In quantum physics, the concept of the matter wave provides

an apparatus for the mathematical description of the position

of the particle and its Fourier conjugation momentum char-

acterizes the motion of the particle, position, and motion of

the particle intricately undividedly entangled and cannot be

sharply simultaneously known. According to Heisenberg, the

standard deviation of position ∆xand the standard deviation

of momentum ∆pcannot be known simultaneously more

precisely than ℏ

2, where ℏis the reduced Planck constant,

from an analytical perspective, it means that the pair position

and momentum related via Pontryagin conjugation, and the

wave functions in two dual orthonormal bases in the Hilbert

space are Fourier transform of one another, more generally, the

mathematical variant of the uncertainty principle states that a

function ψand its Fourier transform F(ψ)cannot be highly

concentrated at the same time [2, 5, 8, 12]. The uncertainty

principle can be formulated in several different versions that

were proved by Hardy, Narayanan [22, 23], Morgan, Cowling-

Price [16], etc. see the references therein.

Since harmonic analysis is a classical brunch of mathemat-

ical science, there exists extensive literature on the subject

[2, 5, 8], however, mathematical aspects of the uncertainty

principle (the Heisenberg principle of quantum mechanics) are

still demanding additional investigations [1-7].

Following notations of D.L. Donoho and P.B. Stark [2], we

denote an integrable function on the locally compact Hausdorff

group Gby ψand its Fourier transform F(ψ)by ˆ

ψ. Assume

that ψ∈Lp(G)is ε-concentrated on a measurable set T

and Fourier transform ˆ

ψ δ-concentrated on Wthen we have

obtained the uncertainty estimation in the form

1 + δ

1−ε−δp

µG(T) ˆµ(U)≥1

where µGis measure Haar on Gand ˆµis a Plancherel measure

on the set of irreducible unitary representations of G.

To deﬁne the Fourier transform F:L2(G/K)→

⟨V(·)⟩(P, ˆµ)= Υ on G/K we need to deﬁne a direct

integral decomposition ⟨V(·)⟩(P, ˆµ)=Υ=R⊕V(ω)dˆµ(ω)

of L2(G/K). The direct integral R⊕V(ω)dˆµ(ω)of the set

{V(ω)}ωof Hilbert spaces with respect to the measure ˆµis

a space of measurable vector ﬁelds v∈V(ω)of the variable

ωsuch that

∥v∥L2

2=ZP

∥v(ω)∥V(ω)

2dˆµ(ω).

Let Gbe a locally compact group and let π(ω)be a unitary

representation Gon a Hilbert space V(ω)deﬁned for every

ω, and mapping ω7→ π(g, ω)be a measurable ﬁeld of

mapping for every g∈G, then a unitary representation of

Gon ⟨V(·)⟩(P, ˆµ)is expressed by

π(ω) = Z⊕

(g, ω)dˆµ(ω).

The Fourier transform F:L2(G/K)→ ⟨V(·)⟩(P, ˆµ)= Υ

is deﬁned by F(ψ) (ω) = π(ψ, ω)v(ω), where unit vector

v(ω)∈V(π(ω)) = V(ω).

In its most general form, the Plancherel theory estab-

lishes that presume (G, K)is communicative and ˆµis

its Plancherel measure then ∥ψ∥L2(G/K)=∥F()∥L2(G/K)

and the inverse Fourier transform is given by ψ(g) =

⟨⟨F(ψ), π (g, ·)v(·)⟩V⟩(P, ˆµ).

We are going to compare loosely the uncertainty principle

for G/K with the classical uncertainty inequality of quantum

The Plancherel theory and the uncertainty principle

MYKOLA IVANOVICH YAREMENKO

The National Technical University of Ukraine

Kyiv, UKRAINE

Abstract: In this article, we are revising the Plancherel theory for a unimodular locally compact Hausdorff group with a

Haar measure. Let be a connected semisimple real Lie group such that there exists an analytic diffeomorphism from the

manifold to group according to the rule , decomposition is the Iwasawa decomposition of the group , dim(A) = rank(G).

The group center(G) ⊂ K is closed, under the adjoint representation of is a maximal compact subgroup of the adjoint of ;

subgroups and are simply connected. The associated minimal parabolic subgroup of is . Let and be Lies algebras of and ,

respectively, the norms correspond to the and dual algebra relative to the inner product induced by the Killing form of .Let

an irreducible unitary representation of being presented as a left translation on where is finite-dimensional. Let be an

element of the complexification of .Loosely said the Hardy uncertainty principle maintains that the function and its Fourier

transform cannot be simultaneously both rapidly decreasing. The uncertainty principle is considered from several points of

view: first, we consider the uncertainty principle in the case of a locally compact Hausdorff group G equipped with a

probabilistic Haar measure μG and K be a maximal compact subgroup of G with a probabilistic Haar measure μK then we

establish (1 + δ)p μG (T) 𝜇 (U) ≥ (1 − ε − δ)p, where T is ε-concentration for ψ ∈ Lp (G); second, we establish several

variants of the statement that a function and its Fourier transform cannot be too rapidly decreasing namely |ψ (g)| ≤ c1 exp

( −c2 ∥g∥2) and ∥π (ψ, u, 𝑢)∥ ≤ 𝑐1 (u) exp ( −𝑐2 ∥ 𝑢∥2) for all g ∈ G, on semisimple Lie group with the finite center.

Keywords: Hausdorff groups, Heisenberg principle, uncertainty principle, Fourier transform, Wigner function, compact

group, Peter-Weyl theorem.

Received: November 24, 2022. Revised: June 16, 2023. Accepted: July 23, 2023. Published: August 28, 2023.

1. Introduction

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mechanics. Let Aand Bbe a pair of Hermitian operators

and let Ψbe a physical state of the quantum system. The

uncertainties of the operators Aand Bin the state Ψis denoted

by ∆Aand ∆B, respectively. Then, the classical uncertainty

inequality can be presented in the form

Ψ

2i[A, B]

Ψ

≤∆A·∆B

or if we take A= ˆxand B= ˆpthen obtain

Ψ

2i[ˆx, ˆp]

Ψ2

≤(∆x)2·(∆p)2

since 1

2i[ˆx, ˆp] = ℏ

2we have

ℏ

2≤∆x·∆p.

If p= 2 then the uncertainty principle for G/K can be

rewritten as

1−ε−δ≤µG(T) ˆµ(U).

So, loosely, since position and momentum are conjugate

variables the right part of the classical equation is a special

case of the general theory, and for the left side, we have that

1−ε−δin the case of G/K corresponds to ℏ

2of classical

case.

In the simple case of Rn, the Hardy uncertainty principle

can be considered for the Fourier transform given by

F(ψ) (χ) = ˆ

ψ(χ) = (2π)−n

2ZRn

exp (−iχ ·x)ψ(x)dx,

and if ψ∈L1(Rn)and

ZRnZRn

exp (|χ·x|)|F(χ)| |ψ(x)|dxdχ < ∞

then we have that necessary ψ= 0 almost everywhere.

The broader instance of Hardy’s principle is given by the

following statement: let Gbe a connected semisimple Lie

group with a ﬁnite center having a uniquely deﬁned class of

Cartan subgroups, let Kbe a maximal compact subgroup of

G. The centralizer of the exponent Aof a maximal abelian

subspace of positively deﬁned Cartan-Killing form on the Lie

algebras of G, in Kis denoted by M. Let ψbe a measurable

function on Gsuch that

|ψ(g)| ≤ c1exp −c2∥g∥2

for all g∈G, and the estimation

∥π(ψ, v, ˜v)∥ ≤ ˜c1(v) exp −˜c2∥˜v∥2

holds for all v∈ˆ

Mand ˜v∈a∗, where c1, c2,˜c1(v),˜c2are

constants. If the product c2˜c2>1

4then the function ψequals

zero almost everywhere.

The proof of the broader Hardy uncertainty principle em-

ploys that the Plancherel measure is supported on ˆ

M×a∗

then from the condition inequalities, we have π(ψ, v, ˜v)=0

on ˆ

M×a∗therefore the Hardy principle is proven.

Let Gbe a locally compact Hausdorff group equipped

with a Haar measure µ. We deﬁne the character χof the

group Gas a topological continuous group homomorphism

χ:G→U(1). For instance, if we assume that Gis

an additive group then the character satisﬁes the condition

χ(g−h) = χ(g) (χ(h))−1for all g, h ∈G.

Deﬁnition 1. The Fourier transform Fof a function ψ∈

L2(G)TL1(G)is given by

F(ψ) (χ) = ˆ

ψ(χ) = ZG

ψ(g)χ(g)dµ (g).(1)

Deﬁnition 2.The topological group ˆ

Gconsisting of all

characters χ:G→U(1) on Gwith its natural operation

on multiplication is called a dual group of G.

Let Gbe a locally compact Hausdorff group equipped with

a Haar measure µthen one can uniquely deﬁne a Haar measure

ˆµon the dual group ˆ

G. The measure ˆµis deﬁned by ˆµ(χ) =

RGχ(g)dµ (g), this is a Fourier-Stieltjes transformation of

the measure µ.

Deﬁnition 3. The Fourier inversion transform F−1of a

function ˆ

ψ∈L2ˆ

GTL1ˆ

Gis deﬁned by

ψ(g) = F−1ˆ

ψ(g) = ˆ

ψ∨

(g) =

=Rˆ

Gˆ

ψ(χ)χ(g)dˆµ(χ).

(2)

We denote M (G)the associative Banach algebra of all

measures on the σ-algebra of all Borel sets of a Hausdorff

topological locally compact group G. Let µ, η ∈M (G), the

convolution of measures µand ηis deﬁned by

(µ∗η) (D) = ZG×G

ϕD(g, h)dµ (g)dη (h),(3)

where ϕDis an indicator of D, namely, ϕD(g, h) =

1if g, h ∈D

0if g, h ∈G\D.

Convolution of the elements of L1(G)is deﬁned by

(ψ∗φ) (g) = ZG

ψ(h)φh−1gdµ (h),(4)

and agrees with the convolution of the measures when L1(G)

is naturally embedded in M (G).

Straightforward calculation yields

F(µ∗η) (χ) = F(µ) (χ)F(η) (χ) = ˆµ(χ) ˆη(χ).(5)

Let Gbe a unimodular locally compact Hausdorff group.

If the continuous homomorphism πfrom Ginto the group

U(H)of the unitary operators on the separable Hilbert space

H, such that mapping π:G→U(H)satisﬁes conditions:

π(gh) = π(g)π(h)and πg−1= (π(g))−1= (π(g))∗,

(??)

and the mapping g→π(g)xis continuous for all x∈H,

then the homomorphism πis called a unitary representation

of Gin H.

2. Fourier Transform

3. The Plancherel Theory

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Let us denote the right and left regular representations of

Gon L2(G)by

ρ(g)ψ(h) = ψ(hg)(6)

and

λ(g)ψ(h) = ψg−1h(7)

respectively.

Assume that the group Gis such that every primary

representation of the group Gis a direct sum of copies of

irreducible representations. The matrix coefﬁcient Λof the

unitary representation πis a function Λ (ψ, φ) (g)given by

Λ (ψ, φ) (g) = ⟨π(g)ψ, φ⟩(8)

for ψ, φ ∈L2(G) = Hand g∈G. Let ψ=ej, φ =ek

then denote

πjk (g) = Λ (ej, ek) (g) = ⟨π(g)ej, ek⟩,(9)

where the system {ej}is a basis of L2(G) = H. Each element

Λπis uniquely corresponded with a continuous function such

that for each ﬁnite-dimensional representation πthere exists

a decomposition Λπ=⊕1≤k≤n(π)Λπ∗mkwhere mkis an

irreducible idempotent, and so that ϕπ=Pk=1,..,n(π)mkand

ϕπ=Pk=1,..,n(π)ek. Let {ak}1≤k≤n(π)be a Hilbert basis in

Λπ∗m1such that the condition ak∈mk∗Λπ∗m1holds.

For every ﬁnite-dimensional representation π, we deﬁne a

matrix Mπ(g)of n(π)×n(π)-dimension with coefﬁcients

aij (g) = (n(π))−1ai(g)∗aj(g−1)(10)

for 1≤i≤n(π)and 1≤j≤n(π). So, we have aii =mi.

We deﬁne a linear span θπof the matrix coefﬁcients πjk,

which is a subspace of L2(G).

We denote the set of all ﬁnite linear combinations of the

matrix coefﬁcients of irreducible representations by θ, so θis

a linear span of Λπoverall ﬁnite-dimensional representations

of the group G.

Theorem 1.The set θconstitutes an algebra.

Proof. Let π, ˜π∈ˆ

Gas equivalence classes of irreducible

representations then we have

πjk (g) = ⟨π(g)ej, ek⟩(11)

and

˜πmq (g) = ⟨˜π(g) ˜em,˜eq⟩.(12)

The spaces Hand ˜

Hare deﬁned by choices of bases {ek}

and {˜em}for πjk and ˜πmq. The spaces Hand ˜

Hcan be

identiﬁed with Cnand C˜n, where n= dim (π)and ˜n=

dim (˜π). Let Cn,˜nbe a space of all matrices over Cof n×˜n

dimension. Let Tbe an operator of unitary equivalence of π

and ˜πso that ˜π(g) = T π (g)T−1. The tensor product π⊗˜π

of representations πand ˜π, on Cn,˜nis given by

(π⊗˜π) (g)T= (π) (g)T˜πg−1.(13)

Since ˜πmq g−1= ˜πqm (g)we obtain the statement of the

theorem

⟨(π⊗˜π) (g)ekq, ejm⟩=πjk (g) ˜πmq (g) (g).

Theorem 2. The algebra θis uniformly dense in Lp(G)

in the Lpnorm for 1<p<∞.

The proof of this theorem follows from the density of θin

C(G).

The Peter-Weyl theorem states:

ψ(g) = F−1ˆ

ψ(g) = ˆ

ψ∨

(g) =

=Rˆ

Gˆ

ψ(χ)χ(g)dˆµ(χ)

.(14)

ﬁrst statement. The mapping F:L2(G)→L2ˆ

G

deﬁned by

F(ψ) (π) = Zψ(g)Mπg−1dµ (g)(15)

is an isometric isomorphism. For each element ψ∈L2(G),

we have a representation

ψ=Pπn(π)Pi,k=1,...,n(π)

⟨⟨F(ψ) (π) (ei(π)) ,(ek(π))⟩⟩ ϕik (π),(16)

where {ei(π)}i=1,...,n(π)is an orthonormal basis in Cn(π)and

coordinate functions ϕik are deﬁned as

ϕik (π) (g) = ⟨Mπ(g)ei(π), ek(π)⟩(17)

for all g∈Gand i, k = 1, ..., n (π).

second statement. Let Gbe a compact group then the inverse

Fourier transform F−1:L2ˆ

G→L2(G)is deﬁned by

ψ(g) = X

n(π)tr (F(ψ) (π)Mπ(g)) (18)

for any Fourier transform F(ψ)∈L2ˆ

Gof ψ∈L2(G)

and the series converges in L2.

Now, we can formulate an analog of the Plancherel theory.

Theorem (analog of Plancherel) 3. Let Gbe a uni-

modular locally compact Hausdorff group with a Haar

measure µ. Then a measure ˆµon the dual group ˆ

Gis

uniquely deﬁned by the measure µ. The Fourier transform

F:L2(G)TL1(G)→L2(G)satisﬁes the equality

ψ(g)φ(g)dg =ZG

T r ˆ

ψ(π) ( ˆφ(π))∗dˆµ(π)(19)

for all ψ, φ ∈L2(G)TL1(G).

More precisely, let ψ, φ ∈L2(G)TL1(G)then

F(ψ∗φ) (π) = F(φ(π) ) F(ψ(π)) is a classical trace class

for almost everywhere πso the trace of F(φ(π) ) F(ψ(π))

is integrable on ˆ

The equality (19) expresses the unitarity of the Fourier

transform and can be rewritten in the form of the statement

that for each F(ζ), the inverse Fourier transform is given by

ζ(g) = ZG

T r ˆπ(g)ˆ

ζ(π)dˆµ(π)(20)

since if we take ζ=φ∗∗ψand g= 1 then we obtain (19).

Lemma. Let ψ∈Lp(G)for all 1<p<∞then F(ψ)∈

p−1(G)so that ∥F(ψ)∥L

p−1≤ ∥ψ∥Lp.

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Below we will follow notations of the David L. Donoho

and Philip B. Stark [2] when it is convenient.

Let Gbe a locally compact Hausdorff group equipped with

a probabilistic Haar measure µGand Kbe a maximal compact

subgroup of Gwith a probabilistic Haar measure µK.

For any ψ∈Lp(G), we deﬁned a measurable set Tsuch

that

∥ψ−ψ1T∥ ≤ ε∥ψ∥,(21)

where 1Tis a characteristic function of the set T. The set Tis

called ε-concentration set for the function ψ∈Lp(G), loosely

speaking it means that the support of the function ψ∈Lp(G)

is ε-close to the set T. Let the Fourier transform F(ψ)be

δ-concentrated on the measurable set W.

We deﬁne pair of operators

(PT(ψ)) (g) = ψ(g), g ∈T,

0, g /∈T(22)

and

(QW(ψ)) (g) = F−11Wˆ

ψ(g),(23)

where F−1denotes the inverse Fourier transform. The op-

erator QWpartially returns the function ψneglecting all

frequency information outside of the set Wso that the function

QW(ψ)is the nearest function to ψ.

The convolution ψ∗φof functions ψand φis given by

(ψ∗φ) (g) = ZG

ψ(h)φh−1gdµG(h).(24)

Let K\G/K be a double coset of Gthen the convolution

algebras C0(K\G/K)and L1(K\G/K)are subalgebras of

algebras C0(G)and L1(G)respectively.

Deﬁnition 4.For g∈G, the measure µggiven by

ψ(h)dµg(h) = ZKZK

ψkg˜

kdµK(k)dµK˜

k(25)

is called a K-dually invariant probability measure.

Deﬁnition 5. If equality µg∗µh=µh∗µgholds for all

g, h ∈Gthen (G, K)is called a Gelfand pair.

Straightforward consideration shows that (G, K)is a

Gelfand pair if and only if equality KgK ·KhK =KhK ·

KgK holds for all g, h ∈G.

The space M (G, K)is a Banach convolutive subalgebra of

Radon measures on Gthat are dual K-invariant.

A measure µ∈M (G, K)is a positive type relative Kif

and only if µψ(g)∗ψ(g−1)≥0for all ψ∈C0(G, K).

The projection of C0(G),M (G), and Lp(G)onto its

subspace C0(K\G/K),M (K\G/K), and Lp(K\G/K)of

dual K-invariant functions and measure, respectively, we will

denote by ψ7→ ψ#for functions and µ7→ µ#for measures.

Deﬁnition 6. Let Gbe a locally compact communicative

group, the mapping Lp(G)7→ Lqˆ

Ggiven by

ψ(χ) = ZG

ψ(g)χg−1dµG(g)(26)

is called a spherical transport of the function ψ, here χ∈ˆ

p+q=pq, p > 1.

Now, we are going to deﬁne the class of continuous func-

tions that are quasi-weights for spherical measures.

Deﬁnition 7. A continuous function ω:G→Cis called

a zonal spherical function if the Radon measure dµ (g) =

ωg−1dµG(g)satisﬁes the following conditions:

1. measure µis dual K-invariant namely µkE˜

k−1=

µ(E)for all measurable subsets E⊆G;

2.µ (ψ∗φ) = µ(φ)ZG

ψ(g)dµ (g).

The set of all zonal spherical functions is denoted by

S(G, K)and the subset positive functions of S(G, K)by

P(G, K).

Such measures dµ (g) = ωg−1dµG(g)are called spher-

ical.

For ψ∈C0(K\G/K)we deﬁne D(ψ) =

{z(ψ)∈Cψ:|z(ψ)| ≤ ∥ψ∥L1}. Since, for

ψ∈C0(K\G/K), the mapping

ω7→ ZG

ψ(g)ωg−1dµG(g) = ˆ

ψ(ω)(27)

is an injection, we have P(G, K)⊂TD(ψ).

Theorem (Godement) 4.

1. Let µ∈M (G, K)be a positive type relative Kmeasure

then there exists a uniquely deﬁne positive Radon measure

ˆµthat coincides with the spherical Fourier transform F(µ)

of the measure µ.

2. Let ψ∈C0(K\G/K)then there exists a uniquely

deﬁne positive Radon measure ˆµon P(G, K)such that

∥ψ∥L2=



ψ



L2.

From deﬁnitions and simple considerations, we obtain the

following statement.

Statement. 1. Assume φ∈P(G)such that φ(1) = 1 then

there exist a uniquely deﬁne representation space (V, π),π

is a unitary irreducible representation of G, and a uniquely

deﬁne unit cyclic vector v∈Vsuch that φ(g) = ⟨v, π (g)v⟩

holds for all g∈G. 2. Assume πis a unitary irreducible

representation of Gand a unit vector v∈Vis spanned by

(V, K, π)then ⟨v, π (g)v⟩ ∈ P(G)and ⟨v, π (1) v⟩= 1.

So, each density-function ω∈P(G)deﬁnes representation

space (V(ω), π (ω)), and a vector v(ω)∈V(ω)such that

equality ω(g) = ⟨v(ω), π (g, ω)v(ω)⟩Vholds for all g∈G.

Deﬁnition 8. The mapping F:L1(G/K)→

⟨V(·)⟩(P, ˆµ)= Υ given by F(ψ) (ω) = π(ψ, ω)v(ω)is

called the Fourier transform of the function ψ∈L1(G/K)

on G/K.

In the last deﬁnition, ⟨V(·)⟩(P, ˆµ)is understood as a direct

integral with the Plancherel measure ˆµ.

Theorem (Plancherel-Godement) 5. Let (G, K)be

Gelfand and Abelian then the Fourier transform satisﬁes

the equalities

ψ(g) = ⟨⟨F(ψ), π (g, ·)v(·)⟩V⟩(P, ˆµ)(28)

4. The Generalized Heisenberg Principle

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and

∥F(ψ)∥Υ=∥ψ∥L2(G/K)(29)

for all ψ∈L2(G/K).

Deﬁnition 9. The mapping F−1: Υ →L1(G/K)given

by F−1(ζ) (g) = ⟨⟨ζ(·), π (g, ·)v(·)⟩V⟩(P, ˆµ)is called the

inverse Fourier transform of the function ζon G/K.

Theorem (Heisenberg principle) 6.

Let ε, δ ≥0. Let function ψ∈Lp(G/K)be ε-

concentrated on T=T K ⊂Gin Lp-norm and satisﬁes

the condition there exists a function ψU∈Lp(G/K)such

that sup p(F(ψU)) ⊂Uand ∥ψ−ψU∥Lp≤δ∥ψ∥Lp. Then

1 + δ

1−ε−δp

µG(T) ˆµ(U)≥1.(30)

Proof. First, we show that

∥P Q∥p

Lp≤µG(T) ˆµ(U)

where operators are deﬁned by

(PT(ψ)) (g) = ψ(g), g ∈T,

0, g /∈T

and

(QW(ψ)) (g) = F−1(1W(F ψ (g))) .

Indeed, we have

P Qψ (g)=1T(g)F−1(1W(F ψ) (g)) =

= 1T(g)RGRPψ(h) 1W(ω)

v(ω), π h−1g, ωv(ω)V(ω)dˆµ(ω)dµG(h) =

=Dψ(·),1T(g)F−1((ω7→ 1W(ω)v(ω)) (h−1g))EL2,

since, by Riesz-Thorin theorem, for all ψ∈Lp(G/K)and

v∈Vp,we have ∥F(ψ)∥Lq≤ ∥ψ∥Lpand 

F−1(v)

Lq≤

∥v∥Lp, thus Holder and Titchmarsh inequalities yield

∥P Qψ∥p

Lp≤

≤ ∥ψ∥p

Lp

F−1((ω7→ 1W(ω)v(ω)))



LqµG(T)≤

≤µG(T) ˆµ(U)∥ψ∥p

Lp.

Second, there is the estimation

1−ε−δ

1 + δ≤ ∥P Q∥Lp.

Indeed, using the conditions, we estimate

∥ψ∥Lp− ∥P Qψ∥Lp≤ ∥ψ−P Qψ∥Lp≤

≤ ∥ψ−P Qψ∥Lp+∥P ψ −P ψU∥Lp+

+∥P QψU−P Qψ∥Lp≤

≤ε∥ψ∥Lp+δ∥ψ∥Lp+δ∥ψ∥Lp∥P Qψ∥Lp,

so, we obtain 1−ε−δ≤(1 + δ)∥P Q∥Lpthat proves the

Heisenberg principle.

Let Gbe a connected semisimple real Lie group such that

there exists an analytic diffeomorphism from the manifold K×

A×Nto group Gaccording to the rule (k, a, n)7→ kan,

decomposition KAN is called the Iwasawa decomposition of

the group G, where the dimension of Ais equal to the real

rank of G. The group Kis closed and contains the center

of G,Imag (K)under the adjoint representation of Gis a

maximal compact subgroup of the adjoint of G; subgroups A

and Nare simply connected. The associated minimal parabolic

subgroup of Gis MAN . Let gand abe Lies algebras of

Gand A, respectively, the norms ∥·∥ correspond to the aand

dual algebra a∗relative to the inner product induced by the

Killing form of g.

Let an irreducible unitary representation vof Mbeing

presented as a left translation on Vv⊂C(M)where Vvis

ﬁnite-dimensional. Let ˜vbe an element of the complexiﬁcation

a∗Cof a∗.

Loosely said the Hardy uncertainty principle maintains that

the function ψand its Fourier transform F(ψ)cannot be

simultaneously both rapidly decreasing.

Theorem 7. Let a measurable on Gfunction ψsatisﬁes

the conditions:

|ψ(k1ak2)| ≤ c1exp −c2|log (a)|2(31)

for all k1, k2∈Kand a∈A, and the following estimation

∥π(ψ, v, ˜v)∥ ≤ ˜c1(v) exp −˜c2|˜v|2(32)

for all v∈ˆ

Mand ˜v∈a∗, where c1, c2,˜c1(v),˜c2are

constants. If the product c2˜c2>1

4then the function ψ

equals identically to zero.

Proof. We denote

ψρ,˜ρ(g) = dim (ρ) dim (˜ρ)

ZKZK

χρ(k1)χ˜ρ(k2)ψ(k1gk2)dµK(k1)dµK(k2),

where ρand ˜ρare irreducible representations of K.

Employing conditions and remarks π(ψρ,˜ρ, v, ˜v) =

Pρπ(ψ, v, ˜v)P˜ρhere Pρand P˜ρare the projections of L2(K)

on the sum of all submodules, which are isomorphic to the ρ1

and ρ2weight modules. Let 0< c2(v)< c2and take ˜c2so

that c2(v) ˜c2>1

4, we have

∥π(ψρ,˜ρ, v, ˜v)∥ ≤ c5exp |˜v|2

4c2(v)!

for v∈ˆ

Mand ˜v∈a∗C.

Applying the Naimark equivalent, there is a quotient repre-

sentation ˜π(v, ˜v)of π(v, ˜v)to close quotient subset V1/V0,

there exists a densely deﬁne intertwining operator (π, Vπ)7→

(˜π(v, ˜v), V1/V0)on the domain of which π(ψρ,˜ρ)=0, from

the properties of quotient representation, by continuity, we

have π(ψρ,˜ρ) = 0 on Vπ. By summing over all ρand ˜ρ,

we have π(ψ)=0for all representations of G, applying

Plancherel theory, we obtain ψ= 0 [19].

5. The Hardy Uncertainty Principle

EQUATIONS

DOI: 10.37394/232021.2023.3.6

Mykola Ivanovich Yaremenko

E-ISSN: 2732-9976

Volume 3, 2023

[1] N. Arkani-Hamed, T.-C. Huang, and Y. Huang, Scattering amplitudes

for all masses and spins, JHEP 11 (2021), 070.

[2] D. L. Donoho & P. B. Stark, Uncertainty principles and signal recovery,

SIAM J. Applied Math. 49 (1989), 906–931.

[3] V. Shtabovenko, R. Mertig, and F. Orellana, New features and improve-

ments, Comput. Phys. Commun. 256 (2020), 107478.

[4] L. de la Cruz, B. Maybee, D. O’Connell, and A. Ross, Classical Yang-

Mills observables from amplitudes, JHEP 12 (2020), 076.

[5] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9

(1983), 129–206.

[6] R. Monteiro, D. O’Connell, D. P. Veiga, and M. Sergola, Classical

solutions and their double copy in split signature, JHEP 05 (2021), 268.

[7] Mariusz P. Dabrowski and Fabian Wagner, Asymptotic Generalized

Extended Uncertainty Principle. Eur. Phys. J. C, 80:676, (2020).

[8] M. J. Lake, A New Approach to Generalised Uncertainty Relations. 8

(2020).

[9] P. and F. Wagner, Gravitationally induced uncertainty relations in curved

backgrounds. Phys. Rev. D, 103:104061, (2021).

[10] L. Petruzziello, Generalized uncertainty principle with maximal observ-

able momentum and no minimal length indeterminacy, Class. Quant.

Grav. 38 no. 13, (2021), 135005.

[11] S. P. Kumar and M. B. Plenio, On Quantum Gravity Tests with

Composite Particles, Nature Commun. 11 no. 1, (2020), 3900.

[12] K. T. Smith, The uncertainty principle on groups, SIAM J. Appl. Math.

50 (1990), 876–882.

[13] M.I. Yaremenko, Calderon-Zygmund Operators and Singular Inte-

grals, Applied Mathematics & Information Sciences: Vol. 15: Iss. 1,

Article 13, (2021).

[14] B. Krotz, J. Kuit, E. Opdam, and H. Schlichtkrull, The inﬁnitesimal

characters of discrete series for real spherical spaces, Geom. Funct. Anal.

30 (2020), 804–857.

[15] K. Smaoui, K. Abid, Heisenberg uncertainty inequality for Gabor

transform on nilpotent Lie groups. Anal Math 48, 147–171 (2022).

[16] Cowling, M.; Sitaram, A.; Sundari, M. Hardy’s uncertainty principle on

semisimple groups. Paciﬁc J. Math. 192 (2000), no. 2, 293–296.

[17] B.K. Germain, K. Kinvi, On Gelfand Pair Over Hypergroups, Far East

J. Math. 132 (2021), 63–76.

[18] L. Sz´

ekelyhidi, Spherical Spectral Synthesis on Hypergroups, Acta

Math. Hungar. 163 (2020), 247–275.

[19] K. Vati, Gelfand Pairs Over Hypergroup Joins, Acta Math. Hungar. 160

(2019), 101–108.

[20] Harish-Chandra, Harmonic analysis on real reductive Lie groups I, The

theory of constant term. J. Funct. Anal., 19, (1975), 104-204.

[21] G.B. Folland and A. Sitaram, The uncertainty principle: a mathematical

survey, J. Fourier Anal. Appl., 3 (1997), 207-238.

[22] E.K. Narayanan, S.K. Ray, The heat kernel and Hardy’s theorem on

symmetric spaces of noncompact type. Proc. Indian Acad. Sci. Math.

Sci. 112 (2002), no. 2, 321–330.

[23] E.K. Narayanan, S.K. Ray, L∧p version of Hardy’s theorem on semisim-

ple Lie groups. Proc. Amer. Math. Soc. 130 (2002), no. 6, 1859–1866.

References

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EQUATIONS

DOI: 10.37394/232021.2023.3.6

Mykola Ivanovich Yaremenko

E-ISSN: 2732-9976

Volume 3, 2023