### A characterization of bi-invariant Schwartz space multipliers on nilpotent Lie groups

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Let G be a locally compact abelian group and M(G) its measure algebra. Two measures μ and λ are said to be equivalent if there exists an invertible measure ϖ such that ϖ*μ = λ. The main result of this note is the following: A measure μ is invertible iff |μ̂| ≥ ε on Ĝ for some ε > 0 and μ is equivalent to a measure λ of the form λ = a + θ, where a ∈ L¹(G) and θ ∈ M(G) is an idempotent measure.

We prove an ${L}^{p}$-boundedness result for a convolution operator with rough kernel supported on a hyperplane of a group of Heisenberg type.

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function ${\phi}_{a}\left(t\right):=\phi \left({\alpha}_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma}_{w}*\left({\phi}_{a}\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${\u0245}_{\phi}^{a}$ to be the union of all...

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{NL}}^{1,Q}$ by ${L}^{\infty}$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for $$ on the Heisenberg group ${\mathbb{H}}^{n}$.