##### Document Text Contents

Page 1

International Conference

H A R M O N I C A N A L Y S I S

A N D

A P P R O X I M A T I O N S, VI

12 - 18 September, 2015

Tsaghkadzor, Armenia

Yerevan, 2015

Page 2

Organizers:

Host Institutions

Institute of Mathematics of the National Academy of Sciences,

Yerevan State University

Programme Committee

N. H. Arakelian (Armenia), P. Gauthier (Canada), B. S. Kashin (Russia),

M. Lacey (USA), W. Luh (Germany), A. M. Olevskii (Israel), A. A. Talalian

(Armenia), V. N. Temlyakov (USA), P. Wojtaszczyk (Poland)

Organizing Committee

G. Gevorkyan, A. Sahakian, A. Hakobyan, M. Poghosyan

Sponsors:

• The Research Mathematics Fund

• Promethey Science Foundation

• State Committee of Science, Armenia

• "Hyur Service" LLC

Page 47

a) it has a vertex in the origin of coordinates and in all coordinate axes;

b) outward normals of all (n− 1)-dimensional non-coordinate faces

are positive.

Let µi be the outward normal of the face ℵ

(n−1)

i

such that ∀α ∈ ℵ(n−1)

i

,

(α, µi) = α1µ

i

1 + · · ·+ αnµ

i

n = 1;

∣∣µi∣∣ = ∣∣µi1∣∣+ · · ·+ ∣∣µin∣∣. Let us denote by

Wℵp (R

n) the set of all measurable functions in Rn for which f ∈ Lp(Rn)

and for any ∀αi ∈ ℵ(n−1)

i

, Dαi f ∈ Lp(Rn), i = 1, · · · ,M.

In the present work an integral representation through the differentia-

tion operator is offered, which is generated via the polyhedron ℵ, and, ap-

plying the obtained integral representation, embedding of the setWℵp (R

n)

in Lq(Rn) is proved.

Theorem. Let ℵ be a convex polyhedron and f ∈ Lp(Rn) and ∀α ∈ ∂

′ℵ,

Dα1 f ∈ Lp(Rn). Let multi-index β and numbers 1 ≤ p ≤ q ≤ ∞ be such

that (β; µ) +

(

1

p

− 1

q

)

|µ| < 1, for any normal µ of the (n − 1) -dimensional

hyper-plane of the polyhedron ℵ.

Let

max(β; µ) +

(

1

p

−

1

q

)

|µ| = (β; µ0) +

(

1

p

−

1

q

)

|µ0| .

Then DβWℵp (R

n) is embedded Lq(Rn), i.e. for any f ∈ Wℵp (Rn), the derivative

Dβ f ∈ Lq(Rn) exists, and the following estimate is true:

∥Dα f ∥Lq(Rn) ≤ C1h

1−

(

(β;µ0)+

(

1

p−

1

q

)

|µ0|

)

M

∑

i=1

∥Dα f ∥Lp(Rn)

+C2h

−

(

(β;µ0)+

(

1

p−

1

q

)

|µ0|

)

∥ f ∥Lp(Rn) ,

where C1,C2 are numbers independent of f , h , and h is a parameter, which varies

in 0 < h < h0.

47

Page 48

Mixed norm variable exponent Bergman space

on the unit disc

A. Karapetyants

(Southern Federal University and Don State Technical University, Russia)

[email protected]

This is a joint work with S. Samko.

We introduce and study the mixed norm variable order Bergman space

Aq,p(·)(D), 1 6 q < ∞, 1 6 p(r) 6 ∞, on the unit disk D in the complex

plane. The mixed norm variable order Lebesgue - type space Lq,p(·)(D) is

de�ned by the requirement that the sequence of variable exponent Lp(·)(I)

- norms of the Fourier coef�cients of the function f belongs to lq. Then

Aq,p(·)(D) is de�ned to be the subspace of Lq,p(·)(D) which consists of

analytic functions. We prove the boundedness of the Bergman projection

and reveal the dependence of the nature of such spaces on possible growth

of variable exponent p(r) when r → 1 from inside the interval I = (0, 1).

The situation is quite different in the cases p(1) < ∞ and p(1) = ∞. In

the case p(1) < ∞ we also characterize the introduced Bergman space

A2,p(·)(D) as the space of Flett's fractional derivatives of functions from

the Hardy space H2(D). The case p(1) = ∞ is specially studied, and an

open problem is formulated in this case. We also reveal a condition on the

growth from below of p(r) when r → 1, under which A2,p(·)(D) = H2(D)

up to norm equivalence, and also �nd a condition on the growth from

above of p(r) when this is not longer true.

48

mailto: [email protected]

Page 93

nal subgroups, intuitively viewed as the monad of in�nitesimal elements

and the galaxy of �nite elements, respectively. It can be shown that ev-

ery LCA group G is isomorphic to the observable trace or nonstandard hull

G♭ = Gf/G0 of such a triplet. The dual triplet of (G,G0,Gf) is de�ned as(

Ĝ,G∼

f

,G∼0

)

, where Ĝ is the group of all internal homomorphisms (char-

acters) G → ∗T, and the in�nitesimal annihilators

G∼

f

=

{

χ ∈ Ĝ

∣∣ ∀ a ∈ Gf : χ(a) ≈ 1} ,

G∼0 =

{

χ ∈ Ĝ

∣∣ ∀ a ∈ G0 : χ(a) ≈ 1}

of the subgroups Gf, G0 consist of characters which are in�nitesimally

close to 1 on the galaxy Gf or continuous in the intuitive sense backed by

NSA, respectively.

GC1 states that for G = G♭ its dual group Ĝ is canonically isomorphic

to the observable trace Ĝ♭ = G∼0 /G

∼

f

of the dual triplet

(

Ĝ,G∼

f

,G∼0

)

. It

turns out to be equivalent to the Triplet Duality Theorem according to which

the dual triplet

(

G,G∼∼0 ,G

∼∼

f

)

of the dual triplet

(

Ĝ,G∼

f

,G∼0

)

coincides

with the original triplet (G,G0,Gf).

GC2 states certain natural duality relation, partly akin to the Uncer-

tainty Principle, between �normalizing coef�cients� or �elementary char-

ges� on both the triplets, by means of which the Haar measures on their

nonstandard hulls can be de�ned using the Loeb measure construction.

Representing the pair of dual LCA groups G, Ĝ by a pair of dual

triplets enables to approximate the Fourier-Plancherel transform L2(G) →

L2

(

Ĝ

)

by means of the hyper�nite dimensional DFT ∗CG → ∗CĜ. GC3

states that such an approximation is in�nitesimally precise almost ev-

erywhere. Essentially the same is true also for the Fourier transform

L1(G) → C0

(

Ĝ

)

and even for the Fourier-Stieltjes transform M(G) →

Cbu

(

Ĝ

)

extending it, as well as for the generalized Fourier transforms

Lp(G) → Lq

(

Ĝ

)

, for any pair of adjoint exponents 1 < p ≤ 2 ≤ q < ∞.

Standard interpretations of these results imply the existence of �arbi-

trarily good� approximations of all the above Fourier transforms on every

LCA group G by the DFT on some �nite abelian group G.

93

Page 94

Depending on time, we will survey most of the above mentioned con-

structions and results.

References

[1] E. I. Gordon, Nonstandard analysis and locally compact abelian groups,

Acta Appl. Math. 25 (1991), 221�239.

[2] E. I. Gordon, Nonstandard Methods in Commutative Harmonic Analy-

sis, Translations of Mathematical Monographs, vol. 164, Amer. Math.

Soc., Providence, R. I., 1997.

[3] B. Green, I. Ruzsa, Freiman's theorem in an arbitrary abelian group,

J. London Math. Soc. (2) 75 (2007), 163�175.

[4] T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press,

Cambridge-New York, etc. 2006.

[5] P. Zlato�, Gordon's Conjectures: Pontryagin-vanKampen Duality

and Fourier Transform in Hyper�nite Ambience, arXiv:1409.6128v2

[math.CA], (81 pages), submitted to Amer. J. Math.

94

International Conference

H A R M O N I C A N A L Y S I S

A N D

A P P R O X I M A T I O N S, VI

12 - 18 September, 2015

Tsaghkadzor, Armenia

Yerevan, 2015

Page 2

Organizers:

Host Institutions

Institute of Mathematics of the National Academy of Sciences,

Yerevan State University

Programme Committee

N. H. Arakelian (Armenia), P. Gauthier (Canada), B. S. Kashin (Russia),

M. Lacey (USA), W. Luh (Germany), A. M. Olevskii (Israel), A. A. Talalian

(Armenia), V. N. Temlyakov (USA), P. Wojtaszczyk (Poland)

Organizing Committee

G. Gevorkyan, A. Sahakian, A. Hakobyan, M. Poghosyan

Sponsors:

• The Research Mathematics Fund

• Promethey Science Foundation

• State Committee of Science, Armenia

• "Hyur Service" LLC

Page 47

a) it has a vertex in the origin of coordinates and in all coordinate axes;

b) outward normals of all (n− 1)-dimensional non-coordinate faces

are positive.

Let µi be the outward normal of the face ℵ

(n−1)

i

such that ∀α ∈ ℵ(n−1)

i

,

(α, µi) = α1µ

i

1 + · · ·+ αnµ

i

n = 1;

∣∣µi∣∣ = ∣∣µi1∣∣+ · · ·+ ∣∣µin∣∣. Let us denote by

Wℵp (R

n) the set of all measurable functions in Rn for which f ∈ Lp(Rn)

and for any ∀αi ∈ ℵ(n−1)

i

, Dαi f ∈ Lp(Rn), i = 1, · · · ,M.

In the present work an integral representation through the differentia-

tion operator is offered, which is generated via the polyhedron ℵ, and, ap-

plying the obtained integral representation, embedding of the setWℵp (R

n)

in Lq(Rn) is proved.

Theorem. Let ℵ be a convex polyhedron and f ∈ Lp(Rn) and ∀α ∈ ∂

′ℵ,

Dα1 f ∈ Lp(Rn). Let multi-index β and numbers 1 ≤ p ≤ q ≤ ∞ be such

that (β; µ) +

(

1

p

− 1

q

)

|µ| < 1, for any normal µ of the (n − 1) -dimensional

hyper-plane of the polyhedron ℵ.

Let

max(β; µ) +

(

1

p

−

1

q

)

|µ| = (β; µ0) +

(

1

p

−

1

q

)

|µ0| .

Then DβWℵp (R

n) is embedded Lq(Rn), i.e. for any f ∈ Wℵp (Rn), the derivative

Dβ f ∈ Lq(Rn) exists, and the following estimate is true:

∥Dα f ∥Lq(Rn) ≤ C1h

1−

(

(β;µ0)+

(

1

p−

1

q

)

|µ0|

)

M

∑

i=1

∥Dα f ∥Lp(Rn)

+C2h

−

(

(β;µ0)+

(

1

p−

1

q

)

|µ0|

)

∥ f ∥Lp(Rn) ,

where C1,C2 are numbers independent of f , h , and h is a parameter, which varies

in 0 < h < h0.

47

Page 48

Mixed norm variable exponent Bergman space

on the unit disc

A. Karapetyants

(Southern Federal University and Don State Technical University, Russia)

[email protected]

This is a joint work with S. Samko.

We introduce and study the mixed norm variable order Bergman space

Aq,p(·)(D), 1 6 q < ∞, 1 6 p(r) 6 ∞, on the unit disk D in the complex

plane. The mixed norm variable order Lebesgue - type space Lq,p(·)(D) is

de�ned by the requirement that the sequence of variable exponent Lp(·)(I)

- norms of the Fourier coef�cients of the function f belongs to lq. Then

Aq,p(·)(D) is de�ned to be the subspace of Lq,p(·)(D) which consists of

analytic functions. We prove the boundedness of the Bergman projection

and reveal the dependence of the nature of such spaces on possible growth

of variable exponent p(r) when r → 1 from inside the interval I = (0, 1).

The situation is quite different in the cases p(1) < ∞ and p(1) = ∞. In

the case p(1) < ∞ we also characterize the introduced Bergman space

A2,p(·)(D) as the space of Flett's fractional derivatives of functions from

the Hardy space H2(D). The case p(1) = ∞ is specially studied, and an

open problem is formulated in this case. We also reveal a condition on the

growth from below of p(r) when r → 1, under which A2,p(·)(D) = H2(D)

up to norm equivalence, and also �nd a condition on the growth from

above of p(r) when this is not longer true.

48

mailto: [email protected]

Page 93

nal subgroups, intuitively viewed as the monad of in�nitesimal elements

and the galaxy of �nite elements, respectively. It can be shown that ev-

ery LCA group G is isomorphic to the observable trace or nonstandard hull

G♭ = Gf/G0 of such a triplet. The dual triplet of (G,G0,Gf) is de�ned as(

Ĝ,G∼

f

,G∼0

)

, where Ĝ is the group of all internal homomorphisms (char-

acters) G → ∗T, and the in�nitesimal annihilators

G∼

f

=

{

χ ∈ Ĝ

∣∣ ∀ a ∈ Gf : χ(a) ≈ 1} ,

G∼0 =

{

χ ∈ Ĝ

∣∣ ∀ a ∈ G0 : χ(a) ≈ 1}

of the subgroups Gf, G0 consist of characters which are in�nitesimally

close to 1 on the galaxy Gf or continuous in the intuitive sense backed by

NSA, respectively.

GC1 states that for G = G♭ its dual group Ĝ is canonically isomorphic

to the observable trace Ĝ♭ = G∼0 /G

∼

f

of the dual triplet

(

Ĝ,G∼

f

,G∼0

)

. It

turns out to be equivalent to the Triplet Duality Theorem according to which

the dual triplet

(

G,G∼∼0 ,G

∼∼

f

)

of the dual triplet

(

Ĝ,G∼

f

,G∼0

)

coincides

with the original triplet (G,G0,Gf).

GC2 states certain natural duality relation, partly akin to the Uncer-

tainty Principle, between �normalizing coef�cients� or �elementary char-

ges� on both the triplets, by means of which the Haar measures on their

nonstandard hulls can be de�ned using the Loeb measure construction.

Representing the pair of dual LCA groups G, Ĝ by a pair of dual

triplets enables to approximate the Fourier-Plancherel transform L2(G) →

L2

(

Ĝ

)

by means of the hyper�nite dimensional DFT ∗CG → ∗CĜ. GC3

states that such an approximation is in�nitesimally precise almost ev-

erywhere. Essentially the same is true also for the Fourier transform

L1(G) → C0

(

Ĝ

)

and even for the Fourier-Stieltjes transform M(G) →

Cbu

(

Ĝ

)

extending it, as well as for the generalized Fourier transforms

Lp(G) → Lq

(

Ĝ

)

, for any pair of adjoint exponents 1 < p ≤ 2 ≤ q < ∞.

Standard interpretations of these results imply the existence of �arbi-

trarily good� approximations of all the above Fourier transforms on every

LCA group G by the DFT on some �nite abelian group G.

93

Page 94

Depending on time, we will survey most of the above mentioned con-

structions and results.

References

[1] E. I. Gordon, Nonstandard analysis and locally compact abelian groups,

Acta Appl. Math. 25 (1991), 221�239.

[2] E. I. Gordon, Nonstandard Methods in Commutative Harmonic Analy-

sis, Translations of Mathematical Monographs, vol. 164, Amer. Math.

Soc., Providence, R. I., 1997.

[3] B. Green, I. Ruzsa, Freiman's theorem in an arbitrary abelian group,

J. London Math. Soc. (2) 75 (2007), 163�175.

[4] T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press,

Cambridge-New York, etc. 2006.

[5] P. Zlato�, Gordon's Conjectures: Pontryagin-vanKampen Duality

and Fourier Transform in Hyper�nite Ambience, arXiv:1409.6128v2

[math.CA], (81 pages), submitted to Amer. J. Math.

94