Hardy's Theorem Research Articles

The Lp –Lq version of Hardy's Theorem on nilpotent Lie groups

Let p, q be such that 2≤ p, q ≤ + ∞. We prove in this paper the Lp – Lq version of Hardy's Theorem for an arbitrary nilpotent Lie group G extending then earlier cases and the classical Hardy theorem proved recently by E. Kaniuth and A. Kumar. The case where 1 ≤ p, q ≤ + ∞ is studied for a restricted class of nilpotent Lie groups.

On theorems of Beurling and Hardy for the Euclidean Motion group

We establish an analogue of Beurling's uncertainty principle for the group Fourier transform on the Euclidean motion group. We also prove the most general version of Hardy's theorem on it which characterises functions on the motion group that are controlled by the heat kernel associated to the Laplacian of the Euclidean space.

An uncertainty principle like Hardy's theorem for nilpotent Lie groups

AbstractWe extend an uncertainty principle due to Cowling and Price to threadlike nilpotent Lie groups. This uncertainty principle is a generalization of a classical result due to Hardy. We are thus extending earlier work on Rnand Heisenberg groups.

AN ANALOGUE OF COWLING–PRICE'S THEOREM AND HARDY'S THEOREM FOR THE GENERALIZED FOURIER TRANSFORM ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR

Using the harmonic analysis associated with the spherical mean operator R and its dualtR we establish an analogue of Cowling–Price's theorem and Hardy's theorem for the spherical mean operator R.

On Theorems of Hardy, Gelfand–Shilov and Beurling for Semisimple Groups

In this paper we prove a strong version of Hardy’s theorem for the group Fourier transform on semisimple Lie groups which characterises the Fourier transforms of all functions satisfying Hardy type conditions. In the particular case of \mathit{SL}(2, ℝ) we characterise all such functions and conjecture that the same is true for all rank one semisimple groups. We also establish an analogue of a theorem of Gelfand and Shilov in the context of semisimple groups. A version of Beurling’s theorem which assumes a Cowling–Price condition on the function is also proved. We show that these results yield most of the earlier results as corollaries.

An L p version of Hardy's theorem for the Jacobi–Dunkl transform

In this paper, we give a generalization of Hardy's theorem for the Jacobi–Dunkl transform ℱ on ℝ. More precisely for all a > 0, b > 0 and p, q ∈ [1, +∞], we determine the measurable functions f on ℝ such that E 1/4a −1 f ∈ L α,β p (ℝ) and e bλ2 ℱf ∈ L σ q (ℝ), where E t , t > 0, L α,β p (ℝ), p ∈ [1, +∞], and L σ q (ℝ), q ∈ [1, +∞], are respectively the heat kernel and the Lebesgue spaces associated with the Jacobi–Dunkl operator. *E-mail: fredj.chouchane@ipeim.rnu.tn †E-mail: maher.mili@fsm.rnu.tn ‡E-mail: mohamed.sifi@fst.rnu.tn

The Hardy–Littlewood–Sobolev inequality for ( β, γ)-distance Riesz potentials

The generalized with respect to ( β, γ)-distance Riesz potential defined on Sobolev space W p l( R n) is constructed and for this potential the theorem of Hardy–Littlewood–Sobolev type has been established.

The ( p, q, l)-properties of a generalized Riesz potential

The generalized Riesz potential defined on Sobolev space W p l ( R n ) is constructed and for this potential the theorem of Hardy–Littlewood–Sobolev type have established.

Hardy's theorem for the Jacobi transform

Let ab 1⁄4 1 4 for positive constants a; b. Hardy’s theorem states that the function f ðxÞ 1⁄4 e ax is the only function (modulo constants) satisfying the decay conditions f ðxÞ 1⁄4 Oðe ax Þ and f ðxÞ 1⁄4 Oðe bx Þ, where f denotes the Fourier transform of f . We generalise this theorem and its L p analogues to the Jacobi transform. We then consider the Fourier transform on the real hyperbolic spaces SOoðm; nÞ=SOoðm 1; nÞ, m; n A N, and show, as an application of our results for the Jacobi transform, that Hardy’s theorem only can be generalised to the Riemannian ðm 1⁄4 1Þ case. It can, in particular, not be generalised to SLð2;RÞFSUð1; 1ÞF SOoð2; 2Þ=SOoð1; 2Þ.

Heat Kernel and Hardy's Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform by using the sharp estimate of the heat kernel.

The Isoperimetric Inequality and a Theorem of Hardy and Littlewood

(2003). The Isoperimetric Inequality and a Theorem of Hardy and Littlewood. The American Mathematical Monthly: Vol. 110, No. 6, pp. 532-536.

The Isoperimetric Inequality and a Theorem of Hardy and Littlewood

The Isoperimetric Inequality and a Theorem of Hardy and Littlewood

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL2(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝn or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.

Un analogue d'un théorème de Hardy pour la transformation de Dunkl

In this Note we give a generalization of Hardy's theorem for the Dunkl transform F D on R d . More precisely, for all a>0, b>0 and p, q∈[1,+∞], we determine the measurable functions f such that e a||x|| 2 f∈ L k p( R d) and e b||y|| 2 F D(f)∈ L k q( R d) , where L k p( R d) are the L p spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimèche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X=G/K be the associated symmetric space and assume that X is of rank one . Let M be the centraliser of A in K and consider an orthonormal basis {Y δ,j :δ ∈ K^ 0 ,1 ≤ j ≤ d δ } of L 2 (K/M) consisting of K-finite functions of type δ on K/M. For a function ƒ on X let ƒ˜ (λ,b), λ ∈ C, be the Helgason Fourior transform. Let h t be the heat kernel associated to the Laplace-Beltrami operator and let Q δ (iλ+ϱ) bethe Kostant polynomials. We establish the following version of Hardy's theorem for the Helgason Fourier transform: Let ƒ be a function on G/K whcih satisfies |ƒ(ka r )| ≤ Ch t (r). Further assume that for every δ and j the functions F δ,j (λ)=Q δ (iλ+ϱ) -1 ∫ K/M ƒ˜(λ,b)Y δj (b)db Satisfy the estimates |F δ,j (λ) ≤ C δj e -tλ 2 for λ ∈ R. Then ƒ is a constant multiple of heat kernel h t .

A complete analogue of Hardy's theorem on semisimple Lie groups

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f are O(|x|e 2 ) and O(|x|e 2/(4α)) respectively for some m,n ≥ 0 and α > 0, then f and f are P (x)e 2 and P (x)e /(4α) respectively for some polynomials P and P . If in particular f is as above, but f is o(e 2/(4α)), then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

Formulas of Ramanujan for the power series coefficients of certain quotients of Eisenstein series

In their last published paper [9], [16, pp. 310–321], G. H. Hardy and S. Ramanujan derived infinite series representations for the coefficients of certain modular forms of negative weight which are not analytic in the upper half-plane. In particular, they examined in detail the coefficients of the reciprocal of the Eisenstein series E6(τ). While confined to the sanitarium, Matlock House, in 1918, Ramanujan wrote several letters to Hardy about the coefficients in the power series expansions of certain quotients of Eisenstein series. These letters are photocopied in [18, pp. 97–126], and printed versions with commentary can be found in [6, pp. 175–191]. In these letters, Ramanujan recorded formulas for the coefficients of several quotients of Eisenstein series not examined by Hardy and him in [9]. These claims fall into two related classes. In the first class are formulas for coefficients that arise from the main theorem of Hardy and Ramanujan, or a slight modification of it, and these results have been proved in a paper by Berndt and Bialek [5]. Those in the second class, which we prove in this paper, are much harder to prove. To establish the first main result, we need an extension of Hardy and Ramanujan’s theorem due to H. Petersson [11]. To prove the second primary result, we need to first extend work of H. Poincare [14], Petersson [11], [12], [13], and J. Lehner [10] to cover double poles. In all cases, the formulas have a completely different shape from those arising from modular forms analytic in the upper half-plane, such as the famous infinite series for the partition function p(n) arising from the reciprocal of the Dedekind eta-function. As we shall see in the sequel, the series examined in this paper are very rapidly convergent, even more so than those arising from modular forms analytic in the upper half-plane, so that truncating a series, even with a small number of terms, provides a remarkable approximation. Using Mathematica, we calculated several coefficients and series approximations for the two primary functions 1/B(q) and 1/B(q) (defined below) examined by Ramanujan. As will be seen from the first table, the coefficient of q in 1/B(q), for example, has 17 digits, while just two terms of Ramanujan’s infinite series representation calculate this coefficient with

Hardy's Theorem for simply connected nilpotent Lie groups

We prove an analogue of Hardy's Theorem for Fourier transform pairs in ℝ for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on ℝn and the Heisenberg groups ℍn.

An analogue of Hardy's theorem for the Harish-Chandra transform

A theorem of Hardy asserts that a function and its Fourier transform cannot both be very small. We prove analogues of Hardy’s theorem for the Harish- Chandra transform for spherical functions on a non-compact semisimple Lie group and the Helgason transform on a Riemannian symmetric space of the non-compact type.

Uncertainty principles on two step nilpotent Lie groups

We extend an uncertainty principle due to Cowling and Price to two step nilpotent Lie groups, which generalizes a classical theorem of Hardy. We also prove an analogue of Heisenberg inequality on two step nilpotent Lie groups.