<span class="mathjax-formula">$L_{p,q}$</span>-cohomology of warped cylinders

BLAISE PASCAL

Yaroslav Kopylov

L_p,q-cohomology of warped cylinders Volume 16, n^o2 (2009), p. 321-338.

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L

-cohomology of warped cylinders

Yaroslav Kopylov

Abstract

We extend some results by Gol⁰dshtein, Kuz⁰minov, and Shvedov about theLp- cohomology of warped cylinders toLp,q-cohomology forp6=q. As an application, we establish some sufficient conditions for the nontriviality of theLp,q-torsion of a surface of revolution.

Cohomologie L_p,q des cylindres tordus

Résumé

On généralise quelques résultats par Gol⁰dshtein, Kuz⁰minov et Shvedov sur la cohomologie Lp des cylindres tordus à cohomologieLp,q pourp 6=q. Comme application, on établit des conditions suffisantes pour la non-nullité de la torsion Lp,q d’une surface de révolution.

1. Introduction

Let M be a Riemannian manifold. For 1 ≤ p ≤ ∞ and a positive con- tinuous function σ : M → R, denote by L^j_p(M, σ) the Banach space of measurable forms of degree j on M with the finite norm

kωkL^jp(M,σ)=







R

M|ω(x)|^pσ^p(x)dx 1/p

if1≤p <∞, ess sup_x∈M|ω(x)|σ(x) ifp=∞.

Here dx stands for the volume element of M and |ω(x)| is the modulus of the exterior form ω(x). In the usual way, we also define the spaces L_p,loc(M).

Denote by D^j(M) = C₀^∞,j(M) the space of smooth forms of degree j on M having compact support included in IntM. A form ψ∈L^j+1_1,loc(M)

Keywords:Differential form,Lp,q-cohomology,Lp,q-torsion, warped cylinder.

Math. classification:58A12, 46E30.

is called the (weak) differentialdω of ω∈L^j_1,loc(M) if Z

U

ω∧du= (−1)^j+1 Z

U

ψ∧u

for every orientable domainU ⊂IntMand every formu∈D^dimM−j−1(M) having support in U.

For two weights σ_j,σ_j+1 onM, put

W_p,q^j (M, σ_j, σ_j+1) ={ω∈L^j_p(M, σ_j)|dω∈L^j+1_q (M, σ_j+1)}.

The space W_p,q^j (M, σj, σj+1) is endowed with the norm kωk_Wj

p,q(M,σj,σj+1)=kωk_Lj

p(M,σj)+kdωk_Lj+1

q (M,σj+1).

If p=q then it is often more convenient to consider the equivalent norm kωk_Wj

p(M,σj,σj+1) =

kωk^p

L^jp(M,σj)+kdωk^p

L^j+1p (M,σj+1)

1/p

.

In the sequel we let V_p,q^j (M, σj, σj+1) denote the closure of D^j(M) in the norm of W_p,q^j (M, σ_j, σ_j+1).

Given an arbitrary subset A ⊂ M, let W_p,q^j (M, A, σ_j, σ_j+1) be the closure in W_p,q^j (M, σj, σj+1) of the subspace spanned by all forms ω ∈ W_p,q^j (M, σ_j, σ_j+1) which vanish on some neighborhood of A (depending on ω).

Let Z_q^j(M, σj) be the subspace in W_q,q^j (M, σj, σj) that consists of all forms ω such thatdω= 0 and let

B_p,q^j (M, σj−1, σ_j) ={θ∈W_q,q^j (M, σ_j, σ_j)

|θ=dψ for some ψ∈W_p,q^j−1(M, σj−1, σj)}.

The spaces

H_p,q^j (M, σj−1, σj) =Z_q^j(M, σj)/B_p,q^j (M, σj−1, σj) and

H^j_p,q(M, σj−1, σj) =Z_q^j(M, σj)/B^j_p,q(M, σj−1, σj),

where B^j_p,q(M, σj−1, σ_j) is the closure of B_p,q^j (M, σj−1, σ_j) in L^j_q(M, σ_j) (equivalently, in W_q,q^j (M, σ_j, σ_j)) are called the jth L_p,q-cohomology and

the jth reduced Lp,q-cohomology of the Riemannian manifold M with weights σj−1 and σ_j. The quotient space

T_p,q^j (M, σj−1, σ_j) =B^j_p,q(M, σj−1, σ_j)/B_p,q^j (M, σj−1, σ_j)

will be referred to as theLp,q-torsionofM with the given weights. Clearly, the space T_p,q^j (M, σj−1, σ_j) is isomorphic to the closure of the zero in H_p,q^j (M, σj−1, σj).

Given a subset A ⊂ M, the relative nonreduced and reduced L_p,q- cohomology spaces H_p,q^j (M, A, σj−1, σj) and H^j_p,q(M, A, σj−1, σj) are de- fined as

H_p,q^j (M, A, σj−1, σj) =Z_q^j(M, A, σj)/B^j_p,q(M, A, σj−1, σj) and

H^j_p,q(M, A, σj−1, σ_j) =Z_q^j(M, A, σ_j)/B^j_p,q(M, A, σj−1, σ_j),

where the relative spaces Z_q^j(M, A, σ_j) and B^j_p,q(M, A, σj−1, σ_j) are de- fined as their absolute analogs above with the spaces W_p,q^j (M, σ_j, σ_j) and W_p,q^j−1(M, σj−1, σj) replaced by the spaces

W_p,q^j (M, A, σj, σj) and W_p,q^j−1(M, A, σj−1, σj).

For p = q, we write the subscript p instead of p, p throughout. If the weights involved in the definition of the corresponding space are equal to1 then they will be omitted.

The spaces Wp,q and Lp,q-cohomology were introduced at the begin- ning of the 1980’s by Gol⁰dshtein, Kuz⁰minov, and Shvedov [3, 4, 5, 6, 7, 8], who obtained many results concerning W_p,q-forms and especially L_p- cohomology. Later Lp,q-cohomology was considered in [11, 12, 13, 14, 15, 17, 22].

In this paper, we, following [9, 10], look for conditions of the nontrivial- ity of theL_p,q-cohomology andL_p,q-torsion on warped cylinders, a class of warped products of Riemannian manifolds. By thewarped productX×_f Y of two Riemannian manifolds (X, g_X) and (Y, g_Y) with the warping func- tion f : X → R+ we mean the product manifold X×Y endowed with the metric gX +f²(x)gY. If X = [a, b[ is a half-interval on the real line then X×_f Y is referred to as the warped cylinder. The study of the L2- cohomology of warped cylinders was initiated by Cheeger [2].

The structure of the article is as follows. In Section 2, we adapt the results of [9] about the Lp-cohomology of a half-interval to the casep6=q.

After that, using these Lp,q-results, in Section 3, we prove a partialLp,q- generalization of Theorem 1 of [9] about the L_p-cohomology of a warped cylinder[a, b[×_fY depending on the analytic properties of the functionf. As an application, we obtain an extension of the necessary condition for the triviality of the Lp,q-torsion of a surface of revolution in Rⁿ⁺² [16]

from the case p=q to arbitrary p,q such that ¹_q −¹_p < _n+1¹ . 2. Weighted L_p,q-cohomology of a half-interval

Consider a half-interval [a, b[, −∞< a < b≤ ∞ and positive continuous functions v0, v1 : [a, b[→R. For1< p, q <∞, the space W_p,q⁰ ([a, b[, v0, v1) can be identified with the space of the functions g ∈ Lp([a, b[, v0) whose weak derivativeg⁰ ∈L_q([a, b[, v1). As above, endowW_p,q⁰ ([a, b[, v0, v1)with the norm

kgk_W0

p,q([a,b[,v0,v1)= Z _b

a

|g(t)|^pv^p₀dt 1/p

+ Z _b

a

|g⁰(t)|^qv^q₁dt 1/q

.

From the classical Sobolev Embedding Theorem it follows that the func- tions of the class W_p,q⁰ ([a, b[, v0, v1) are continuous on[a, b[. Consider also the space

W_p,q⁰ ([a, b[,{a}, v₀, v1) ={f ∈W_p,q⁰ ([a, b[,{a}, v₀, v1)|f(a) = 0}.

We have

H_p,q¹ ([a, b[, v0, v1) =W_q¹([a, b[, v1, v1)/dW_p,q⁰ ([a, b[, v0, v1);

H_p,q¹ ([a, b[,{a}, v0, v1) =W_q¹([a, b[,{a}, v1, v1)/dW_p,q⁰ ([a, b[,{a}, v0, v1).

The spacesH¹_p,q([a, b[, v0, v1)andH¹_p,q([a, b[,{a}, v0, v1)are described sim- ilarly.

We call the following assertion the lemma about the Hardy inequality [1, 10, 21]:

Lemma 2.1. Suppose that1≤p, q≤ ∞, ¹_q+_q¹0 = 1,α, β ∈[−∞,∞],Iα,β

is the interval with endpoints α and β, v0 and v1 are continuous positive functions on I_α,β. Then for the existence of a global constantC such that

Z β α

v0(t) Z τ

α

g(t)dt

p

dτ

1/p

≤C

Z β α

|v₁(t)g(t)|^qdt

1/q

for every g∈Lq(I_α,β, v1), it is necessary and sufficient that χ_p,q(α, β, v0, v1)<∞.

Here

χp,q(α, β, v0, v1) = sup

τ∈I_α,β

Z β τ

|v0(t)|^pdt

1/p

Z τ α

|v1(t)|^−q0dt

1/q⁰

if p≥q;

χp,q(α, β, v0, v1)

=

Z β α

Z τ α

|v₁(t)|^−q0dt

p−1

Z β τ

|v₀(t)|^pdt

q q−p

|v₁(τ)|^−q0dτ

q−p pq

if p < q.

If p = 1 (q⁰ =∞) then the corresponding integral must be replaced by ess sup.

The constant χp,q(α, β, v0, v1)will be referred to as theHardy constant.

The following lemma was proved in [9] forp=qandv0 =v1. The proof given in [9] holds for differentp and q and different v0 and v1.

Lemma 2.2. Suppose that α, β ∈ [−∞,∞], v0, v1 : Iα,β → R are posi- tive continuous functions, and χ_p,q(α, β, v0, v1) =∞. Then there exists a nonnegative function h such that

Z β α

v^q₁(t)h^q(t)dt|

<∞,

Z β α

v^p₀(τ)

Z τ α

h(t)dt

p

dτ

=∞.

As in [9], Lemma 2 yields the following assertion.

Theorem 2.3. If v0, v1 are positive continuous functions on [a, b[ and 1< p, q <∞ then

(1) H_p,q¹ ([a, b[,{a}, v₀, v1) = 0⇐⇒χp,q(a, b, v0, v1)<∞;

(2)H_p,q¹ ([a, b[, v0, v1) = 0⇐⇒χ_p,q(a, b, v0, v1)<∞ or χ_p,q(b, a, v0, v1)<∞.

Let

0→A→^ϕ B →^ψ C→0 (2.1)

be an exact sequence of Banach complexes, i.e., complexes in the category of Banach spaces and bounded linear operators. Sequence (2.1) yields an exact sequence of the cohomology spaces

· · · →H^k−1(C)→^∂ H^k(A) ^ϕ

∗

→H^k(B) ^ψ

∗

→H^k(C)→. . .

with continuous operators ∂^∗, ϕ^∗, ψ^∗ and a semi-exact sequence of the reduced cohomology spaces

· · · →H^k−1(C)→^∂ H^k(A) ^ϕ

∗

→H^k(B)^ψ

∗

→H^k(C)→. . . (2.2) Under certain conditions, sequence (2.2) is exact at some terms (see [10, 18, 20]). In particular, Gol⁰dshtein, Kuz⁰minov, and Shvedov proved the following assertion in [10, Theorem 1(1)]:

Lemma 2.4. If H^k(C) is separated and dim∂(H^k−1(C))< ∞ then the sequence H^k−1(C)→^∂ H^k(A) ^ϕ

∗

→H^k(B) ^ψ

∗

→H^k(C) is exact.

As was explained in [12], we can describe thejth weightedLp,q-cohomo- logy of ann-dimensional Riemannian manifoldM with given weightsσj−1

and σ_j in terms of Banach complexes. To this end, consider an arbitrary sequence π = {p0, p1, . . . , pn} ⊂ [1,∞] with pj−1 = p and pj = q and a sequence of positive continuous weights σ ={σ_k}ⁿ_k=0 with the givenσj−1

and σj. Given a subsetA⊂M, put

W_π^k(M, A, σ) =W_pk,pk+1(M, A, σ_k, σ_k+1).

Here we have assumed that p_n+1 =p_n and σ_n+1=σ_n. Since the exterior differential is a bounded operator d^k−1:W_π^k−1(M, A, σ)→W_π^k(M, A, σ), we obtain a Banach complex

0→W_π⁰(M, A, σ)→^d0 W_π¹(M, A, σ)→. . .^dn−1→ W_πⁿ(M, A, σ)→0. (2.3) By the k-th L_π-cohomology H_π^k(M, A, σ) (reduced k-th L_π-cohomology H^k_π(M, A, σ))of the Riemannian manifoldM with respect toAwith weight σ we mean the cohomology (reduced cohomology) of (2.3). Thus,

H_π^k(M, A, σ) =H_p^k_k−1,p

k(M, A, σk−1, σ_k) and

H^k_π(M, A, σ) =H^k_pk−1,p

k(M, A, σk−1, σ_k) for all k. In particular,

H_π^j(M, A, σ) =H_p,q^j (M, A, σj−1, σj), H^j_π(M, A, σ) =H^j_p,q(M, A, σj−1, σj).

Take M = [a, b[, A = {a}, 1 < p, q < ∞, π = {p, q}, and a pair of weights v = {v₀, v1}. We have the following exact sequence of Banach complexes:

0→W_π^∗([a, b[,{a}, v)→^j W_π^∗([a, b[, v)→ⁱ H^∗({a})→0,

whereH^∗({a})is the complex with the only nontrivial termH⁰({a}) =R. Here the mappingsiandjare defined as follows:jis the inclusion mapping;

if g ∈ W_π⁰([a, b[, v) then ig = g(a) (recall that g is continuous) and in dimension one jis zero. Lemma 2.4 yields the exact sequence

R=H⁰({a})→^∂ H¹_p,q([a, b[,{a}, v0, v1) ^j

∗

→H¹_p,q([a, b[, v0, v1).

Thus, we infer the following assertion, proved for p = q in [9]. With what has been said above, the proof of [9] extends to the case of p 6= q without change.

Theorem 2.5. If v0, v1 are positive continuous functions on [a, b[, 1 <

p <∞, 1< q <∞, ¹_q+_q¹0 = 1 then (1) H¹_p,q([a, b[, v0, v1) = 0;

(2) H¹_p,q([a, b[,{a}, v₀, v1) = 0 if and only if

b

R

a

v^−q₁ ⁰(t)dt = ∞ or

b

R

a

v₀^p(t)dt <∞;

(3) If Hp,q([a, b[,{a}, v₀, v1)6= 0 then

∂:R=H⁰({a})→H¹_p,q([a, b[,{a}, v₀, v1) is an isomorphism.

3. L_p,q-cohomology of the warped cylinder C_a,b^f

Let Y be an orientable manifold of dimension n,C_a,b^f Y = [a, b[×_fY. Put Ya ={a} ×Y. Generally speaking, C_a,b^f is a Lipschitz Riemannian mani- fold in the sense of [3] but we will assume throughout for simplicity that

∂Y = ∅to makeC_a,b^f smooth, which will be enough for our purposes.

Suppose that 1< p <∞ and 1< q <∞.

In [9], Gol⁰dshtein, Kuz⁰minov, and Shvedov introduced the bilinear mapping

ν :L^j−1_p (Y)×L¹_p([a, b[, f^np−j+1)→L^j_p(C_a,b^f Y),

ν(ϕ, gdt) = gdt∧ ϕ. In [9] it was proved that ν is continuous and if ϕ ∈ Z_p^j−1(Y) then νϕ =ν(ϕ,·) : L¹_p([a, b[, f^np−j+1) → L^j_p(C_a,b^f Y) induces continuous mappings

ν_ϕ^∗ :H_p¹([a, b[, f^np−j+1)→H_p^j(C_a,b^f Y);

ν˜_ϕ^∗ :H_p¹([a, b[,{a}, f^np−j+1)→H_p^j(C_a,b^f Y, Ya).

Supposing thatϕ∈Z_p^j−1(Y)∩Z_q^j−1(Y), we similarly become convinced that the mapping νϕ =ν(ϕ,·) induces continuous mappings

ν_ϕ^∗ :H_p,q¹ ([a, b[, f^np−j+1, f^nq−j+1)→H_p,q^j (C_a,b^f Y);

ν˜_ϕ^∗ :H_p,q¹ ([a, b[,{a}, f^np−j+1, f^nq−j+1)→H_p,q^j (C_a,b^f Y, Ya).

Now, assume thatψ∈L^n+1−j_p0 (Y)(p⁰ = _p−1^p ) andω ∈L^j_p(Ca,bY). Write ω in the form ω = ω_A+dt∧ω_B, where ω_A, ω_B do not contain dt [10].

Following [9], introduce the continuous operator µ_ψ: L^j_p(C_a,b^f Y)→L¹_p([a, b[, f^np−j+1) by the formula

µ_ψω = Z

Y

ω_B(t)∧ψ

dt.

The following lemma was proved in [9] for p =q and ψ∈ V_p^n−j+10 (Y).

The proof in [9] easily extends to p6=q:

Lemma 3.1. If ψ∈D^n−j+1(Y) and dψ= 0 thenµ_ψ induces continuous mappings

µ^∗_ψ :H_p,q^j (C_a,b^f Y)→H_p,q¹ ([a, b[, f^np−j+1, f^nq−j+1);

µ˜^∗_ψ :H_p,q^j (C_a,b^f Y, Ya)→H_p,q¹ ([a, b[,{a}, f^np−j+1, f^nq−j+1)

We have the following theorem partially generalizing item 7 of Theo- rem 1 in [9]:

Theorem 3.2. Suppose thatY is an orientablen-dimensional Riemann- ian manifold, ∞ < a < b ≤ ∞, f : [a, b[→ R is a positive continu- ous function, 1 < p < ∞, 1 < q < ∞. Assume that there exists ϕ ∈

Z_p^j−1(Y)∩Z_q^j−1(Y) such that^R_Y ϕ∧γ 6= 0for some form γ ∈D^n−j+1(Y), dγ = 0.

The following hold:

(1) if χ_p,q(a, b, f^np−j+1, f^nq−j+1) =∞ then H_p,q^j (C_a,b^f Y, Y_a)6= 0;

(2) if χ_p,q(a, b, f^np−j+1, f^nq−j+1) = ∞ and χ_p,q(b, a, f^np−j+1, f^nq−j+1)=

∞ thenT_p,q^j (C_a,b^f Y)6= 0 and, hence, dimH_p,q^j (C_a,b^f Y) =∞.

Proof. Let ϕ ∈ Z_p^j−1(Y)∩ Z_q^j−1(Y) be a cocycle having the property mentioned in the theorem and let γ ∈ D^n−j+1(M) be a form such that R

Y ϕ∧γ = 1. Then µ^∗_γ ◦ν_ϕ^∗ = id, µ˜^∗_γ ◦ν˜_ϕ^∗ = id [9]. Consequently, the mappings

ν_ϕ^∗ :H_p,q¹ ([a, b[, f^np−j+1, f^nq−j+1)→H_p,q^j (C_a,b^f Y) and

ν˜_ϕ^∗ :H_p,q¹ ([a, b[,{a}, f^np−j+1, f^nq−j+1)→H_p,q^j (C_a,b^f Y, Ya) are injective.

Suppose that χp,q(a, b, f^np−j+1, f^nq−j+1) = ∞. Then, by Theorem 2.3, H_p,q¹ ([a, b[,{a}, f^np−j+1, f^nq−j+1)6= 0. Therefore,H_p,q^j (C_a,b^f Y, Ya)6= 0.

Assume now that

χp,q(a, b, f^np−j+1, f^nq−j+1) =∞ and

χ_p,q(b, a, f^np−j+1, f^nq−j+1) =∞.

Then, by Theorem 2.3, H_p,q¹ ([a, b[, f^np−j+1, f^nq−j+1) 6= 0. Since, by Theo- rem 2.5, H_p,q([a, b[, f^np−j+1, f^nq−j+1) = 0, we have

T_p,q¹ ([a, b[, f^np−j+1, f^nq−j+1)6= 0.

Now, if we had T_p,q^j (C_a,b^f Y) = 0,ν_ϕ^∗ would be a continuous injective map- ping with values in the Hausdorff space H_p,q^j (C_a,b^f Y), and so the coho- mology space H_p,q¹ ([a, b[, f^np−j+1, f^nq−j+1) would also be Hausdorff, i.e., without torsion. Thus, T_p,q^j (C_a,b^f Y)6= 0. The theorem is proved.

L_p,q-torsion of a surface of revolution

Let M be a surface of revolution in Rⁿ⁺², i.e., the (n+ 1)-dimensional surface defined by the equation

f²(x1) =x²₂+· · ·+x²_n+2, (x1, . . . , x_n+2)∈Rⁿ⁺², x1 ≥0, (3.1) wheref : [0,∞[→Ris a positive smooth function. The manifoldM is the product [0,∞[×Sⁿ endowed with the metric

gM = (1 +f⁰²(x1))dx²₁+f²(x1)dy²

induced from Rⁿ⁺², where dx²₁ and dy² are the conventional Riemannian metrics on [0,∞[ and the sphere Sⁿ. In other words, M may be consid- ered as the warped product [0,∞[×_FSⁿ, where F = f ◦G⁻¹, G(x) = Rx

0

p1 +f⁰²(t)dt.

In [17], we have proved the following fact:

Theorem 3.3. Suppose that f is unbounded, p, q ∈[1,∞[, ¹_q −¹_p < _n+1¹ , 1≤j ≤n+ 1. Then T_p,q^j (M)6= 0.

Kuz⁰minov and Shvedov [19] established that whenf is bounded from above, T_p^j(M) is zero for all j, 2 ≤ j ≤ n and that, for j = 1, n+ 1, the triviality of T_p^j(M) depends on the finiteness of some Hardy con- stants. This is due to the connection between the Lp-cohomology of the warped product C_a,b^f Y and the weighted L_p-cohomology of [a, b[ given in the mentioned papers [9, 10]. Above we have shown that there is a con- nection of this type for Lp,q-cohomology. Namely, by Theorem 3.2, since Sⁿis compact and the de Rham cohomology H^j−1(Sⁿ) ofSⁿ is nontrivial if j = 1, n+ 1, for T_p,q^j (M) (j = 1, n+ 1) to be zero, it is necessary that χp,q(0,∞, F^np−j+1, F^nq−j+1)<∞ orχp,q(∞,0, F^np−j+1, F^nq−j+1)<∞.

The main result of this section is a generalization of Theorems 2 and 2⁰ of [16] and is formulated as follows:

Theorem 3.4. Let M be the surface of revolution (3.1). Suppose that 1< p <∞, 1< q <∞, ¹_q −¹_p < _n+1¹ , j ∈ {1, n+ 1}. If T_p,q^j (M) = 0 then

x→∞lim f(x) = 0 and volM <∞.

Proof. Putk=j−1,q⁰= _q−1^q .

We have the following equalities:

χ⁰_p,q≡χp,q(0,∞, F^np−k, F^nq−k)

= sup

τ >0

Z^∞

τ

f^n−kp(t)p

1 +f⁰²(t)dt

1/pZ^τ

0

f^{−(nq−k)q0}(t)p

1 +f⁰²(t)dt 1/q⁰

;

χ^∞_p,q≡χp,q(∞,0, F^np−k, F^nq−k)

= sup

τ >0

Z^τ

0

f^n−kp(t)p

1 +f⁰²(t)dt

1/pZ^∞

τ

f^{−(nq−k)q0}(t)p

1 +f⁰²(t)dt 1/q⁰

ifp≥q;

χ⁰_p,q≡χ_p,q(0,∞, F^np−k, F^nq−k)

=

∞

Z

0

H(x)

Z

0

f^{−(nq−k)q0}(t) q

1 +f⁰²(t)dt

^p−1 Z^∞

H(x)

f^n−kp(t) q

1 +f⁰²(t)dt

q q−p

×f^{−(nq−k)q0}(x) q

1 +f⁰²(x)dx

!

q−p qp

; (3.2)

χ^∞_p,q≡χp,q(∞,0, F^np−k, F^nq−k)

=

∞

Z

0

Z^∞

H(x)

f^{−(nq−k)q0}(t) q

1 +f⁰²(t)dt ^p−1

H(x)

Z

0

f^n−kp(t) q

1 +f⁰²(t)dt

q q−p

×f^{−(nq−k)q0}(x) q

1 +f⁰²(x)dx

!

q−p qp

if p < q. Here H(x) is the function inverse to the arc length function G(x) =^R₀^xp1 +f⁰²(t)dt.

The main element in the proof of Theorem 3.4 is the following lemma which has some independent interest.

Lemma 3.5. If ¹_q − ¹_p < _n+1¹ , 1 < p < ∞, 1< q < ∞, 0≤k ≤n, then the following hold:

(1) if χ⁰_p,q <∞ or χ^∞_p,q <∞ then lim

t→∞f(t) = 0;

(2) if ⁿ_p −k≤0 then χ⁰_p,q=∞;

(3) if ⁿ_q −k≥0 then χ^∞_p,q=∞.

Proof. Suppose first thatp≥q.

Assume that χ⁰_p,q<∞. Then

Z ∞ 0

f^n−kp(t) q

1 +f⁰²(t)dt <∞ (3.3)

Since

f^n−kp(t) q

1 +f⁰²(t)≥f^n−kp(t)|f⁰(t)|,

it follows that the integral

Z ∞ 0

f^n−kp(t)f⁰(t)dt

=







1

n−kp+1 lim

t→∞(f^n−kp+1(t)−f^n−kp+1(0)) ifn−kp6=−1,

t→∞lim log_f(0)^f(t) ifn−kp=−1 (3.4)

is finite.

There appear several possibilities:

(a) ⁿ_p−k >0. The above implies that there exists a finite limit lim

t→∞f(t), which is zero by (3.3).

(b) ⁿ_p −k= 0. This is impossible in view of (3.3).

(c) −¹_p < ⁿ_p −k <0. Then n−kp+ 1>0and f(t) has a finite limit as t→ ∞, which contradicts (3.3).

(d) ⁿ_p −k=−¹_p. A contradiction to (3.3).

(e) ⁿ_p −k <−¹_p. In this case,n−kp <−1. Hence, lim

t→∞f(t) =∞.

Note that, since ¹_q−¹_p < _n+1¹ , we havek+ 1> ⁿ⁺¹_p + 1> ⁿ⁺¹_q , whence

−(ⁿ_q −k)q⁰+ 1>0. We infer

Z ∞ τ

f^n−kp(t)^q1 +f⁰²(t)dt

^1/pZ τ 0

f^{−(nq−k)q0}(t)^q1 +f⁰²(t)dt ^1/q0

≥ Z ∞

τ

f^n−kp(t)|f⁰(t)|dt

^1/pZ _τ

0

f^{−(nq−k)q0}(t)|f⁰(t)|dt ^1/q0

≥

Z ∞ τ

f^n−kp(t)f⁰(t)dt

1/p

Z τ 0

f^{−(nq−k)q0}(t)f⁰(t)dt

1/q⁰

≥

f^n−kp+1(τ)

|n−kp+ 1|

1/p

f^{−nq−k)q0+1}(τ)−f^{−(nq−k)q0+1}(0)

−(ⁿ_q −k)q⁰+ 1

1/q⁰

=C·f

n+1

p −ⁿ⁺¹_q +1

(τ)|1−f^{−(nq−k)q0+1}(0)f^{(nq−k)q0−1}(τ)|^1/q0. (3.5)

The last quantity in (3.5) is equivalent toCf

n+1

p −ⁿ⁺¹_q +1

(τ)asτ → ∞and, hence, tends to infinity. Therefore, χ⁰_p,q =∞, and we obtain a contradic- tion.

Thus, ifχ⁰_p,q<∞ then lim

t→0f(t) = 0 and ⁿ_p −k >0.

Suppose now that χ^∞_p,q <∞. Then

Z ∞ 0

f^{−(nq−k)q0}(t) q

1 +f⁰²(t)dt <∞ (3.6)

and, hence, there exists a finite integral

Z ∞ 0

f^{−(nq−k)q0}(t)f⁰(t)dt

=











lim

t→∞f⁻⁽

n q−k)q0+1

(t)−f^{−(nq−k)q0+1}(0)

−(ⁿ

q−k)q⁰+1 if−(ⁿ_q −k)q⁰ 6=−1,

t→∞lim log_f^f₍₀₎^(t) if−(ⁿ_q −k)q⁰ =−1.

(3.7)