Norm inequalities for fractional integrals of Laguerre and Hermite expansions

George Gasper and Walter Trebels

Abstract. Suppose the fractional integration operator I^{σ} is generated by the sequence {(k + 1)^{−σ}} in the setting of Laguerre and Hermite expansions. Then, via projection formulas, the problem of the norm boundedness of I^{σ} is reduced to the well-known fractional integration on the half-line. A corresponding result with respect to the modified Hankel transform is derived and its connection with the Laguerre fractional integration is indicated.

1 Introduction

The aim of this note is to present a new approach for the derivation of boundedness properties of fractional integration operators generated by multipliers with respect to Laguerre and Hermite expansions as well as with respect to modified Hankel transforms. The method of proof consists in reducing the problem to classical fractional integration via projection formulas as given, e.g., in Askey and Fitch [3] and then applying weighted norm inequalities for the resulting integral operators as given in Samko, Kilbas and Marichev [12] and in Andersen and Heinig [1], thus separating two difficulties: The orthogonal setting is handled via the projection formulas, and the behavior of classical fractional integration on the half-line in a weighted setting is essentially known.

Let us mention two different approaches in the case of Laguerre series.

(i) In Gasper, Stempak and Trebels [6] the delicate convolution structure for Laguerre expansions ∑ a_{k}L_{k}^{α} is essentially used (thus restricting the parameter α to nonnegative values) to obtain an unweighted fractional integration theorem which is then extended to power weights by a method due to E. M. Stein and G. Weiss. (ii) Kanjin and Sato [10] consider Laguerre expansions ∑ a_{k}L_{k}^{α} with respect to the orthonormalized Laguerre functions, apply the intricate transplantation theorem due to Kanjin (cf. [10, Theorem A]) to reduce the problem to the case α = 0 and, by a transplantation theorem between ∑ a_{k}L_{k}^{0} and one-dimensional Fourier series, can switch to fractional integration for Fourier series (generated via a multiplier sequence). The advantage of this method is that negative values of α are also admitted.

01991 Mathematics Subject Classification. Primary 26A33; Secondary 33C45, 42A38, 42C10.

Key words and phrases. Fractional integration, Laguerre and Hermite expansions, Hankel transforms, multipliers

The benefits of the present approach are the following: (a) It is elementary; (b) it improves, e.g., the known fractional integration theorems for Laguerre expansions in two ways: more power weights are admitted and sharper norm inequalities in the following sense are derived (σ > 0)

∥ ∑(k + 1)^{−σ}a_{k}L_{k}^{α}∥_{r} ≲ ∥ ∑(k + 1)^{−σ}a_{k}L_{k}^{α+σ}∥_{q} ≲ ∥ ∑ a_{k}L_{k}^{α}∥_{p}

(the notation A ≲ B is used to indicate that A ≤ c · B with a constant c independent of significant quantities; the parameters r, p, σ, α satisfy the standard relation 1/r = 1/p − σ/(α + 1) in the “natural” weight case with respect to the convolution structure); (c) in principle those other than power weights are admitted; (d) the method should work for further expansions or integral transforms whenever relevant projection formulas and relevant results on classical fractional integration are available.

2 Laguerre expansions

We take over the notation from [9] and consider the Lebesgue spaces

L_{w(γ)}^{p} = {f : ∥f∥_{L_{w(γ)}^{p}} = ( ∫_{0}^{∞} |f(x)e^{−x/2}|^{p}x^{γ} dx)^{1/p} < ∞} , 1 ≤ p < ∞, γ > −1.

We denote the Laguerre polynomials by L_{k}^{α}, α > −1, k ∈ N_{0} (see [16, p. 100]), and normalize them by

R_{k}^{α}(x) = L_{k}^{α}(x)/L_{k}^{α}(0), L_{k}^{α}(0) = A_{k}^{α} = \binom{k + α}{k} = \frac{Γ(k + α + 1)}{Γ(k + 1)Γ(α + 1)} .

If γ < p(α + 1) − 1, one can formally expand f ∈ L_{w(γ)}^{p} into a Laguerre series

f(x) ∼ (Γ(α + 1))^{−1} ∑_{k=0}^{∞} f̂_{α}(k)L_{k}^{α}(x), f̂_{α}(k) = ∫_{0}^{∞} f(x)R_{k}^{α}(x)x^{α}e^{−x} dx .

Since the polynomials are dense in these L_{w(γ)}^{p}-spaces, we restrict ourselves in the following to considering polynomials f = ∑ a_{k}L_{k}^{α}(x) with only finitely many non-zero coefficients a_{k}. Motivated by the definition of fractional integration for Fourier series discussed by Hardy and Littlewood, we define a fractional integral operator I^{σ}, σ > 0, for Laguerre expansions by

I^{σ}( ∑_{k=0}^{∞} a_{k}L_{k}^{α}(x)) = ∑_{k=0}^{∞}(k + 1)^{−σ}a_{k}L_{k}^{α}(x) .

In order to use also related fractional integrals, equivalent to I^{σ} (with respect to the mapping behavior), we need the idea of multipliers. A sequence m = {m_{k}} is called a (bounded) multiplier from L_{w(γ)}^{p} into L_{w(γ)}^{r} , denoted by m ∈ M_{w(γ)}^{p,r}, if

∥T_{m}( ∑_{k=0}^{∞} a_{k}L_{k}^{α})∥_{L_{w(γ)}^{r}} := ∥ ∑_{k=0}^{∞} m_{k}a_{k}L_{k}^{α}∥_{L_{w(γ)}^{r}} ≤ C ∥ ∑_{k=0}^{∞} a_{k}L_{k}^{α}∥_{L_{w(γ)}^{p}}

for all polynomials; the smallest constant C for which this holds is called the multiplier norm ∥m∥_{M_{w(γ)}^{p,r}} which coincides of course with the operator norm of T_{m} . We mention that I^{σ} extends to a bounded operator from L_{w(γ)}^{p} to L_{w(γ)}^{p} when 0 ≤ αp/2 ≤ γ ≤ α, 1 ≤ p ≤ ∞, (see [6]) or when −1 < γ < (α + 1)p − 1, 1 ≤ p ≤ ∞, and −1 < α ≤ min {1 − 2/(3p), 1/3 + 2/(3p)}; this follows from a Corollary due to Poiani [11, p. 11] and [17, Theorem 3.3]. By [4, 1.18], [9, II, Theorem 3.1] and [17, Theorem 3.3] it is then clear that, e.g., the multiplier sequences

\{\frac{Γ(k + α + 1)}{Γ(k + α + σ + 1)}\}_{k∈N_{0}} , \{(k^{δ} + c + 1)^{−σ/δ}\}_{k∈N_{0}} , c > −1 , δ > 0,

generate fractional integral operators on L_{w(γ)}^{p} , 1 ≤ p ≤ ∞ , which are equivalent to the above I^{σ}. As a consequence, without loss of generality we can work with any of the fractional integrals generated by the preceding sequences.

Theorem 1. Let α > −1 and 1 < p ≤ r < ∞ . Assume further that 0 < σ < α + 1, a < (α + 1)/p′, b < (α + 1)/r, a + b ≥ −σ(2α + 1). Then

∥I^{σ}f∥_{L_{w(α−br)}^{r}} ≲ ∥f∥_{L_{w(α+ap)}^{p}} , \frac{1}{r} = \frac{1}{p} − \frac{σ − a − b}{α + 1} .

Remark 1. Not only is Theorem 1 proved by a new method, but this method also extends the parameter range α ≥ 0, a + b ≥ 0 in [6, Theorem 1.1] to α > −1, a + b ≥ −σ(2α + 1). Moreover, the result of Kanjin and Sato [10, Theorem] is completely recovered without the use of transplantation theorems. We only have to choose a = α(1/2 − 1/p), b = α(1/r − 1/2); then 1/r = 1/p − σ; the condition a + b ≥ −σ(2α + 1) causes no new restriction, and the assumptions a < (α + 1)/p′, b < (α + 1)/r, lead to the restriction of the (p, r)-range as given in [10].

Remark 2. The proof of Theorem 1 also gives the following result which makes precise the norm estimates indicated in the Introduction. Corollary 2. Set 2/q = 1/p+1/r. Then, under the hypotheses of Theorem 1, there holds

∥ ∑(k + 1)^{−σ}a_{k}L_{k}^{α}∥_{L_{w(α−br)}^{r}} ≲ ∥ ∑(k + 1)^{−σ}a_{k}L_{k}^{α+σ}∥_{L_{w(α+σ)}^{q}} ≲ ∥ ∑ a_{k}L_{k}^{α}∥_{L_{w(α+ap)}^{p}} .

Proof of Theorem 1. From [3, (3.30) and (3.31)] we have that if α > −1, σ > 0, then the following projection formulas for Laguerre polynomials hold

\frac{x^{α+σ}L_{k}^{α+σ}(x)}{Γ(k + α + σ + 1)} = \frac{1}{Γ(σ)} ∫_{0}^{x} \frac{(x − y)^{σ−1}y^{α}L_{k}^{α}(y) dy}{Γ(k + α + 1)} , (1)

e^{−x}L_{k}^{α}(x) = \frac{1}{Γ(σ)} ∫_{x}^{∞}(y − x)^{σ−1}e^{−y}L_{k}^{α+σ}(y) dy . (2)

On the one hand, we have by (1)

(∫_{0}^{∞} |∑_{k=0}^{∞} \frac{Γ(k + α + 1)}{Γ(k + α + σ + 1)}a_{k}L_{k}^{α+σ}(x)e^{−x/2}|^{q} x^{α+σ}dx)^{1/q}

≲ (∫_{0}^{∞} |∫_{0}^{x}(x − y)^{σ−1}e^{−(x−y)/2}( ∑ a_{k}L_{k}^{α}(y)y^{α}e^{−y/2}) dy|^{q} x^{−(α+σ)q/q′}dx)^{1/q} =: J_{1} .

Also, by (2),

(∫_{0}^{∞} |∑_{k=0}^{∞} b_{k}L_{k}^{α}(x)e^{−x/2}|^{r} x^{α−br}dx)^{1/r}

≲ (∫_{0}^{∞} |∫_{x}^{∞}(y − x)^{σ−1}e^{−(y−x)/2}( ∑ b_{k}L_{k}^{α+σ}(y)e^{−y/2}) dy|^{r} x^{α−br}dx)^{1/r} =: J_{2} .

Observing that e^{−|x−y|/2} ≤ 1, we see that the problem is reduced to finding weighted L^{p}-L^{q} estimates for the Riemann- Liouville fractional integral

I_{+}^{σ}h(t) = \frac{1}{Γ(σ)} ∫_{0}^{t}(t − s)^{σ−1}h(s) ds

and for the Weyl fractional integral

I_{−}^{σ}h(t) = \frac{1}{Γ(σ)} ∫_{t}^{∞}(s − t)^{σ−1}h(s) ds .

Convenient norm estimates can be found in the book of Samko, Kilbas and Marichev [12] from which we now quote Theorem 5.4. Theorem A. Let σ > 0, p ≥ 1, and p ≤ q ≤ p/(1 − pσ) when 1 ≤ p < 1/σ (in the case p = 1 the right endpoint p/(1 − pσ) is excluded), or p ≤ q < ∞ when p ≥ 1/σ . Suppose also that −∞ < N < ∞ and M < p−1 when we consider I_{+}^{σ} , or M > σp−1 in the case I_{−}^{σ} , and

\frac{N + 1}{q} = \frac{M + 1}{p} − σ . (3)

Then

( ∫_{0}^{∞} |I_{±}^{σ}h(t)|^{q}t^{N}dt)^{1/q} ≲ ( ∫_{0}^{∞} |h(t)|^{p} t^{M}dt)^{1/p}.

In order to estimate J_{1} choose the parameters N = −(α + σ)q/q′ and M = α(1 − p) + ap in Theorem A. Then the condition M < p − 1 is equivalent to a < (α + 1)/p′; in the case 1 < p < 1/σ the condition q ≤ p/(1 − pσ) is equivalent to σ ≥ 1/p − 1/q. Furthermore (3) leads to

\frac{α + σ + 1}{q} = \frac{α + 1}{p} + a . (4)

Thus, if p ≤ q, we have under the above conditions

∥ ∑(k + 1)^{−σ}a_{k}L_{k}^{α+σ}∥_{L_{w(α+σ)}^{q}} ≲ J_{1} ≲ ∥ ∑ a_{k}L_{k}^{α}∥_{L_{w(α+ap)}^{p}} . (5)

Similarly, we handle J_{2} by replacing p, q in Theorem A by q, r, respectively, and choosing N = α − br, M = α + σ . Then (3) leads to

\frac{α + 1}{r} − b = \frac{α + σ + 1}{q} − σ , (6)

which in combination with M > σq − 1 gives b < (α + 1)/r. In the case 1 < q < 1/σ the condition r ≤ q/(1 − qσ) is now equivalent to σ ≥ 1/q − 1/r. Thus, if q ≤ r, we have ∥ ∑ b_{k}L_{k}^{α}∥_{L_{w(α−br)}^{r}} ≲ J_{2} ≲ ∥ ∑ b_{k}L_{k}^{α+σ}∥_{L_{w(α+σ)}^{q}} . (7)

We can combine (5) and (7) if σ ≥ max{1/p − 1/q, 1/q − 1/r}; the maximum is assumed with 1/q = (1/p + 1/r)/2, thus, for this choice of q, we have p ≤ q ≤ r since by hypothesis p ≤ r. Finally, since (4) and (6) lead to the relation between r, p, α, σ, a, b, as asserted in Theorem 1, we arrive at the condition σ ≥ (1/p − 1/r)/2 = (σ − a − b)/(2α + 2) which implies a + b ≥ −σ(2α + 1). This finishes the proof of Theorem 1 and its Corollary.

Remark 3. In [1] and [2] there are admitted more general weights than those of power type as in Theorem A. We note that the conditions in [1] and [2] lead to more restrictions (e.g., the order of fractional integration is assumed to be ≤ 1). In the last years, a number of papers have been published which deal with the precise description of norm inequalities with two weights for fractional integrals or fractional maximal functions; the derived characterizations seem to be hard to apply in our situation.

3 Hermite expansions

We follow again [9] and consider for 1 ≤ p < ∞ the Lebesgue space

L_{w(H)}^{p} = {f : ∥f∥_{L_{w(H)}^{p}} = ( ∫_{−∞}^{∞} |f(x)e^{−x^{2}/2}|^{p}dx)^{1/p} < ∞} .

The Hermite polynomials (see [16, p. 106])

H_{n}(x) = (−1)^{n}e^{x^{2}}(\frac{d}{dx})^{n}e^{−x^{2}} , x ∈ R , n ∈ N_{0} ,

belong to L_{w(H)}^{p}; we associate to f ∈ L_{w(H)}^{p} its Hermite expansion

f(x) ∼ ∑_{k=0}^{∞} f̂_{H}(k)H_{k}(x) ,  f̂_{H}(k) = h_{k}^{H} ∫_{−∞}^{∞} f(t)H_{k}(t)e^{−t^{2}}dt ,

where h_{k}^{H} = ∥H_{k}∥_{L_{w(H)}^{2}}^{−2} = (\sqrt{π}2^{k}k!)^{−1}. Without loss of generality we can work with

polynomials f in the following, since the polynomials are dense in L_{w(H)}^{p}. As in the Laguerre series case we define a fractional integral operator I_{H}^{σ}, σ > 0, for Hermite expansions by (f = ∑ a_{k}H_{k} being a polynomial)

I_{H}^{σ}( ∑_{k=0}^{∞} a_{k}H_{k}(x)) = ∑_{k=0}^{∞}(k + 1)^{−σ}a_{k}H_{k}(x) .

This time we derive results on the mapping behavior of I_{H}^{σ} by reducing the problem to the corresponding one for Laguerre expansions. Observe that H_{2n} and H_{2n+1} are even and odd polynomials, respectively, and that one can uniquely decompose f ∈ L_{w(H)}^{p} into its even and odd parts:

f_{e}(x) = \frac{1}{2}(f(x) + f(−x)) , f_{o}(x) = \frac{1}{2}(f(x) − f(−x)) .

Then ∥f∥_{L_{w(H)}^{p}} ≤ ∥f_{e}∥_{L_{w(H)}^{p}} + ∥f_{o}∥_{L_{w(H)}^{p}} ≤ 2∥f∥_{L_{w(H)}^{p}} , (8)

and for their Hermite coefficients we obtain

(f_{e})_{ˆ H}(k) = \{ \begin{matrix}f̂_{H}(k) if & k is even \\ 0  if & k is odd \end{matrix} , (f_{o})_{ˆ H}(k) = \{ \begin{matrix}0  if & k is even \\ f̂_{H}(k) if & k is odd \end{matrix} .

Quadratic transformations [16, (5.6.1)] relate the Hermite and the Laguerre polynomials in the following way

H_{2k}(x) = (−1)^{k}2^{2k}k!L_{k}^{−1/2}(x^{2}) , H_{2k+1}(x) = (−1)^{k}2^{2k+1}k!xL_{k}^{1/2}(x^{2}) .

As a consequence we have

∑_{k=0}^{∞} a_{2k}H_{2k}(x) = ∑_{k=0}^{∞} b_{k}L_{k}^{−1/2}(x^{2}) , b_{k} = (−1)^{k}2^{2k}k!a_{2k} ,

∑_{k=0}^{∞} a_{2k+1}H_{2k+1}(x) = ∑_{k=0}^{∞} c_{k}x L_{k}^{1/2}(x^{2}) , c_{k} = (−1)^{k}2^{2k+1}k!a_{2k+1} .

Therefore,

∥(I_{H}^{σ}f)_{e}∥_{L_{w(H)}^{r}} ≲ ∥ ∑_{k=0}^{∞}(2k + 1)^{−σ}b_{k}L_{k}^{−1/2}∥_{L_{w(−1/2)}^{r}} ≲ ∥ ∑_{k=0}^{∞} b_{k}L_{k}^{−1/2}∥_{L_{w(−1/2)}^{p}}

≲ ∥f_{e}∥_{L_{w(H)}^{p}} , \frac{1}{r} = \frac{1}{p} − 2σ ,

by Theorem 1. Similarly,

∥(I_{H}^{σ}f)_{o}∥_{L_{w(H)}^{r}} ≲ ∥ ∑_{k=0}^{∞}(2k + 2)^{−σ}c_{k} L_{k}^{1/2}∥_{L_{w((r−1)/2)}^{r}}

≲ ∥ ∑_{k=0}^{∞} c_{k} L_{k}^{1/2}∥_{L_{w((p−1)/2)}^{p}} ≲ ∥f_{o}∥_{L_{w(H)}^{p}} , \frac{1}{r} = \frac{1}{p} − 2σ ,

by Theorem 1 with a = 1/2 − 1/p, b = 1/r − 1/2 and a + b = 1/r − 1/p ≥ −2σ by hypothesis. On account of (8) we can combine these two estimates to obtain the following result.

Theorem 3. Let 1 < p < r < ∞ and 0 < σ < 1/2. Then

∥ ∑_{k=0}^{∞}(k + 1)^{−σ}a_{k}H_{k}∥_{L_{w(H)}^{r}} ≲ ∥ ∑_{k=0}^{∞} a_{k}H_{k}∥_{L_{w(H)}^{p}} , \frac{1}{r} = \frac{1}{p} − 2σ .

4 Hankel transforms

Consider the weighted Lebesgue spaces L_{v(γ)}^{p} , 1 ≤ p < ∞, γ > −1, consisting of those measurable functions on (0, ∞) which satisfy

∥f∥_{L_{v(γ)}^{p}} = ( ∫_{0}^{∞} |f(t)|^{p}t^{2γ+1}dt)^{1/p} < ∞ .

For fixed α ≥ −1/2 the (modified) Hankel transform of f ∈ L_{v(α)}^{1} is defined by

H_{α}f(τ) = ∫_{0}^{∞} \frac{J_{α}(τt)}{(τt)^{α}}f(t) t^{2α+1}dt, τ > 0 .

Here J_{α} denotes the Bessel function of the first kind of order α (see [16, (1.71.1)]). We mention the connection with the multidimensional Fourier transform: If α = (n − 2)/2, n ∈ N, the Hankel transform H_{α}f(τ) coincides with the Fourier transform of the radial extension f(|x|) to R^{n} of f(t). One convenient substitute of the set of polynomials in the Laguerre case is now described by H_{α}(C_{0}^{∞}), the set of Hankel transforms of C^{∞}-functions with compact support in (0, ∞) , since by [15, Theorem 4.7] we have that H_{α}(C_{0}^{∞}) is dense in L_{v(α+γ)}^{p} for γ > −1/2, 1 < p < ∞.

We note that the Hankel transform is self inverse on H_{α}(C_{0}^{∞}), i.e.,

H_{α}[H_{α}f] = f , f ∈ H_{α}(C_{0}^{∞}) .

Motivated by the definition of the Riesz fractional integral on L^{p}(R^{n}) via its Fourier symbol, we define a fractional integral operator on H_{α}(C_{0}^{∞}) by

I^{σ}f = H_{α}(τ^{−σ}H_{α}(g)(τ)) .

To obtain a norm estimate for I^{σ} we proceed as in the Laguerre series case and work with the parameters of the Hankel transform. We slightly extend Theorems 3.1 and 3.2 of [15] and interpret them in a new way. In the “natural” weight case γ = α, we can at once conclude from [15, Theorem 3.1/2] that

∥I^{σ}f∥_{L_{v(α)}^{r}} ≲ ∥f∥_{L_{v(α)}^{p}} , \frac{1}{r} = \frac{1}{p} − \frac{σ}{2(α + 1)} , f ∈ H_{α}(C_{0}^{∞}) .

For α = (n − 2)/2 this reduces to the well-known result for Riesz-potentials restricted to the radial functions of L^{p}(R^{n}). More generally there holds the following.

Theorem 4. Let α ≥ −1/2 and σ > 0, 1 < p ≤ q ≤ r < ∞, 2/q = 1/p + 1/r. Assume further that a < 2(α + 1)/p′, b < 2(α + 1)/r and a + b ≥ −σ(2α + 1). Then

∥I^{σ}f∥_{L_{v(α−br/2)}^{r}} ≲ ∥H_{α+σ/2}(H_{α}(I^{σ}f))∥_{L_{v(α+σ/2)}^{q}} ≲ ∥f∥_{L_{(α+ap/2)}^{p}} , \frac{1}{r} = \frac{1}{p} − \frac{σ − a − b}{2(α + 1)} .

For the proof of Theorem 4 we observe that the relevant projection formulas for the Bessel functions are given by [5, 8.5(32) and 8.5(33)]. Formula [5, 8.5(32)] leads to [15, (3.9)] which, when taking g = H_{α+σ}(H_{α}(I^{2σ}f)), may be rewritten in the form

I^{2σ}f(t) = c_{α,σ} ∫_{t}^{∞} s(s^{2} − t^{2})^{σ−1}H_{α+σ}(H_{α}(I^{2σ}f))(s) ds , f ∈ H_{α}(C_{0}^{∞}),

provided α ≥ −1/2, σ > 0. Also for these parameters, formula [5, 8.5(33)] yields (all double integrals are absolutely convergent – see [15, p. 57])

H_{α+σ}(H_{α}(I^{2σ}f))(t) = \frac{c_{α,σ}^{∗}}{t^{2(α+σ)}} ∫_{0}^{t}(t^{2} − s^{2})^{σ−1}f(s)s^{2α+1} ds , σ > 0, f ∈ H_{α}(C_{0}^{∞}) .

Variable changes lead to

∥H_{α+σ}(H_{α}(I^{2σ}f))∥_{L_{v(α+σ)}^{q}} ≲ (∫_{0}^{∞} | ∫_{0}^{t}(t − s)^{σ−1}f(s^{\frac{1}{2}} ) s^{α}ds|^{q}t^{−(α+σ)q/q′}dt)^{1/q} =: I_{1} ,

∥I^{2σ}f∥_{L_{v(α−br/2)}^{r}}

≲ (∫_{0}^{∞} | ∫_{t}^{∞}(s − t)^{σ−1}H_{α+σ}(H_{α}(I^{2σ}f))(s^{\frac{1}{2}} ) ds|^{r}t^{−br/2+α}dt)^{1/r} =: I_{2} .

We are now in a situation to apply Theorem A and follow the pattern of the proof to Corollary 2. Concerning I_{1} choose N = −(α + σ)q/q′, M = (1 − p)α + ap/2; the restrictions σ ≥ 1/p − 1/q, a/2 < (α + 1)/p′ and (α + σ + 1)/q = (α + 1)/p + a/2 then result from Theorem A. Analogously, when dealing with I_{2}, we obtain the restrictions σ ≥ 1/q − 1/r, b/2 < (α + 1)/r and (α + σ + 1)/q = (α + 1)/r + σ − b/2. Taking again the maximum over the lower bounds of σ leads to a + b ≥ −2σ(2α + 1). Finally, since in the proof we worked with I^{2σ} instead of I^{σ}, we only have to replace σ by σ/2 to establish Theorem 4.

Remark 4. Let us apply Theorem 4 to fractional integration of order σ > 0 with respect to the classical Hankel transform given by

I^{σ}f = H_{α}(τ^{−σ}H_{α}(f)(τ)) , H_{α}f(τ) = ∫_{0}^{∞}(sτ)^{\frac{1}{2}} J_{α}(sτ)f(s) ds .

Here we may restrict ourselves to elements of the (self-explaining) set H_{α}(C_{0}^{∞}) which is dense in L^{p}(0, ∞), 1 < p < ∞ (see [15, Cor. 4.8]). If we choose in Theorem 4 the parameters a = (2α + 1)(1/2 − 1/p), b = (2α + 1)(1/r − 1/2), then we arrive at the following result (∥ · ∥_{p} denotes the standard L^{p}(0, ∞)-norm ).

Corollary 5. Let α ≥ −1/2 and 1 < p < r < ∞. Then, for all f ∈ L^{p}(0, ∞) there holds

∥I^{σ}f∥_{r} ≲ ∥f∥_{p} , \frac{1}{r} = \frac{1}{p} − σ .

An alternative proof of Corollary 5 in the spirit of Kanjin and Sato would be to prove the assertion for the cosine transform (thus for α = −1/2 which is classical) and then use Guy’s transplantation theorem for the Hankel transform (cf. [15, p. 53]).

Remark 5. Obviously, Theorem 4 and its proof are quite analogous to the Laguerre expansion case. That this is not accidental may be seen by a result due to Stempak [13, Theorem 1.1]. Though Stempak’s theorem is only formulated for the case r = p its proof actually shows the following.

Theorem B. Let 1 ≤ p ≤ r < ∞ and α > −1. Assume that m is a function on R_{+} , which is continuous except on a set of Lebesgue measure zero. Then

∥m∥_{M^{p,r}(H_{α})} ≲ lim inf_{ε→0+} ∥ε^{σ}\{m(εk^{1/2})\}_{k∈N_{0}}∥_{M_{w(α)}^{p,r}} , \frac{1}{r} = \frac{1}{p} − \frac{σ}{2(α + 1)} , (9)

whenever the right hand side is finite. Here the (modified ) Hankel multiplier norm is given by

∥m∥_{M^{p,r}(H_{α})} = inf{C : ∥H_{α}(mH_{α}f)∥_{L_{v(α)}^{r}} ≤ C ∥f∥_{L_{v(α)}^{p}} , f ∈ H_{α}(C_{0}^{∞})} .

Observe that the relation 1/r = 1/p − σ/2(α + 1) reflects the mapping behavior of a fractional integral operator of order σ/2 in the Laguerre setting. In order to apply Theorem B we face the problem to ensure that the right hand side of (9) is bounded. Here the case 1 < p < r < 2 is the essential one. For, by duality, it is equivalent to the problem for 2 < p < r < ∞; if 1 < p ≤ 2 ≤ q < ∞ we observe that on account of M_{w(α)}^{2,2} = ℓ^{∞} we have (see [6, p. 69]) that

∥ε^{σ}\{m(εk^{1/2})\}_{k∈N_{0}}∥_{M_{w(α)}^{p,r}} ≲ ∥\{ε^{σ}k^{σ/2}m(εk^{1/2})\}_{k∈N_{0}}∥_{ℓ^{∞}} , \frac{1}{r} = \frac{1}{p} − \frac{σ}{2(α + 1)} ,

which is an easy to verify condition.

Now, in the case 1 < p < r < 2, the extension of [6, Cor. 1.2] from α ≥ 0 to α ≥ −1/2 is obvious by Theorem 1 for a = b = 0; note that Corollaries 1.2 and 4.5 in [14] give sufficient M_{w(α)}^{p,p}-multiplier criteria for all α > −1 and that the assumption a + b = 0 ≥ −σ(2α + 1) leads to the restriction α ≥ −1/2. Thus, using the notation s = [α + 1] + 2, Δm_{k} = m_{k} − m_{k+1}, Δ^{s} = Δ Δ^{s−1}, and choosing m(τ) = (1 + τ^{2})^{−σ/2} in Theorem B we can estimate as follows:

sup_{ε>0} ∥ε^{σ}{m(εk^{1/2})}_{k∈N_{0}}∥_{M_{w(α)}^{p,r}}^{2}

≲ sup_{ε>0} (∥ε^{σ}k^{σ/2}{m(εk^{1/2})}_{k∈N_{0}}∥_{∞}^{2}

+ sup_{N∈N} ∑_{k=N}^{2N} |(k + 1)^{s+σ/2}Δ^{s}ε^{σ}m(εk^{1/2})|^{2}(k + 1)^{−1} )

≲ sup_{ε>0} (O(1) + sup_{N>0} ∫_{N}^{2N} |τ^{s+σ/2}(\frac{d}{dτ} )^{s} \frac{ε^{σ}}{(1 + ε^{2}t)^{σ/2}} |^{2}dt) ,

the latter inequality being valid on account of a slight variation of [7, (5.1)]. A straightforward computation at once yields that the last integral is uniformly bounded in ε > 0. Therefore, by Theorem B, the mapping behavior of the fractional integration, generated by the (modified) Hankel symbol (1 + τ)^{−σ}, can be deduced from the corresponding one for fractional integration of order σ/2 in the Laguerre series setting, i.e., from Theorem 1 in the case a = b = 0.

Of course, when 1 < p < r < ∞, α ≥ −1/2,

∥H_{α}((1 + τ)^{−σ}H_{α}(g)(τ))∥_{L_{v(α)}^{r}} ≲ ∥f∥_{L_{v(α)}^{p}} , \frac{1}{r} = \frac{1}{p} − \frac{σ}{2(α + 1)} ,

also follows from Theorem 4, since τ^{σ}/(1 + τ)^{σ} generates a bounded operator on all L_{v(α)}^{p}-spaces, 1 ≤ p ≤ ∞.

References

[1] K. F. Andersen and H. P. Heinig, Weighted norm inequalities for certain integral operators, SIAM J. Math. Anal. 14 (1983), 834 – 844.

[2] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), 547 – 558.

[3] R. Askey and J. Fitch, Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl. 26 (1969), 411 – 437.

[4] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.

[5] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954.

[6] G. Gasper, K. Stempak and W. Trebels, Fractional integration for Laguerre expansions, Methods Appl. Anal. 2 (1995), 67 – 75.

[7] G. Gasper and W. Trebels, A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers, Studia Math. 65 (1979), 243 – 278.

[8] G. Gasper and W. Trebels, Necessary conditions for Hankel multipliers, Indiana Univ. Math. J. 31 (1982), 403 – 414.

[9] G. Gasper and W. Trebels, Applications of weighted Laguerre transplantation theorems, Methods Appl. Anal. 6 (1999), 337 – 346.

[10] Y. Kanjin and E. Sato, The Hardy-Littlewood theorem on fractional integration for Laguerre series, Proc. Amer. Math. Soc. 123 (1995), 2165 – 2171.

[11] E. L. Poiani, Mean Cesàro summability of Laguerre and Hermite series, Trans. Amer. Math. Soc. 173 (1972), 1 – 31.

[12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Sci. Publ., Langhorne, PA, 1993.

[13] K. Stempak, On connections between Hankel, Laguerre and Heisenberg multipliers, J. London Math. Soc. (2) 51 (1995), 286 – 298.

[14] K. Stempak and W. Trebels, On weighted transplantation and multipliers for Laguerre expansions, Math. Ann. 300 (1994), 203 – 219.

[15] K. Stempak and W. Trebels, Hankel multipliers and translation operators, Studia Math. 126 (1997), 51 –66.

[16] G. Szegö, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I., 1975.

[17] W. Trebels, Multipliers for (C, α)-Bounded Fourier Expansions in Banach Spaces and Approximation Theory, Lecture Notes in Math., Vol. 329, Springer, Berlin, 1973.

George Gasper

Department of Mathematics Northwestern University Evanston, IL 60208 USA

E-mail address: g-gasper@nwu.edu Walter Trebels

AG 5, Fachbereich Mathematik Technische Universität Darmstadt D – 64289 Darmstadt Germany

E-mail address: trebels@ mathematik.tu-darmstadt.de