\right)\le exp\left(O\right(dlogd\left)\right)\mu \left(A\right)$.