Displaying similar documents to “ L p - L q estimates for some convolution operators with singular measures on the Heisenberg group”

On L p - L q boundedness for convolutions with kernels having singularities on a sphere

Alexey N. Karapetyants (2001)

Studia Mathematica

Similarity:

For the convolution operators A a α with symbols a ( | ξ | ) | ξ | - α e x p i | ξ | , 0 ≤ Re α < n, a ( | ξ | ) L , we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from L p to L q .

L p -improving properties of certain singular measures on the Heisenberg group

Pablo Rocha (2022)

Mathematica Bohemica

Similarity:

Let μ A be the singular measure on the Heisenberg group n supported on the graph of the quadratic function ϕ ( y ) = y t A y , where A is a 2 n × 2 n real symmetric matrix. If det ( 2 A ± J ) 0 , we prove that the operator of convolution by μ A on the right is bounded from L ( 2 n + 2 ) ( 2 n + 1 ) ( n ) to L 2 n + 2 ( n ) . We also study the type set of the measures d ν γ ( y , s ) = η ( y ) | y | - γ d μ A ( y , s ) , for 0 γ < 2 n , where η is a cut-off function around the origin on 2 n . Moreover, for γ = 0 we characterize the type set of ν 0 .

Convolution operators with anisotropically homogeneous measures on 2 n with n-dimensional support

E. Ferreyra, T. Godoy, M. Urciuolo (2002)

Colloquium Mathematicae

Similarity:

Let α i , β i > 0 , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let t x = ( t α x , . . . , t α x ) , t x = ( t β x , . . . , t β x ) and | | x | | = i = 1 n | x i | 1 / α i . Let φ₁,...,φₙ be real functions in C ( - 0 ) such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on 2 n given by μ ( E ) = χ E ( x , φ ( x ) ) | | x | | γ - α d x , where α = i = 1 n α i and dx denotes the Lebesgue measure on ℝⁿ. Let T μ f = μ f and let | | T μ | | p , q be the operator norm of T μ from L p ( 2 n ) into L q ( 2 n ) , where the L p spaces are taken with respect to the Lebesgue measure. The type set E μ is defined by E μ = ( 1 / p , 1 / q ) : | | T μ | | p , q < , 1 p , q . In the case α i β k for 1 ≤ i,k ≤ n we characterize the...

Characterization of surjective convolution operators on Sato's hyperfunctions

Michael Langenbruch (2010)

Banach Center Publications

Similarity:

Let μ ( d ) ' be an analytic functional and let T μ be the corresponding convolution operator on Sato’s space ( d ) of hyperfunctions. We show that T μ is surjective iff T μ admits an elementary solution in ( d ) iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are 0 μ ( d ) ' such that T μ is not surjective on ( d ) .

The type set for homogeneous singular measures on ℝ ³ of polynomial type

E. Ferreyra, T. Godoy (2006)

Colloquium Mathematicae

Similarity:

Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by μ ( E ) = D χ E ( x , φ ( x ) ) d x with D = x ∈ ℝ ²:|x| ≤ 1 and let T μ be the convolution operator with the measure μ. Let φ = φ e φ e be the decomposition of φ into irreducible factors. We show that if e i m / 2 for each φ i of degree 1, then the type set E μ : = ( 1 / p , 1 / q ) [ 0 , 1 ] × [ 0 , 1 ] : | | T μ | | p , q < can be explicitly described as a closed polygonal region.

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

A convolution operator, bounded on L q ( n ) , is bounded on L p ( n ) , with the same operator norm, if p and q are conjugate exponents. It is well known that this fact is false if we replace n with a general non-commutative locally compact group G . In this paper we give a simple construction of a convolution operator on a suitable compact group G , wich is bounded on L q ( G ) for every q [ 2 , ) and is unbounded on L p ( G ) if p [ 1 , 2 ) .

Multiplication operators on L ( L p ) and p -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

Similarity:

A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 p < 2 , and T | X is an isomorphism for some reflexive subspace X of L p , then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.

On certain G L ( 6 ) form and its Rankin-Selberg convolution

Amrinder Kaur, Ayyadurai Sankaranarayanan (2024)

Czechoslovak Mathematical Journal

Similarity:

We consider L G ( s ) to be the L -function attached to a particular automorphic form G on G L ( 6 ) . We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg L -function L G × G ( s ) . As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of L G × G ( s ) .

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze, Michael Lin (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n * f - f d m p 0 for every f L p , every (some) p [ 1 , ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f L p whenever p g t ; 1 . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of μ , for the strong sweeping out property (existence of a Borel set B with lim sup μ n * 1 B = 1 a.e. and lim inf μ n * 1 B = 0 a.e.). The results are extended to general compact Abelian groups...

On nearly radial marginals of high-dimensional probability measures

Bo&#039;az Klartag (2010)

Journal of the European Mathematical Society

Similarity:

Suppose that μ is an absolutely continuous probability measure on R n, for large n . Then μ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if n ( C / ε ) C d , then there exist d -dimensional marginals of μ that are ε -far from being sphericallysymmetric, in an appropriate sense. Here C > 0 is a universal constant.

Approximation properties of β-expansions

Simon Baker (2015)

Acta Arithmetica

Similarity:

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ( ϵ i ) i = 1 0 , 1 a β-expansion for x if x = i = 1 ϵ i β - i . We call a finite sequence ( ϵ i ) i = 1 n 0 , 1 n an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given Ψ : 0 , we introduce the following subset of ℝ: W β ( Ψ ) : = m = 1 n = m ( ϵ i ) i = 1 n 0 , 1 n [ i = 1 n ( ϵ i ) / ( β i ) , i = 1 n ( ϵ i ) / ( β i ) + Ψ ( n ) ] In other words, W β ( Ψ ) is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities 0 x - i = 1 n ( ϵ i ) / ( β i ) Ψ ( n ) . When n = 1 2 n Ψ ( n ) < , the Borel-Cantelli lemma tells us that the Lebesgue measure...

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We consider C 2 families t f t of  C 4 unimodal maps f t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure μ t of  f t depends differentiably on  t , as a distribution of order 1 . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of  μ t for a Benedicks-Carleson map f t , in terms of a single smooth function and the...

Measure-geometric Laplacians for partially atomic measures

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer (2020)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure η supported on a compact subset of U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator η and a second order differential operator Δ η , with respect to η . We generalize this approach to measures of the form η : = ν + δ , where ν is non-atomic and δ is finitely supported. We determine...

Theoretical analysis for 1 - 2 minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

Similarity:

We investigate the recovery of k -sparse signals using the 1 - 2 minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k -sparse signals 𝐱 with the prior support T which is composed of g true indices and b wrong...

The sizes of the classes of H ( N ) -sets

Václav Vlasák (2014)

Fundamenta Mathematicae

Similarity:

The class of H ( N ) -sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of H ( N ) -sets are the same for all N ∈ ℕ. To prove our result we also present a new description of H ( N ) -sets.

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

Similarity:

Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there...

The small Ree group 2 G 2 ( 3 2 n + 1 ) and related graph

Alireza K. Asboei, Seyed S. S. Amiri (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let G be a finite group. The main supergraph 𝒮 ( G ) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o ( x ) o ( y ) or o ( y ) o ( x ) . In this paper, we will show that G 2 G 2 ( 3 2 n + 1 ) if and only if 𝒮 ( G ) 𝒮 ( 2 G 2 ( 3 2 n + 1 ) ) . As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group 2 G 2 ( 3 2 n + 1 ) .

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...