Startseite Nonlinear operations and factorizations on a class of affine modulation spaces
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Nonlinear operations and factorizations on a class of affine modulation spaces

  • Md Hasan Ali Biswas EMAIL logo und Ramesh Manna
VerĂśffentlicht/Copyright: 30. November 2024
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 37 Heft 5

Abstract

In the first part of the paper, we obtain a necessary and sufficient condition on a complex-valued function 𝐹 on R 2 such that F ⁢ ( f ) ∈ M θ r , s ⁢ ( R n ) for all f ∈ M θ r , 1 ⁢ ( R n ) , 1 ≤ r < ∞ , 1 ≤ s < 2 , where M θ r , s ⁢ ( R n ) is the affine modulation space associated with the fractional Fourier transform. In this context, we obtain that if 𝐹 acts on M θ r , 1 ⁢ ( R n ) by composition, then 𝐹 is necessarily real analytic and F ⁢ ( 0 ) = 0 . As M θ r , 1 ⁢ ( R n ) is neither a Banach algebra nor closed under complex conjugation, we prove sufficiency of the condition for a subclass of M θ r , 1 ⁢ ( R n ) under some additional hypothesis on 𝐹. We provide an example to show that we cannot drop the additional hypothesis. Thus we characterize the functions that operate in the affine modulation space setting. In the second part, we study the factorization problem in the context of M θ r , s ⁢ ( R n ) . We obtain factorization of M θ r , s ⁢ ( R 2 ⁢ n ) , viewing it as an L 1 ⁢ ( R 2 ⁢ n ) module with respect to convolution and 𝜆-twisted convolution associated with the fractional Fourier transform. The problems mentioned above originated in the work of Helson, Kahane, Katznelson, Rudin, Cohen, and Hewitt respectively.

Keywords: Nonlinear operations; factorizations; affine modulation spaces; special affine Fourier transform
MSC 2020: 42B35; 46H25; 42B10; 42A85

Funding source: Department of Science and Technology, Ministry of Science and Technology, India

Award Identifier / Grant number: DST/INSPIRE/04/2019/001914

Funding statement: Ramesh Manna is thankful for the research grants (DST/INSPIRE/04/2019/001914).

A Appendix

In Example 3.4, we outlined the key steps. Although the verification is elementary, here we provide a detailed justification for the convenience of the reader.

Example A.1

Let 1 < r < 2 and define

F ⁢ ( x , ω ) = 1 | x | 1 / r ⁢ ( 1 + ( ln ⁡ | x | ) 2 ) ⋅ 1 1 + ω 2 .

Then F ∈ L r , 1 ⁢ ( R 2 ) . Let g ⁢ ( x ) = e − π ⁢ x 2 and h = V g ∗ ⁢ F . Then, using Proposition 2.13, we get that h ∈ M r , 1 ⁢ ( R ) . Set f = C − θ ⁢ h . Then f ∈ M θ r , 1 ⁢ ( R ) , appealing to Proposition 2.20. Further, using (2.4) and (2.6), we obtain

V C − θ ⁢ g ⁢ f ⁢ ( u , η ) = V C − θ ⁢ g ⁢ C − θ ⁢ h ⁢ ( u , η ) = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ V g ⁢ h ⁢ ( u , η + u ⁢ cot ⁡ θ ) = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ ∫ R 2 F ⁢ ( x , ω ) ⁢ V g ⁢ g ⁢ ( u − x , η + u ⁢ cot ⁡ θ − ω ) ⁢ e − 2 ⁢ π ⁢ i ⁢ x ⁢ ( η + u ⁢ cot ⁡ θ − ω ) ⁢ d x ⁢ d ω = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ ∫ R 2 F ⁢ ( u − x , η + u ⁢ cot ⁡ θ − ω ) ⁢ V g ⁢ g ⁢ ( x , ω ) ⁢ e − 2 ⁢ π ⁢ i ⁢ ( u − x ) ⁢ ω ⁢ d x ⁢ d ω = e π ⁢ i ⁢ u 2 ⁢ cot ⁡ θ ⁢ ∫ R 2 F ⁢ ( u + x , η + u ⁢ cot ⁡ θ − ω ) ⁢ V g ⁢ g ⁢ ( ω , x ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ω ⁢ d x ⁢ d ω ,

applying a change of variables. Similarly,

V g ⁢ C θ ⁢ f ⁢ ( u , η ) = V g ⁢ h ⁢ ( u , η ) = ∫ R 2 F ⁢ ( u + x , η − ω ) ⁢ V g ⁢ g ⁢ ( ω , x ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ω ⁢ d x ⁢ d ω .

Thus, using Remark 2.8 (i), we obtain

V C − θ ⁢ g 2 ⁢ C θ ⁢ f 2 ⁢ ( u , η ) = ( C θ ⁢ f 2 ⁢ T u ⁢ ( C − θ ⁢ g 2 ̄ ) ) ⁢ ̂ ⁢ ( η ) = ( C θ ⁢ f ⁢ T u ⁢ ( g ̄ ) ⁢ f ⁢ T u ⁢ ( C − θ ⁢ g ̄ ) ) ⁢ ̂ ⁢ ( η ) = ( ( C θ ⁢ f ⁢ T u ⁢ ( g ̄ ) ) ⁢ ̂ ⋆ ( f ⁢ T u ⁢ ( C − θ ⁢ g ̄ ) ) ⁢ ̂ ) ⁢ ( η ) = ∫ R ( C θ ⁢ f ⁢ T u ⁢ ( g ̄ ) ) ⁢ ̂ ⁢ ( y ) ⁢ ( f ⁢ T u ⁢ ( C − θ ⁢ g ̄ ) ) ⁢ ̂ ⁢ ( η − y ) ⁢ d y = ∫ R V g ⁢ C θ ⁢ f ⁢ ( u , y ) ⁢ V C − θ ⁢ g ⁢ f ⁢ ( u , η − y ) ⁢ d y .

Now, taking

F 1 ⁢ ( x ) = 1 | x | 1 / r ⁢ ( 1 + ( ln ⁡ | x | ) 2 ) and F 2 ⁢ ( ω ) = 1 1 + ω 2 ,

we obtain

V C − θ ⁢ g ⁢ C θ ⁢ f 2 ⁢ ( u , η ) = ∫ R 2 ∫ R 2 V g ⁢ g ⁢ ( ω 1 , x 1 ) ⁢ V g ⁢ g ⁢ ( ω 2 , x 2 ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ( ω 1 + ω 2 ) ⁢ F 1 ⁢ ( u + x 1 ) ⁢ F 1 ⁢ ( u + x 2 ) × ∫ R F 2 ( η − y + u cot θ − ω 1 ) F 2 ( y − ω 2 ) d y d x 1 d ω 1 d x 2 d ω 2 = ∫ R 2 ∫ R 2 V g ⁢ g ⁢ ( ω 1 , x 1 ) ⁢ V g ⁢ g ⁢ ( ω 2 , x 2 ) ⁢ e − 2 ⁢ π ⁢ i ⁢ u ⁢ ( ω 1 + ω 2 ) ⁢ F 1 ⁢ ( u + x 1 ) ⁢ F 1 ⁢ ( u + x 2 ) × F 2 ⋆ F 2 ⁢ ( η + u ⁢ cot ⁡ θ − ω 1 − ω 2 ) ⁢ d ⁢ x 1 ⁢ d ⁢ ω 1 ⁢ d ⁢ x 2 ⁢ d ⁢ ω 2 .

Hence, for s > r , appealing to Minkowski’s integral inequality, we have

I = ∫ R ( ∫ R | V C − θ ⁢ g ⁢ C θ ⁢ f 2 ⁢ ( u , η ) | s ⁢ d u ) 1 / s ⁢ d η = ∫ R ( ∫ R | ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ u ⁢ ( ω 1 + ω 2 ) F 1 ( u + x 1 ) F 1 ( u + x 2 ) × F 2 ⋆ F 2 ( η + u cot θ − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 | s d u ) 1 / s d η = ∫ R ( ∫ R | ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 | s d u ) 1 / s d η ≥ ( ∫ R ( ∫ R | ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 | d η ) s d u ) 1 / s ≥ ( ∫ R | ∫ R ∫ R 2 ∫ R 2 V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η | s d u ) 1 / s ≥ ( ∫ R ( ∫ R ∫ R 2 ∫ R 2 Re ( V g g ( ω 1 , x 1 ) V g g ( ω 2 , x 2 ) e − 2 ⁢ π ⁢ i ⁢ ( ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) ) F 1 ( ( u − η ) tan θ + x 1 ) × F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η ) s d u ) 1 / s = 1 2 ( ∫ R ( ∫ R ∫ R 2 ∫ R 2 e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) Re ( e 2 ⁢ π ⁢ i ⁢ ( ω 1 ⁢ x 1 + ω 2 ⁢ x 2 − ( u − η ) ⁢ ( ω 1 + ω 2 ) ⁢ tan ⁡ θ ) ) × F 1 ( ( u − η ) tan θ + x 1 ) F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η ) s d u ) 1 / s = 1 2 ( ∫ R ( ∫ R ∫ R 2 ∫ R 2 e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − ( u − η ) ( ω 1 + ω 2 ) tan θ ) ) × F 1 ( ( u − η ) tan θ + x 1 ) F 1 ( ( u − η ) tan θ + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d x 1 d ω 1 d x 2 d ω 2 d η ) s d u ) 1 / s ∼ 1 2 ( ∫ R ( ∫ R 2 ∫ R 2 ∫ R e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − η ( ω 1 + ω 2 ) ) ) F 1 ( η + x 1 ) × F 1 ( η + x 2 ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s = 1 2 ( ∫ R ( ∫ R 2 ∫ R 2 ∫ R e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − ( η − x 2 ) ( ω 1 + ω 2 ) ) ) × F 1 ( η + x 1 − x 2 ) F 1 ( η ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s ,

using | x + i ⁢ y | ≥ x and applying a change of variables. Our aim is to show that the integral 𝐼 is not finite.

In order to prove the claim, let

X = { ( x 1 , ω 1 , x 2 , ω 2 ) ∈ R 4 : 0 ≤ ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 < 1 3 , ω 1 + ω 2 > 0 ,  0 ≤ x 2 ≤ x 1 ≤ 1 } .

and for fixed N ∈ N , ( x 1 , ω 1 , x 2 , ω 2 ) ∈ X , set

Y N = { η ∈ R : | η ⁢ ( ω 1 + ω 2 ) − ω 1 ⁢ ( x 1 + x 2 ) − 2 ⁢ ω 2 ⁢ x 2 | < 1 3 , x 1 − x 2 < 1 N } .

Then, for any ( x 1 , ω 1 , x 2 , ω 2 ) ∈ X and η ∈ Y N , we have

1 ω 1 + ω 2 ⁢ ( − 1 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 ) < η < 1 ω 1 + ω 2 ⁢ ( 1 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 )

and

1 ω 1 + ω 2 ⁢ ( − 1 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 ) < 0 .

Taking η 0 = 1 ω 1 + ω 2 ⁢ ( 1 / 3 + ω 1 ⁢ ( x 1 + x 2 ) + 2 ⁢ ω 2 ⁢ x 2 ) , we obtain

∫ Y N F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ cos ⁡ ( 2 ⁢ π ⁢ ( ω 1 ⁢ x 1 + ω 2 ⁢ x 2 − ( η − x 2 ) ⁢ ( ω 1 + ω 2 ) ) ) ⁢ d η ≥ 1 2 ⁢ ∫ Y N F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ≥ 1 2 ⁢ ∫ 0 η 0 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = 1 2 ⁢ ∫ 0 η 0 1 ( η + x 1 − x 2 ) 1 / r ⁢ ( 1 + ( ln ⁡ ( η + x 1 − x 2 ) ) 2 ) ⋅ 1 η 1 / r ⁢ ( 1 + ( ln ⁡ η ) 2 ) ⁢ d η ≥ 1 2 ⁢ ∫ 0 η 0 1 ( η + x 1 − x 2 ) 2 / r ⁢ ( 1 + ( ln ⁡ ( η + x 1 − x 2 ) ) 2 ) 2 ⁢ d η = 1 2 ⁢ ∫ ln ⁡ ( x 1 − x 2 ) ln ⁡ ( η 0 + x 1 − x 2 ) e ( 1 − 2 / r ) ⁢ y ( 1 + y 2 ) 2 ⁢ d y ,

applying a change of variables. Since ( 1 − 2 / r ) < 0 , the integral

∫ Y N F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ cos ⁡ ( 2 ⁢ π ⁢ ( ω 1 ⁢ x 1 + ω 2 ⁢ x 2 − ( η − x 2 ) ⁢ ( ω 1 + ω 2 ) ) ) ⁢ d η

diverges to infinity as N → ∞ .

Now, for any δ > 0 , we compute

I δ = ( ∫ R ( ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) cos ( 2 π ( ω 1 x 1 + ω 2 x 2 − ( η − x 2 ) ( ω 1 + ω 2 ) ) ) × F 1 ( η + x 1 − x 2 ) F 1 ( η ) F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s ≤ ( ∫ R ( ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) F 1 ( η + x 1 − x 2 ) F 1 ( η ) × F 2 ⋆ F 2 ( u − ω 1 − ω 2 ) d η d x 1 d ω 1 d x 2 d ω 2 ) s d u ) 1 / s ≤ ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) ⁢ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) × ( ∫ R ( F 2 ⋆ F 2 ⁢ ( u − ω 1 − ω 2 ) ) s ⁢ d u ) 1 / s ⁢ d ⁢ η ⁢ d ⁢ x 1 ⁢ d ⁢ ω 1 ⁢ d ⁢ x 2 ⁢ d ⁢ ω 2 ≲ ∫ R 2 ∫ R 2 ∫ | δ | > 0 e − π 2 ⁢ ( x 1 2 + x 2 2 + ω 1 2 + ω 2 2 ) ⁢ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ⁢ d x 1 ⁢ d ω 1 ⁢ d x 2 ⁢ d ω 2 ,

using Minkowski’s integral inequality. For x 1 , x 2 ∈ R , we get

∫ | δ | > 0 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = ∫ − ∞ − δ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η + ∫ δ ∞ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = ∫ − ∞ x 2 − x 1 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η + ∫ x 2 − x 1 − δ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η + ∫ δ ∞ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η .

We note that the second integral in the above line is zero if δ > x 1 − x 2 . Now, we estimate each of the three integrals separately. Consider

I δ 1 = ∫ − ∞ x 2 − x 1 F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η = ∫ − ∞ x 2 − x 1 1 ( x 2 − x 1 − η ) 1 / r ⁢ ( 1 + ( ln ⁡ ( x 2 − x 1 − η ) ) 2 ) ⋅ 1 ( − η ) 1 / r ⁢ ( 1 + ( ln ⁡ ( − η ) ) 2 ) ⁢ d η
≲ ∫ x 1 − x 2 ∞ 1 ( x 2 − x 1 + η ) 1 / r ⁢ η 1 / r ⁢ d η = ∫ x 1 − x 2 x 1 − x 2 + 1 1 ( x 2 − x 1 + η ) 1 / r ⁢ η 1 / r ⁢ d η + ∫ x 1 − x 2 + 1 ∞ 1 ( x 2 − x 1 + η ) 1 / r ⁢ η 1 / r ⁢ d η
≤ 1 ( x 1 − x 2 ) 1 / r ⁢ ∫ x 1 − x 2 x 1 − x 2 + 1 1 ( x 2 − x 1 + η ) 1 / r ⁢ d η + ∫ x 1 − x 2 + 1 ∞ 1 ( x 2 − x 1 + η ) 2 / r ⁢ d η
≤ 1 ( x 1 − x 2 ) 1 / r ⁢ ∫ 0 1 1 η 1 / r ⁢ d η + ∫ 1 ∞ 1 η 2 / r ⁢ d η ≲ 1 ( x 1 − x 2 ) 1 / r .
Similarly,
I δ 2 = ∫ x 2 − x 1 − δ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ≤ ∫ x 2 − x 1 − δ 1 ( η + x 1 − x 2 ) 1 / r ⁢ ( − η ) 1 / r ⁢ d η ≤ 1 δ 1 / r ⁢ ∫ x 2 − x 1 − δ 1 ( η + x 1 − x 2 ) 1 / r ⁢ d η = 1 δ 1 / r ⁢ ∫ 0 − δ + x 1 − x 2 1 η 1 / r ⁢ d η ≲ 1 δ 1 / r ⁢ ( δ + x 1 − x 2 ) 1 − 1 / r ,
I δ 3 = ∫ δ ∞ F 1 ⁢ ( η + x 1 − x 2 ) ⁢ F 1 ⁢ ( η ) ⁢ d η ≤ ∫ δ ∞ 1 η 2 / r ⁢ d η ≲ δ 1 − 2 / r .
Since e − π 2 ⁢ ( x 1 2 + ω 1 2 + x 2 2 + ω 2 2 ) ∈ S ⁢ ( R 4 ) , the integral I δ < ∞ . This proves that the integral 𝐼 is not finite. In other words, C θ ⁢ f 2 ∉ M s , 1 , which is the same as f 2 ∉ M θ s , 1 . Since s > r , M θ r , 1 ⊂ M θ s , 1 . Thus, in particular, f 2 ∉ M θ r , 1 .

Acknowledgements

The authors thank the National Institute of Science Education and Research Bhubaneswar for providing an excellent research facility. We thank the referee for meticulously reading our manuscript and giving us several valuable suggestions.

  1. Communicated by: Anke Pohl

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Received: 2024-04-25
Revised: 2024-10-15
Published Online: 2024-11-30
Published in Print: 2025-06-01

Š 2024 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(𝔽𝑞) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
https://doi.org/10.1515/forum-2024-0191
SchlagwĂśrter fĂźr diesen Artikel
Nonlinear operations; factorizations; affine modulation spaces; special affine Fourier transform

Artikel in diesem Heft

  1. Frontmatter
  2. Rings of differential operators on (k,s)-th Tjurina algebras of singularities
  3. Cokernels of random matrix products and flag Cohen–Lenstra heuristic
  4. Gradient bounds and Liouville property for a class of hypoelliptic diffusion via coupling
  5. The L p -L q compactness of commutators of oscillatory singular integrals
  6. Determination of a pair of newforms from the product of their twisted central values
  7. Metrical properties of exponentially growing partial quotients
  8. Nonlinear operations and factorizations on a class of affine modulation spaces
  9. On algebraic degrees of certain exponential sums over finite fields
  10. The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
  11. Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
  12. Modular representations of GL2(𝔽𝑞) using calculus
  13. Bounding the number of p'-degree characters from below
  14. Existence and multiplicity of non-radial sign-changing solutions for a semilinear elliptic equation in hyperbolic space
  15. Duality theorems for polyanalytic functions
  16. Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)
  17. Cylindrical ample divisors on Du Val del Pezzo surfaces
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