Fractional integration for Laguerre expansions

George Gasper ^{1}, Krzysztof Stempak ^{2} and Walter Trebels ^{3}

(To appear in Methods and Applications of Analysis)

Key words. Fractional integration, multiplier conditions, Laguerre polynomials

AMS(MOS) subject classifications. 33C45, 42A45, 42C10

1 Introduction

The aim of this note is to provide a fractional integration theorem in the framework of Laguerre expansions. The method of proof consists of establishing an asymptotic estimate for the involved kernel and then applying a method of Hedberg [5]. We combine this result with sufficient (p, p) multiplier criteria of Stempak and Trebels [10]. The resulting sufficient (p, q) multiplier criteria are comparable with necessary ones of Gasper and Trebels [3].

Our notation is essentially that in [10]. Thus we consider the Lebesgue spaces

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

and define the Laguerre function system {l_{k}^{α}} by

l_{k}^{α}(x) = (k!/Γ(k + α + 1))^{1/2}e^{−x/2}L_{k}^{α}(x), α > −1, k ∈ N_{0}.

This system is an orthonormal basis in L^{2}(R_{+}, x^{α}dx) and for γ < p(α + 1) − 1 we can associate to any f ∈ L_{v(γ)}^{p} the Laguerre series

f(x) ∼ ∑_{k=0}^{∞} a_{k}l_{k}^{α}(x), a_{k} = ∫_{0}^{∞} f(x)l_{k}^{α}(x)x^{α}dx.

It is convenient to introduce the vector space

E = {f(x) = p(x)e^{−x/2} : 0 ≤ x < ∞, p(x) a polynomial}

1Dept. of Mathematics, Northwestern University, Evanston, IL 60208, USA. Supported in part by NSF Grant DMS–9103177.

2Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50 –384 Wrocław, Poland. Research of this author was done during a stay at the Fb Mathematik of the TH Darmstadt and was supported by the Deutsche Forschungsgemeinschaft under grant 436POL/115/1/0.

3Fb. Mathematik, TH Darmstadt, Schloßgartenstr. 7, D–64289 Darmstadt, Germany.

which is dense in L_{v(γ)}^{p}. We note that f ∈ E has only finitely many non-zero Fourier-Laguerre coefficients. Analogous to the definition of the Hardy and Littlewood fractional integral operator for Fourier series (see [12, Chap. XII, Sec. 8], we define a fractional integral operator I_{σ}, σ > 0, for Laguerre expansions by

I_{σ}f(x) = ∑_{k=0}^{∞}(k + 1)^{−σ}a_{k}l_{k}^{α}(x) , f ∈ E.

Observing that the l_{k}^{α} are eigenfunctions with eigenvalues λ_{k} of the differential operator

L = −(x\frac{d^{2}}{dx^{2}} + (α + 1)\frac{d}{dx} − \frac{x}{4}) , λ_{k} = k + (α + 1)/2,

(see [11, (5.1.2)]) one realizes that I_{1} is an integral operator essentially inverse to L (see the following Remark 2). As we will see in Section 2 the fractional integral I_{σ}f can be interpreted as a twisted convolution of f with a function

g_{σ}(x) ∼ Γ(α + 1) ∑_{k=0}^{∞}(k + 1)^{−σ}L_{k}^{α}(x)e^{−x/2} .  (1)

From Theorem 3.1 in [3, II] it easily follows that g_{σ} ∈ L_{v(γ)}^{1} when α − γ < σ, so that, by the convolution theorem of Görlich and Markett [4], I_{σ} extends to a bounded operator from L_{v(γ)}^{p} to L_{v(γ)}^{p} when 0 ≤ αp/2 ≤ γ ≤ α, 1 ≤ p ≤ ∞. Our main result is

Theorem 1.1 Let α ≥ 0, 1 < p ≤ q < ∞. Assume further that 0 < σ < α + 1, a < (α + 1)/p′, b < (α + 1)/q, a + b ≥ 0. Then

||I_{σ}f||_{L_{v(α−bq)}^{q}} ≤ C||f||_{L_{v(α+ap)}^{p}}, \frac{1}{q} = \frac{1}{p} − \frac{σ − a − b}{α + 1}.

Remarks. 1) Observe that the upper bounds for a and b imply lower bounds for b and a, respectively.

2) By the above it is clear that the sequence {(k+1)^{−σ}} which generates the fractional integral can be replaced by any sequence {Ω_{σ}(k)} satisfying

Ω_{σ}(k) = ∑_{j=0}^{J} c_{j} (k + 1)^{−σ−j} + O((k + 1)^{−σ−J})

for sufficiently large J, say J ≥ α + 2, thus in particular obtaining the same result for the sequence {Γ(k + 1)/Γ(σ + k + 1)} – see [7].

3) A weaker version of Theorem 1.1 (e.g. in the case a = b = 0) can easily be deduced by the following argument. By a slight modification of the proof of Theorem 3.1 in [3, II] we have:

Let α > −1 and N ∈ N_{0}, N > (2α + 2)(1/r − 1/2) − 1/3. If {f_{k}} is a bounded sequence with lim_{k→∞} f_{k} = 0 and

∑_{k=0}^{∞}(k + 1)^{N+(α+1)/r′}|Δ^{N+1}f_{k}| ≤ K_{r}(f) < ∞, 1 ≤ r < ∞,

then there exists a function f ∈ L_{v(α)}^{r} with

∥f∥_{L_{v(α)}^{r}} ≤ CK_{r}(f), f(x) ∼ ∑_{k=0}^{∞} f_{k}L_{k}^{α}(x)e^{−x/2}.

This applied to the sequence {(k + 1)^{−σ}} gives, by Young’s inequality (see [4]),

||I_{σ}f||_{L_{v(α)}^{q}} ≤ C||f||_{L_{v(α)}^{p}}, \frac{1}{q} > \frac{1}{p} − \frac{σ}{α + 1},

where α ≥ 0, σ > 0 and 1 ≤ p, q ≤ ∞.

Next we indicate how Theorem 1.1 can be used to gain some insight into the structure of M^{p,q}–Laguerre multipliers. For the sake of simplicity let us restrict ourselves to the case γ = α. Consider a sequence m = {m_{k}} of numbers and associate to m the operator

T_{m}f(x) = ∑_{k=0}^{∞} m_{k}a_{k}l_{k}^{α}(x) , f ∈ E.

The sequence m is called a bounded (p, q)-multiplier, notation m ∈ M_{α,α}^{p,q}, if

|| m ||_{M_{α,α}^{p,q}} := inf{C : ||T_{m}f||_{L_{v(α)}^{q}} ≤ C∥f∥_{L_{v(α)}^{p}} for all f ∈ E}

is finite. If p ≤ 2 ≤ q, then sufficient conditions follow at once in the following way: observe that M_{α,α}^{2,2} = l^{∞}, choose σ_{0}, σ_{1} ≥ 0 such that I_{σ_{0}} : L^{p} → L^{2}, I_{σ_{1}} : L^{2} → L^{q}, σ_{0} + σ_{1} = σ, and hence

|| m ||_{M_{α,α}^{p,q}} ≤ ∥k^{−σ_{0}}∥_{M_{α,α}^{p,2}}||{k^{σ}m_{k}}||_{M_{α,α}^{2,2}}∥k^{−σ_{1}}∥_{M_{α,α}^{2,q}};

in particular, m ∈ M_{α,α}^{p,q} when {k^{σ}m_{k}} ∈ l^{∞}. Thus when 1 < p ≤ 2 ≤ q < ∞, and σ is as in Theorem 1.1, {(−1)^{k}(k + 1)^{−σ}} ∈ M_{α,α}^{p,q} generates a bounded operator which does not fall under the scope of the above introduced fractional integral operators. To formulate a Corollary based on a combination of Theorem 1.1 and the multiplier result in [10] we need the notion of a difference operator Δ^{s} of fractional order s given by

Δ^{s}m_{k} = ∑_{j=0}^{∞} A_{j}^{−s−1}m_{k+j} , A_{j}^{t} = \frac{Γ(j + t + 1)}{j! Γ(t + 1)}, t ∈ R,

whenever the sum converges. In view of the remark concerning the case p < 2 < q and on account of duality we may restrict ourselves to the case 1 < p < q < 2.

Corollary 1.2 If α ≥ 0, 1 < p < q < 2, and s > max{(2α + 2)(1/q − 1/2), 1} then, for some constant C independent of the sequence {m_{k}}, there holds

∥{m_{k}}∥_{M_{α,α}^{p,q}}^{2} ≤ C (∥{k^{σ}m_{k}}∥_{∞}^{2} + sup_{N} ∑_{k=N}^{2N} |(k + 1)^{s+σ}Δ^{s}m_{k}|^{2}(k + 1)^{−1}) .

The proof follows as in [2]. The result itself should be compared with the corresponding necessary condition in [3, I] which also shows that {(−1)^{k}(k + 1)^{−σ}} ∉ M_{α,α}^{p,q} provided 1 < p < q < 2 and α is sufficiently large.

The plan of the paper is as follows. In Section 2 we derive an asymptotic estimate of the function g_{σ} defined above by (1). Then the twisted generalized convolution is used to dominate I_{σ}f by a generalized Euclidean convolution of g_{σ} with f. The latter’s mapping behavior is discussed by a method of Hedberg [5] which uses maximal functions; thus giving Theorem 1.1 in the standard weight case a = b = 0. In Section 3 we extend this result to some power weights, modifying an argument in Stein and Weiss [8].

2 Proof of the standard weight case.

We start by deriving the required asymptotic estimate and showing

Lemma 2.1 Let α > −1. Then, for x > 0 and 0 < σ < α + 1, there holds

|g_{σ}(x)| ≤ Cx^{σ−α−1}.  (2)

Proof. First note that by subordination (see [3, I], p. 1234) there holds for N > α + 2, N ∈ N,

g_{σ}(x) = C ∑_{k=0}^{∞} (Δ^{N}(k + 1)^{−σ}) L_{k}^{α+N}(x)e^{−x/2}.

Then the assertion (2) follows after it is proved that

∑_{k=0}^{∞}(k + 1)^{−σ−N} |x^{α+1−σ}L_{k}^{α+N}(x)e^{−x/2}| ≤ C  (3)

uniformly in x > 0. With the notation

L_{k}^{α}(x) = (k!/Γ(k + α + 1))^{1/2}x^{α/2}e^{−x/2}L_{k}^{α}(x), k ∈ N_{0}.

this is equivalent to

∑_{k=0}^{∞}(k + 1)^{(α−N)/2−σ} |L_{k}^{α+N}(x)| ≤ Cx^{σ+(N−α−2)/2}  (4)

uniformly in 0 < x < ∞. We will make use of the pointwise estimates for the Laguerre functions in [1, Sec. 2]:

|L_{k}^{α+N}(x)| ≤ C \{ \begin{matrix}(x(k + 1))^{(α+N)/2} if & 0 ≤ x ≤ c/(k + 1), \\ (x(k + 1))^{−1/4}  if & c/(k + 1) ≤ x ≤ d(k + 1) \end{matrix}  (5)

for fixed positive constants c and d. These and two further estimates in [6, (2.5)] imply that

sup_{x>0} |L_{k}^{α+N}(x)| ≤ C, k ∈ N_{0},  (6)

and so it is obvious that (6) implies (4) for x ≥ 1.

Therefore, decomposing dyadically the interval (0, 1) it suffices to check that

∑_{k=0}^{∞}(k + 1)^{(α−N)/2−σ} |L_{k}^{α+N}(x)| ≤ C(2^{j})^{σ+(N−α−2)/2}

provided 2^{j} ≤ x ≤ 2^{j+1}, j < 0. Using the first line of (5) we get

∑_{k=0}^{2^{−j}}(k + 1)^{(α−N)/2−σ} |L_{k}^{α+N}(x)| ≤ C(2^{j})^{(α+N)/2} ∑_{k=0}^{2^{−j}}(k + 1)^{α−σ} ≤ C(2^{j})^{σ+(N−α−2)/2},

while the second line of (5) gives

∑_{k=2^{−j}}^{∞} k^{(α−N)/2−σ} |L_{k}^{α+N}(x)| ≤ C2^{−j/4} ∑_{k=2^{−j}}^{∞} k^{(α−N)/2−σ−1/4} ≤ C(2^{j})^{σ+(N−α−2)/2}.

This completes the proof of Lemma 2.1.

As already mentioned we will apply Hedberg’s method which involves maximal functions. To make use of the corresponding results in [9] we switch to the system

ψ_{k}^{α}(x) = (2k!/Γ(k + α + 1))^{1/2}e^{−x^{2}/2}L_{k}^{α}(x^{2}), k ∈ N_{0},

which is obviously orthonormal on L^{2}(R_{+}, dμ_{α}), dμ_{α}(x) = x^{2α+1}dx, α > −1. For the sake of simplicity we write the norm on L^{p}(R_{+}, dμ_{α}) as

∥F ∥_{p} = (∫_{0}^{∞} |F (x)|^{p}dμ_{α}(x))^{1/p} .

We adopt the notion of the twisted generalized convolution on L^{1}(R_{+}, dμ_{α}) from [9]

F × G(x) = ∫_{0}^{∞} τ_{x}F (y)G(y) dμ_{α}(y),

where the twisted generalized translation operator τ_{x} is given by

τ_{x}F (y) = \frac{Γ(α + 1)}{π^{1/2}Γ(α + 1/2)} ∫_{0}^{π} F ((x, y)_{θ})J_{α−1/2}(xy sin θ)(sin θ)^{2α} dθ,

J_{β}(x) = Γ(β + 1)J_{β}(x)/(x/2)^{β}, J_{β} denoting the Bessel function of order β, and

(x, y)_{θ} = (x^{2} + y^{2} − 2xy cos θ)^{1/2}.

With respect to the system {ψ_{k}^{α}} this convolution has the following transform property: if F ∼ ∑ c_{k}ψ_{k}^{α} and F × G ∼ ∑ c_{k}d_{k}ψ_{k}^{α}, then G(x) ∼ Γ(α + 1) ∑ d_{k}L_{k}^{α}(x^{2})e^{−x^{2}/2}.

If we set f(y^{2}) = F (y), g_{σ}(y^{2}) = G_{σ}(y), we see that

|I_{σ}f(x^{2})| = |F × G_{σ}(x)|.

Apart from Lemma 2.1 the proof of Theorem 1.1 will be based on the fact that for α ≥ 0 and suitable F and G

|F × G| ≤ |F | ∗ |G| ,

which follows at once from the definition of the generalized Euclidean ∗-convolution

F ∗ G(x) = ∫_{0}^{∞} τ_{x}^{E}F (y)G(y) dμ_{α}(y)

with associated generalized Euclidean translation

τ_{x}^{E}F (y) = ∫_{0}^{π} F ((x, y)_{θ}) dν_{α}(θ), dν_{α}(θ) = \frac{Γ(α + 1)}{π^{1/2}Γ(α + 1/2)}(sin θ)^{2α}dθ .

Therefore we restrict ourselves to fractional integrals defined via the generalized Euclidean convolution.

Theorem 2.2 Let 1 < p < q < ∞, α > −1/2, K_{σ}(x) = x^{2(σ−α−1)}. Then

∥F ∗ K_{σ}∥_{q} ≤ C∥F ∥_{p} , \frac{1}{q} = \frac{1}{p} − \frac{σ}{α + 1} .

By the above it is clear that Theorem 1.1 for the case a = b = 0 follows from Theorem 2.2.

Proof. Following Hedberg [5] we want to estimate F ∗ K_{σ}(x) pointwise by a suitable maximal function which in this setting turns out to be (see Stempak [9, p. 138])

F^{∗}(x) = sup_{ε>0} ε^{−(2α+2)} ∫_{0}^{ε} τ_{x}^{E}(|F |)(y)dμ_{α}(y)

with the usual boundedness property ||F^{∗}||_{r} ≤ C||F ||_{r}, 1 < r ≤ ∞. Now there holds

|F ∗ K_{σ}(x)| ≤ C (∫_{0}^{δ} + ∫_{δ}^{∞}) τ_{x}^{E}(|F |)(y)y^{2(σ−α−1)}dμ_{α}(y) = J_{1} + J_{2},

where δ > 0 will be chosen later appropriately. Clearly,

J_{1} = ∑_{k=0}^{∞} ∫_{2^{−k−1}δ}^{2^{−k}δ} … ≤ C ∑_{k=0}^{∞}(2^{−k}δ)^{2σ}(2^{−k}δ)^{−2α−2} ∫_{2^{−k−1}δ}^{2^{−k}δ} τ_{x}^{E}(|F |)(s)s^{2α+1}ds

≤ Cδ^{2σ} ∑_{k=0}^{∞} 2^{−k2σ}(2^{−k}δ)^{−2α−2} ∫_{0}^{δ2^{−k}} τ_{x}^{E}(|F |)(s)s^{2α+1}ds ≤ Cδ^{2σ}F^{∗}(x).

On the other hand, by Hölder’s inequality,

J_{2} = ∫_{δ}^{∞} τ_{x}^{E}(|F |)(s)s^{2(σ−α−1)}s^{2α+1}ds ≤ ||τ_{x}^{E}(|F |)||_{p} (∫_{δ}^{∞} s^{2(σ−α−1)p′}s^{2α+1}ds)^{1/p′}

≤ Cδ^{2σ−(2α+2)/p}||F ||_{p} ,

since τ_{x}^{E} are contractions on L^{p}(R_{+}, dμ_{α}). Hence

|F ∗ K_{σ}(x)| ≤ C (δ^{2σ}F^{∗}(x) + δ^{2σ−(2α+2)/p}||F ||_{p} ) .

Choosing δ = (F^{∗}(x)/||F ||_{p})^{−p/(2α+2)} (where ||F ||_{p} ≠ 0 is assumed) we obtain

|F ∗ K_{σ}(x)| ≤ CF^{∗}(x)^{1−σp/(α+1)}∥F ∥_{p}^{σp/(α+1)}

and then ∥F ∗ K_{σ}∥_{q} ≤ C||F ||_{p} due to the inequality for the maximal function with r = q (1 − σp/(α + 1)) = p > 1. This finishes the proof of Theorem 2.2.

3 Extension to power weights

The proof of Theorem 1.1 in the general case follows along the lines in Section 2 from

Theorem 3.1 Let α > −1/2, 0 < σ < α + 1 and a < (α + 1)/p′, b < (α + 1)/q, with a + b ≥ 0. If 1 < p ≤ q < ∞, then

∥K_{σ} ∗ F (x) x^{−2b}∥_{q} ≤ C∥F (x) x^{2a}∥_{p}, \frac{1}{q} = \frac{1}{p} − \frac{σ − a − b}{α + 1}.

Proof. An equivalent version of the inequality above is

(∫_{0}^{∞} |Sf(x)|^{q}dμ_{α}(x))^{1/q} ≤ C (∫_{0}^{∞} |f(x)|^{p}dμ_{α}(x))^{1/p}

where f(x) = x^{2a}F (x), Sf(x) = ∫_{0}^{∞} K(x, y)f(y)dμ_{α}(y) and

K(x, y) = x^{−2b} (∫_{0}^{π}(x, y)_{θ}^{2(σ−α−1)}dν_{α}(θ)) y^{−2a}.

We first consider the case p = q. Then σ = a + b and, therefore, the kernel K(x, y) is homogeneous of degree −(2α+2): K(rx, ry) = r^{−(2α+2)}K(x, y), r > 0. It now suffices to check that (cf. Section 2 of [10])

∫_{0}^{∞} K(1, y)y^{−(2α+2)/p}dμ_{α}(y) < ∞.

We first note that the function K(1, y) has at most an integrable singularity at y = 1, since for 1/2 ≤ y ≤ 2 we have

∫_{0}^{π} \frac{(sin θ)^{2α}dθ}{((1 − y)^{2} + 4y sin^{2}(θ/2))^{α+1−σ}} ≤ C(1 + |1 − y|^{σ−1}) .

To deal with the singularity at 0 we note that for y < 1/2 we have (1, y)_{θ} ≈ 1 and therefore ∫_{0}^{π}(1, y)_{θ}^{2(σ−α−1)}dν_{α}(θ) ≈ 1 as well. Hence K(1, y) ≤ Cy^{−2a} and thus

∫_{0}^{1/2} K(1, y)y^{−(2α+2)/p}dμ_{α}(y) < ∞, a < (α + 1)/p′.

For y > 2 we have (1, y)_{θ} ≈ y, ∫_{0}^{π}(1, y)_{θ}^{2(σ−α−1)}dν_{α}(θ) ≈ y^{2(σ−α−1)} and hence

∫_{2}^{∞} K(1, y)y^{−(2α+2)/p}dμ_{α}(y) < ∞, b < (α + 1)/q,

since σ < (α + 1)/p.

Now, consider the case 1 < p < q < ∞. We use still another equivalent version of the inequality to be proved, namely

∫_{0}^{∞} ∫_{0}^{∞} K(x, y)f(y)g(x) dμ_{α}(y) dμ_{α}(x) ≤ C||f||_{p}||g||_{q′} ,

assuming for simplicity that f and g are nonnegative. Writing R_{+}×R_{+} = D_{1}∪D_{2}∪D_{3} where

D_{1} = {y/2 ≤ x ≤ 2y}, D_{2} = {2y < x}, D_{3} = {x < y/2}

it suffices to check for i = 1, 2, 3 that

I_{i} = ∫∫_{D_{i}} K(x, y)f(y)g(x) dμ_{α}(y) dμ_{α}(x) ≤ C∥f∥_{p}∥g∥_{q′} .

Consider I_{1} first. Since a + b ≥ 0, for x, y ∈ D_{1} and θ ∈ (0, π) we have

((x, y)_{θ})^{2(a+b)} ≤ Cx^{2(a+b)} ≤ Cx^{2b}y^{2a};

so

I_{1} ≤ ∫∫_{D_{1}} K̃(x, y)f(y)g(x) dμ_{α}(y) dμ_{α}(x)

with K̃(x, y) = τ_{x}^{E}K_{σ−a−b}(y) . Hence we are reduced to showing that

∫_{0}^{∞} K_{σ−a−b} ∗ f(x) g(x) dμ_{α}(x) ≤ ||f||_{p}||g||_{q′},

which is implied by Theorem 2.2.

To estimate I_{2} and I_{3} we need the following lemma.

Lemma 3.2 Let α > −1 and V_{δ}f(x) = x^{2(δ−α−1)} ∫_{0}^{x} f(y)y^{−2δ}dμ_{α}(y), δ < (α + 1)/p′. Then

||V_{δ}f||_{p} ≤ C||f||_{p} , |V_{δ}f(x)| ≤ C x^{−(2α+2)/p}||f||_{p} .

Proof. We have that V_{δ}f(x) = ∫_{0}^{∞} L(x, y)f(y)dμ_{α}(y), where L(x, y) = x^{2(δ−α−1)}y^{−2δ} for y < x and equals 0 otherwise. Clearly L is homogeneous of degree −(2α + 2). Therefore the norm inequality is implied by

∫_{0}^{∞} L(1, y)y^{−(2α+2)/p}dμ_{α}(y) = ∫_{0}^{1} y^{−2δ−(2α+2)/p+2α+1} dy < ∞

which is finite since δ < (α + 1)/p′. For the pointwise estimate we simply write

|V_{δ}f(x)| ≤ x^{2(δ−α−1)} ∫_{0}^{x} |f(y)| y^{−2δ}dμ_{α}(y)

≤ Cx^{2(δ−α−1)} (∫_{0}^{x} y^{−2δp′}dμ_{α}(y))^{1/p′} ||f||_{p} ≤ Cx^{−(2α+2)/p}||f||_{p}

which finishes the proof of the lemma.

Estimating I_{2}, we note that for x, y ∈ D_{2} and all θ, 0 < θ < π, there holds x < 2(x − y) ≤ 2((x − y)^{2} + 2xy(1 − cos θ))^{1/2} = 2(x, y)_{θ}; hence

I_{2} ≤ C ∫_{0}^{∞} x^{2(σ−α−1−b)}g(x) (∫_{0}^{x} f(y) y^{−2a}dμ_{α}(y)) dμ_{α}(x)

= C ∫_{0}^{∞} g(x)x^{2(σ−a−b)}V_{a}f(x) dμ_{α}(x) ≤ C||g||_{q′}||x^{2(σ−a−b)}V_{a}f||_{q} .

It now suffices to estimate the latter L^{q} norm by ||f||_{p}. We have

∫_{0}^{∞} |x^{2(σ−a−b)}V_{a}f(x)|^{q}dμ_{α}(x) = ∫_{0}^{∞} |V_{a}f(x)|^{p}|V_{a}f(x)|^{q−p}x^{2(σ−a−b)q}dμ_{α}(x) .

Since a < (α + 1)/p′, by the pointwise estimate in the above lemma

|V_{a}f(x)| ≤ Cx^{−(2α+2)/p}||f||_{p};

hence

|V_{a}f(x)|^{q−p}x^{2(σ−a−b)q} ≤ C||f||_{p}^{q−p}

due to the identity from assumptions. Hence

∫_{0}^{∞} |x^{2(σ−a−b)}V_{a}f(x)|^{q}dμ_{α}(x) ≤ C||f||_{p}^{q−p} ∫_{0}^{∞} |V_{a}f(x)|^{p}dμ_{α}(x) ≤ C||f||_{p}^{q} .

by the norm inequality from Lemma 3.2. The estimate of I_{3} is similar and we omit it. The proof of the theorem is complete.

After submitting this paper the authors became acquainted with the following result of Kanjin and Sato [The Hardy-Littlewood theorem on fractional integration for Laguerre series, Proc. Amer. Math. Soc., to appear]:

∥I_{σ}f∥_{L_{v(αq/2)}^{q}} ≤ C∥f∥_{L_{v(αp/2)}^{p}}, 0 < \frac{1}{q} = \frac{1}{p} − σ, p > 1, α ≥ 0,

whereas for −1 < α < 0 there occurs (1 + \frac{α}{2})^{−1} < p, q < −\frac{2}{α} as additional restriction. Combining the above Theorem 1.1 in the case a = b = α = 0 with Kanjin’s transplantation theorem (cf. [10]) one at once recovers the above stated result of Kanjin and Sato .

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