An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function

The generalization of Hardy uncertainty principle for the windowed Bochner–Fourier transform on the Heisenberg group is proved. We consider a Bochner measurable function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:mrow> </m:mrow> </m:math> {Ψ:G→ X} , where X is a completely separable Hilbert space X and Let G be a completely separable, unimodular, connected nilpotent Lie group. We establish that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ϕ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>C</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {φ∈ CC(G)} is a non-trivial window function and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {Ψ∈ L²(G)} satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>ϕ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ψ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mi>HB</m:mi> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mi>exp</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo>∥</m:mo> <m:mi>χ</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> \|Vφ(Ψ)(g,χ)\|HB≤ c₂(g)exp(-β\|χ\|²) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {β>0} , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {Ψ=0} almost everywhere.