Compactness criteria via Laguerre and Hankel transformations

The aim of this paper is to prove Kolmogorov-Riesz type theorems via Bessel and Laguerre translations, and Pego-type theorems by the corresponding transformations.


Introduction
The populous family of compactness theorems is established by Arzelá and Ascoli. Our starting point, the Kolmogorov-Riesz theorem characterizes precompact (totally bounded) sets of L p (R n ), see e.g. [1]. Besides, such a characterization is interesting in itself, it has several applications to differential and integral equations. Compactness criteria were studied in particular non-standard function spaces, e.g. in Sobolev spaces [17], variable Lebesgue spaces [13], or weighted variable exponent amalgam and Sobolev spaces [5], and also in more general circumstances, see e.g. [10], [25] and [14]. Sudakov-type improvements of the classical theorem are derived in [18] (see [26] as well). The useful criterion of compactness given by Pego via Fourier transformation (see [23]) has also great influence, see e.g. [9], [11], [12]. In the inspirating paper [19], compactness criterion is given by Laplace transformation.
Below we offer a new aspect of characterization of precompact sets in certain Banach spaces. Instead of deriving similar theorems by related, for instance Mellin or cosine transformations, we investigate and extend the notion of equicontinuity. Motivated by the effect of translation, τ y f (x) = f (x − y), on exponential functions, different translation operators were introduced by orthogonal systems {ϕ n } n as T y ϕ n (x) = ϕ n (x)ϕ n (y). Hereinafter we deal with Laguerre and Bessel translations.
In the next section a Riesz-Kolmogorov type theorem is derived by Laguerre translation to weighted L p spaces on the half-line. The corresponding transformation, the discrete Laguerre-Fourier transformation implies a remarkable simplification, since the structure of the corresponding l p ′ spaces are simpler than the original L p ones. In this section we also introduce and study Laguerre translation on sequences. In the third section the method is presented by Bessel translation. The corresponding transformation is the Hankel (Bessel-Fourier) transformation which establishes a Pego-type theorem.

Laguerre translation method
Laguerre translation is developed by product formulae, see e.g. [28] and by investigation of the related Cauchy problem, see e.g. [7]. We mention that it is a natural idea to handle Bessel and Laguerre translations in parallel, since the translated functions in both cases are the solutions to very similar Cauchy problems, u xx − u tt + 2α+1 x u x − 2α+1 t u t − ru = 0, where r(x, t) = 0 in Bessel case and r(x, t) = x 2 −t 2 in Laguerre case. The derived convolution structure is examined e.g. in [15]. The norm of the translation operator ensures a maximum principle to the corresponding hyperbolic problem, and it implies Nikol'skii-type norm-estimations on the half-line, see [2]. Convolution is applicable to study of best approximation in certain spaces, see [16]. In this section compactness is investigated by Laguerre translation. The main result of the section is the characterization of precompact sets in L 2 α . 2.1. Laguerre translation and precompact sets in L p α . We introduce weighted L p -spaces on the half-line similar to the ones given in the previous section.
The translation operator acting on the space given above can be defined as follows, see e.g. [15] .
and j α− 1 2 is the entire Bessel function, cf. (30). By symmetry of the definition in t, x ≥ 0, , and again by definition . Denoting by L α n (x) the n th Laguerre polynomial orthogonal on (0, ∞) with respect to the Laguerre weight, that is and taking into consideration that we denote by Thus R α n (0) = 1 and With this notation we have that , see e.g. [15].
According to [15,Theorem 1] for all α ≥ 0 and 1 ≤ p ≤ ∞, considering the translated function as a function of the variable x, Again by [15,Theorem 1], for all α ≥ 0 and 1 ≤ p, q, r ≤ ∞ with 1 The first theorem of this section is a Kolmogorov -Riesz type theorem, where the standard equicontinuity property is replaced by the one based on Laguerre translation. Theorem 1. Let 1 ≤ p < ∞ and α ≥ 0. A bounded set K ⊂ L p α is precompact if and only if the properties below are fulfilled. P a . For all ε > 0 there is an R > 0 such that for all f ∈ K For all ε and M 0 positive numbers there is a δ > 0 such that for all 0 ≤ t ≤ M 0 , 0 ≤ h ≤ δ and f ∈ K Proof. First we assume that K is precompact. Then for an ε > 0 there is a finite That is there is an R = R ε > 0 such that the closed ball B(0, R) contains the support of each Φ ∈ S ε wich ensures P a . To prove P b take an f ∈ L p α , t ∈ [0, M 0 ] with some M 0 , and let Φ be the element of S ε closest to f with suppΦ ∈ (0, R). In view of (1) For sake of simplicity let us denote by Recalling that suppΦ ⊂ (0, R) and t ∈ [0, Considering that difference of the arguments of the three functions above are at most 2 √ xh ≤ c(R, M 0 ) √ h and Ψ(x, t, ϑ) is compactly supported and continuous, there is a δ > 0 such that if Then, because the set of Φ-s in question is finite we can choose R and δ uniformly.
Since the norm of translation is bounded on [0, M 0 ], cf. (4) (and it is a linear operator), we can finish this part with triangle inequality. Vice versa assuming P a and P b let Then, applying Hölder's inequality and the symmetry of translation and similarly By (10) and (4) |e Thus with fixed a and R, F a,R is bounded (by M 1 (9) and assumption P b it is equicontinuous. Thus for an arbitrary ε > 0 there is an ε-net, V a f 1 , . . . , V a f n , in F a,R such that f i ∈ K, i = 1, . . . , n.
Let f ∈ K be arbitrary, and for an ε > 0 choose R according to property P a . Then again by Hölder and Fubini theorems Thus choosing a according to property P b small enough (assume that a < 1) , With these chosen a and R construct F a,R and for the previous ε and f , from the That is by the triangle inequality {f i } n i=1 is a (4 + c(α, p))ε-net in K.

2.2.
Laguerre translation on sequences. Laguerre translation on sequences, to the best of my knowledge, have not appeared explicitly just as convolution of sequences, see e.g. [4]. The corresponding algebras are investigated in [20]. Below we derive the translation from convolution and investigate its properties. Let α > −1. Recalling the notation above we introduce the discrete weights If a ∈ l p α and b ∈ l p ′ α we denote by

Remark.
Certainly, in discrete case the criterion of precompactness is simpler, cf. [17,Theorem 4]. It is as follows.
For any 1 ≤ p < ∞ and α ≥ 0 a set K ⊂ l p α is precompact if and only if it is pointwise bounded (i.e. for all n ∈ N there is an M (n) such that for all a ∈ K a(n) ≤ M (n)), and the next property fulfils.
Indeed, suppose, that K is precompact. Then it is obviously bounded in l p α . Since w(k) ≥ 1, it is pointwise bounded as well. For ε 2 it has a finite ε 2 -net, b 1 , . . . , b n , say. Since the finite sequences are dense in l p α , there are s i finite sequences, such that Thus the maximal length of s i -s is an appropriate choice of N .
Assume now, that K is pointwise bounded and fulfils P as . Choose an N for an arbitrary ε ensured by P as . Let S N := {a N := a(0), . . . , a(N ) : a ∈ K} be the set of the N + 1-long initial parts of the sequences in K. Then the distance of K and and (again by being finite dimensional) S N contains a finite ε-net, and the corresponding sequences form a 2ε-net in K.
To define translation on the spaces of sequences, our starting point is the convolution defined in [4].
Similarly to the L p -cases if 1 ≤ p, q, r ≤ ∞ and 1 see [4, (4.6)]. Thus, fixing α > α 0 , we can define the corresponding translation as By symmetry we immediately have that So (14) can be written as As on functions, Laguerre translation on sequences is also a bounded operator.
Proof. Let a ∈ l ∞ . In view of (16), (12) and (13) |T According to (15) which implies the statement in l 1 α . Finally interpolation ensures the result. To prepare the next subsection, we mention the standard connection between the corresponding spaces of functions and sequences.
For a function f in some L p α let us denote the corresponding sequence byf = a f , where if the series is convergent in some sense. Let 1 ≤ p ≤ 2. Then, as usual, if f ∈ L p α thenf ∈ l p ′ α , and if a ∈ l p α thenǎ ∈ L p ′ α and the operators map L p α to l p ′ α and l p α to L p ′ α are bounded. Indeed, by interpolation it is a consequence of Parseval's formula and the next inequalities: where (2) is considered. Let a and b are in l 1 α , say, then the seriesf a andg b are uniformly convergent. Thus by (14) ( that is see [4]. Similarly (14) ensures that iff = a f then (22) fRα n (k) = T n (a f )(k). Moreover (3) implies that (23) f * g(n) =f (n)ĝ(n), T α t (f )(n) =f (n)R α n (t), provided that α ≥ 0, f ∈ L p α and g ∈ L q α and 1 p + 1 q ≥ 1. 2.3. Pego-type theorem with Laguerre transformation. At first we introduce the notion of equicontinuity in mean with respect to sequences. P bs . A set K ⊂ l p α is equicontinuous in mean if for all ε > 0 there is an N ∈ N such that for all j > N and a ∈ K Subsequently we need the the next extra property with respect to functions.
After this preparation we are in position to state a Pego-type theorem.
Theorem 2. Let 1 ≤ p ≤ 2 and α ≥ 0. (a) If K ⊂ l p α is bounded and fulfils P as , thenǨ ⊂ L p ′ α fulfils P b . (b) If K ⊂ L p α fulfils P b , thenK ⊂ l p ′ α fulfils P as . (c) If K ⊂ l p α fulfils P bs , thenǨ ⊂ L p ′ α fulfils P a . (d) If in addition α > 1 2 , K ⊂ L p α is bounded and fulfils P a and P a0 , thenK ⊂ l p ′ α fulfils P bs .
The computation in proof of (d) ensures the regular behavior of translation on sequences.
Proof. By property P as it is enough to prove the statement for finite sets of finite sequences. Let S := s 1 , . . . , s n and s i = {a ik } ni k=0 and p i = ni k=0 a ik R α k . For any ε > 0 there are δ i and R i such that With the abbreviation a ik = p i (k), as above we have By (27) and (2) if j is large enough. Thus Since N is independent of ε, it is arbitrary small if j is large enough. According to Proposition 1 the proof can be finished with a triangle inequality.
The main theorem of this section os the next corollary.
is precompact if and only if it satisfies P b , and the corresponding setK is pointwise bounded.
Proof. If K is precompact in L 2 α , Theorem 1 ensures equicontinuity in mean. Moreover K is bounded in L 2 α , soK is bonded in l 2 α and as above, pointwise too. If K is equicontinuous in mean, by Theorem 2K is equivanishing in l 2 α , and the pointwise boundedness ensures precompactness ofK and of K too.

Bessel translation method
Bessel translation and Hankel transformation are widely examined by several authors. We mention here only a few examples. One of our main sources is an early paper, [21]. By Bessel translation modulus of smoothness and the related best approximation can be investigated, see [24], and Nikol'skii inequalities for entire functions can be proved, see [3]. For Fourier-Bessel transformation, similarly to the standard Fourier transformation, uncertainty results are derived, see [8], which are somehow in concordance with Pego-type theorems. Hereinafter we concentrate to compactness criteria.
Let L p,α be the space of measurable functions on R + equipped with the norm The dual space of L p,α is denoted by L p ′ ,α , where 1 p + 1 p ′ = 1. First we introduce the Bessel translation of an integrable function, see [21, (5.19)]: where dµ(ϕ) = c α sin 2α ϕdϕ, and c α = We need the entire Bessel functions which are

12Á. P. HORVÁTH
Subsequently the next properties will be necessary. The norm of an entire Bessel function on the half-line is attained at zero: The derivative of j α can be expressed as see [6].
The operator norm of Bessel translation is one, see [24, (2.24)] The Kolmogorov-Riesz type theorem of this section is as follows.
Theorem 3. 1 ≤ p < ∞ K ⊂ L p,α is bounded. K is precompact if and only if the two conditions below are fulfiled.
Proof. To prove property P A we can proceed just as in the proof of Theorem 1.
On the other hand supposing P A and P B , we show that K is precompact. To this we define where A = a 0 t 2α+1 dt. By Hölder's inequality, then applying the symmetry of translation, cf. (29) Similarly, and recalling (33) Let a and R be fixed positive numbers, and let F a,R := {M a f (s) : f ∈ K, s < R}. Then by (37) and (35) F a,R is equicontinuous (in standard sense), and by (38) and boundedness of K F a,R is uniformly bounded. Thus by the theorem of Arzelá and Ascoli for all ε > 0 there is a finite ε-net N ε ⊂ F a,R . Let us denote the elements of N ε = {M a f 1 , . . . , M a f j }, where f i ∈ K, i = 1, . . . , j and j = j(ε), that is for all f ∈ K there is an with a suitable a, L and R (will be given later).
Applying again Hölder's inequality and then Fubini's theorem Similarly Let ε > 0 be arbitrary and let us choose by (34) R so large that ∀f ∈ K ∞ R |f (x)| p x 2α+1 dx 1 p < ε 5 and by (35) let us choose a so small such that for all 0 ≤ h ≤ a and for all f ∈ K, t ∈ [0, R]

The inverse transformation is
(y)j α (xy)x 2α+1 dy, if exists. If K is a set of certain functions, we denote byK the set of the Hankel transforms of the functions in K. Below we need that Hankel transformation fulfils the Hausdorff-Young inequality, that is (41) H α (f ) p ′ ,(α) ≤ C p f p,(α) , 1 ≤ p ≤ 2, see [8]. We also use the next property of the Hankel transform.
Theorem 4. 1 ≤ p ≤ 2. Let K ⊂ L p,α . If K is bounded and satisfies P A in L p,α , K satisfies P B in L p ′ ,α , and if K satisfies P B in L p,α ,K satisfies P A in L p ′ ,α .