Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 02, 1997, pp. 29–42. Submitted: August 25, 1997. Published: December 30, 1997.

EIGENVALUE DISTRIBUTION OF INTEGRAL OPERATORS DEFINED BY BESOV–ORLICZ KERNELS

E. Elshobaky and M. Faragallah

Abstract. The real interpolation method has been generalized to the interpolation method with function parameter (X_{0}, X_{1})_{ρ,p} instead of (X_{0}, X_{1})_{θ,p}, replacing t^{θ} by some concave function ρ, see Gustavsson-Peetre [GP] and Fernandez-Garcia [FG]. The new method is particularly well-suited to interpolation of Besov-Orlicz spaces B_{Φ,q}^{s}. The main aim of this paper is to combine the new interpolation method with König’s eigenvalue theorem of integral operators defined by kernels in Besov and Sobolev spaces to find the optimal results for the asymptotic distribution of the eigenvalues of integral operators defined by kernel in Besov-Orlicz and Sobolev-Orlicz spaces. We derive some lemmas and theorems using the new interpolation method (e.g., the interpolation of L_{p}-spaces, Schatten classes and Besov spaces). Finally, we give a good estimate for the eigenvalues of integral operators defined by kernels belonging to vector-valued Besov-Orlicz spaces B_{Φ,q}^{s}(B_{Ψ,v}^{σ}). We show that the eigenvalues of the integral operators defined by kernels k ∈ B_{Φ,q}^{s}(B_{Ψ,v}^{σ}) belong to some Lorentz-Orlicz space l_{Φ,q}. Moreover, we derive some special cases, e.g., k ∈ W_{Φ}^{s}(W_{Ψ}^{σ}) and k ∈ B_{Φ,q}^{s}(L_{Ψ}).

Keywords: Interpolation, Eigenvalue distribution, integral operators.

Mathematics subject classification numebrs: 46B20, 47B10

1. The Fernandez-Garcia interpolation method.

We now introduce a variant of the Gustavsson-Peetre interpolation method. It will depend on function parameters. The function parameter will be taken in Peetre’s class P^{+−} , i.e., the class of pseudo-concave positive functions ρ on ℝ^{+} which satisfy Department of Mathematics, Faculty of Science,

Ain Shams University, Cairo 11566, Egypt e-mail: nagwamsa@frcu.eun.eg Copyright \begin{matrix}c \\ ◯ \end{matrix}1996 by Cameron University

ρ̄(t) = sup_{s>0} \frac{ρ(st)}{ρ(s)} = ∘(max(1, t)).  (1)

For a function parameter ρ ∈ P^{+−}, we can replace (1) by

ρ̄(t) = ∘(max(t^{ε}, t^{1−ε})),  (2)

provided ε > 0 is small enough. Consequently, we have

∫_{0}^{∞} min(1, 1/t) ρ̄(t)\frac{dt}{t} < ∞,

which implies

∑_{n=0}^{∞} \frac{ρ(2^{n})}{2^{n}} + ∑_{n=0}^{∞} ρ(2^{−n}) < ∞.  (3)

We shall also consider the sequence (r̃_{n}(.))_{n∈ℤ} defined in [0, 1] by

r̃_{n} = \{ \begin{matrix}r_{2n} & if n ≥ 0 \\ r_{2|n|−1} & if n < 0, \end{matrix}

where (r_{n}(.))_{n∈ℤ} is the sequence of Rademacher functions.

Definition 1.1. Let (X_{0}, X_{1}) be a Banach couple. The space ⟨X_{0}, X_{1}⟩_{ρ,p} is the linear space of all x ∈ X_{0} + X_{1} such that there is a sequence (u_{n})_{n∈ℤ} in X_{0} ∩ X_{1} which satisfies

x = ∑_{n=−∞}^{∞} u_{n}  (convergence in X_{0} + X_{1}),  (4)

sup_{J} ∫_{0}^{1} ∥∑_{n∈J} r̃_{n}(t)\frac{2^{kn}}{ρ(2^{n})}u_{n}∥_{X_{k}}^{p} dt < ∞,  (5)

where the supremum is taken over all finite subsets J in ℤ. Moreover, we shall also assume that, for k = 0, 1 , the sequence (r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}u_{n})_{n∈ℤ} is L_{p}([0, 1], X_{k})-summable.

We equip the space ⟨X_{0}, X_{1}⟩_{ρ,p} with the norm

∥x∥_{⟨X_{0},X_{1}⟩_{ρ,p}} := inf max_{k=0,1} sup_{J} (∥∑_{n∈J} r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}u_{n}∥_{L_{p}([0,1],X_{k})}) ,  (6)

where the infimum is taken over all admissible sequence (u_{n})_{n∈ℤ}.

Theorem 1.2. Let (X_{0}, X_{1}) be a Banach couple, ρ ∈ P^{+−} and 1 ≤ p < ∞. Then the space ⟨X_{0}, X_{1}⟩_{ρ,p} is a Banach space.

Proof : See [FG p. 25]

To show that ⟨X_{0}, X_{1}⟩_{ρ,p} is an intermediate space between X_{0} and X_{1}, we need the following two lemmas.

Lemma 1.3. For all Banach spaces X, we have ⟨X, X⟩_{ρ,p} = X.

Proof : The imbedding X ↪ ⟨X, X⟩_{ρ,p} is immediate. On the other hand, we have ⟨X, X⟩_{ρ,p} ⊂ X. It remains to prove that the imbedding is bounded. Let x ∈ ⟨X, X⟩_{ρ,p} and let (u_{n})_{n∈ℤ} be an admissible sequence in X for x, such that

sup_{J} ∥∑_{n∈J} r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}u_{n}∥_{L_{p}([0,1],X)} ≤ 2∥x∥_{⟨X,X⟩_{ρ,p}}.

Hence, for each n ∈ ℤ and k = 0, 1 , we have

∥\frac{2^{kn}}{ρ(2^{n})} u_{n}∥_{X} ≤ 2∥x∥_{⟨X,X⟩_{ρ,p}},

and recalling (3), it follows that

∥x∥_{X} ≤ ∑_{n=−∞}^{∞} ∥u_{n}∥_{X} = ∑_{n=0}^{∞} ∥u_{n}∥_{X} + ∑_{n=−∞}^{−1} ∥u_{n}∥_{X}

≤ 2∥x∥_{⟨X,X⟩_{ρ,p}} (∑_{n=0}^{∞} \frac{ρ(2^{n})}{2^{n}} + ∑_{n=−∞}^{−1} ρ(2^{n}))

≤ c∥x∥_{⟨X,X⟩_{ρ,p}}.

Lemma 1.4. Let (X_{0}, X_{1}) and (Y_{0}, Y_{1}) be Banach couples such that X_{k} ⊂ Y_{k}, (k = 0, 1). Then

⟨X_{0}, X_{1}⟩_{ρ,p} ↪ ⟨Y_{0}, Y_{1}⟩_{ρ,p}.

Proof : Immediate.

Theorem 1.5. If (X_{0}, X_{1}) is a Banach couple we have

X_{0} ∩ X_{1} ↪ ⟨X_{0}, X_{1}⟩_{ρ,p} ↪ X_{0} + X_{1},

i.e., ⟨X_{0}, X_{1}⟩_{ρ,p} is an intermediate space between X_{0} and X_{1},

Proof : The left imbedding is clear. Now since X_{k} ↪ X_{0} + X_{1} we have

⟨X_{0}, X_{1}⟩_{ρ,p} ↪ ⟨X_{0} + X_{1}, X_{0} + X_{1}⟩_{ρ,p} = X_{0} + X_{1},

using the above two lemmas.

One has the fact is that the functor (X_{0}, X_{1}) ↦ ⟨X_{0}, X_{1}⟩_{ρ,p} is an interpolation functor.

Theorem 1.6. Let (X_{0}, X_{1}) and (Y_{0}, Y_{1}) be Banach couples, ρ ∈ P^{+−} and 1 ≤ p < ∞. Then, for all T ∈ L(X_{k}, Y_{k}), (k = 0, 1), we have

T ∈ L(⟨X_{0}, X_{1}⟩_{ρ,p}, ⟨Y_{0}, Y_{1}⟩_{ρ,p}).

Moreover,

∥T ∥ ≤ max(∥T ∥_{0}, ∥T ∥_{1}).

Proof : Let x ∈ ⟨X_{0}, X_{1}⟩_{ρ,p}, let (u_{n})_{n∈ℤ} be an admissible sequence of x, and set M_{k} = ∥T ∥_{L(X_{k},Y_{K})}, (k = 0, 1). Since the convergence in (4) is in X_{0} +X_{1}, we have T x = ∑_{n∈ℤ} T u_{n}. Hence, setting v_{n} = T u_{n}, we see that ∑_{n∈ℤ} v_{n} converges in Y_{0} + Y_{1} and is a representation of T x. Next, for k = 0, 1 , we get

∥∑_{n∈J} r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}v_{n}∥_{L_{p}([0,1],Y_{k})} = ∥T (∑_{n∈J} r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}u_{n})∥_{L_{p}([0,1],X_{k})}

≤ M_{k} ∥∑_{n∈J} r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}u_{n}∥_{L_{p}([0,1],X_{k})} ,

and consequently ∥T ∥_{⟨Y_{0},Y_{1}⟩_{ρ,p}} ≤ c∥x∥_{⟨X_{0},X_{1}⟩_{ρ,p}}. The proof is complete.

Theorem 1.7. (Kahane, see [FG]) For every 1 < r < ∞, for any Banach space X and every finite sequence (u_{j})_{1 ≤ j ≤ n} in X, we have

∫_{0}^{1} ∥∑_{j=1}^{n} r_{j}(t)u_{j}∥_{X} dt ≈ (∫_{0}^{1} ∥∑_{j=1}^{n} r_{j}(t)u_{j}∥_{X}^{r} dt)^{1/r} .

Theorem 1.8. Let (X_{0}, X_{1}) and (Y_{0}, Y_{1}) be Banach couples, ρ ∈ P^{+−} and 1 ≤ p, q < ∞. Then ⟨X_{0}, X_{1}⟩_{ρ,p} = ⟨X_{0}, X_{1}⟩_{ρ,q}, with norm equivalence.

Proof : This follows at once from Kahane’s theorem.

2. Interpolation of L_{p}(X) spaces

We shall show that L_{p}-spaces commute with the interpolation space ⟨X_{0}, X_{1}⟩_{ρ,p}.

Theorem 2.1 (FG). Let (X_{0}, X_{1}) be a Banach couple of U.M.D. spaces (see FG p. 23).Then the interpolation space ⟨X_{0}, X_{1}⟩_{ρ,p} is also a U.M.D. space.

Theorem 2.2. Let L_{Φ} be an Orlicz space such that the Orlicz function Φ and the conjugate function Φ^{∗} satisfy the Δ_{2}-condition. Then L_{Φ} has the U.M.D. property.

Corollary 2.3. The space Y (X) is U.M.D. if and only if Y and X are U.M.D. spaces.

Lemma 2.4 (FG). Let X be a Banach space, ρ ∈ P^{+−} and 1 ≤ p < ∞. Then if (u_{j})_{1≤j≤n} is a finite sequence in X, then

∥∑_{j=1}^{n} |u_{j}|∥_{X} ≤ cR (∥(∑_{j=1}^{n}(\frac{|u_{j}|}{ρ(2^{j})})^{p})^{1/p}∥_{X} , ∥(∑_{j=1}^{n}(\frac{2^{j}|u_{j}|}{ρ(2^{j})})^{p})^{1/p}∥_{X} ) ,

where R(s, t) = sρ(t/s).

Theorem 2.5. Let (X_{0}, X_{1}) be a U.M.D Banach couple. Then for all p, 1 ≤ p < ∞, we have

⟨L_{p}(X_{0}), L_{p}(X_{1})⟩_{ρ,p} = L_{p}(⟨X_{0}, X_{1}⟩_{ρ,p}),  (7)

with equivalent norms.

Proof : See [FG]

Remark 2.6. Using the fact that l_{q}^{s}-spaces are special L_{q}-spaces and the previous theorem, we have

⟨l_{q}^{s}(X_{0}), l_{q}^{s}(X_{1})⟩_{ρ,p} = l_{q}^{s}(⟨X_{0}, X_{1}⟩_{ρ,p}),  (8)

with the same assumption as in theorem (2.5).

The following theorem shows that reflexive Orlicz spaces can be characterized as interpolation spaces between L_{p}-spaces. This is a converse of Gustavsson-Peetre’s theorem [GP theorem (7.3)]. By an Orlicz function we mean a convex and continuous function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0.

Theorem 2.7 (FG). Let Φ be an Orlicz function and let L_{Φ}(E) be the corresponding E-valued Orlicz space on a σ-finite measure space, where E is a U.M.D. Banach space. Suppose that Φ and the conjugate Φ^{∗} satisfy the Δ_{2}-condition. Then there exist p_{0} and p_{1}, with 1 < p_{1} < p_{0} < ∞,

⟨L_{p_{0}}(E), L_{p_{1}}(E)⟩_{ρ,p} = L_{Φ}(E)  ,where  (9)

ρ(t) = \{ \begin{matrix}t^{\frac{p_{1}}{p_{1}−p_{0}}} Φ^{−1}(t^{\frac{p_{0}p_{1}}{p_{0}−p_{1}}} ) & , t > 0 \\ 0 & t = 0 \end{matrix}  (10)

and 1 < p < ∞.

3. Besov-Orlicz and Sobolev-Orlicz spaces

Definition 3.1. Suppose s ∈ ℝ and let (ϕ_{n}) be a system of functions(see T p. 171). Let X be a Banach space and Φ be an Orlicz function satisfy the Δ_{2}-condition,we define for 1 ≤ q ≤ ∞, the Besov-Orlicz spaces

B_{Φq}^{s}(ℝ^{N}; X) = {f ∈ S′ : (ϕ_{n} ∗ f)_{n} ∈ l_{q}^{s}(L_{Φ}(ℝ^{N}; X))}  (11)

We equip the spaces B_{Φq}^{s}(ℝ^{N}; X) with the norm

∥f∥_{B_{Φq}^{s}} = ∥(ϕ_{n} ∗ f)_{n}∥_{l_{q}^{s}(L_{Φ}(ℝ^{N};X))} = ∥(∥ϕ_{n} ∗ f∥_{L_{Φ}(X)})_{n}∥_{l_{q}^{s}},  (12)

Remark 3.2 (FG). The spaces B_{Φq}^{s} do not depend on the particular system of functions used to define them.

We shall prove that the Besov-Orlicz space B_{Φq}^{s}(ℝ^{N}; X) is a retract of the space l_{q}^{s}(L_{Φ}(ℝ^{N}; X)) using the composition

B_{Φq}^{s}(ℝ^{N}; X) ⟶^{I} l_{q}^{s}(L_{Φ}(ℝ^{N}; X)) ⟶^{P} B_{Φq}^{s}(ℝ^{N}; X).

Theorem 3.3. Let Φ be an Orlicz function and let L_{Φ}(X) be the corresponding Xvalued Orlicz space. Then the space B_{Φq}^{s}(ℝ^{N}; X) is a retract of l_{q}^{s}(L_{Φ}(ℝ^{N}; X)) (s ∈ ℝ, 1 ≤ q ≤ ∞).

Proof : Consider the mappings I and P are defined by

(If)_{j} = ϕ_{j} ∗ f  j = 1, 2, …  , (If)_{0} = ψ ∗ f,
P α = ∑_{j=0}^{∞} ϕ̃_{j} ∗ α_{j}  (convergence inS^{∗})  where α = (α_{j})_{j=0}^{∞},

α_{j} ∈ S^{∗}, j = 1, 2, …  and  ϕ̃_{0} = ψ + ϕ_{1}

ϕ̃_{j} = ∑_{l=−1}^{1} ϕ_{j+l} j = 1, 2, … .

Clearly P If = f , f ∈ S^{∗}  , since ϕ̃_{j} ∗ ϕ_{j} = ϕ_{j} j = 1, 2, …  and

ϕ̃_{0} ∗ ψ = ψ.

(i) First,we note that

∥f∥_{B_{Φq}^{s}(ℝ^{N};X)} ∼ (∑_{j=0}^{∞}(2^{js} ∥ϕ_{j} ∗ f∥_{L_{Φ}(ℝ^{N};X)})^{q})^{1/q}

= (∑_{j=0}^{∞}(2^{js} ∥(If)_{j}∥_{L_{Φ}(ℝ^{N};X)})^{q})^{1/q}

= ∥If∥_{l_{q}^{s}(L_{Φ}(ℝ^{N};X))}.

Thus I : B_{Φq}^{s}(ℝ^{N}; X) → l_{q}^{s}(L_{Φ}(ℝ^{N}; X)). (ii) Since ϕ_{j} ∗ P α = ϕ_{j} ∗ α_{j} (j ≥ 1), we have

∥P α∥_{B_{Φq}^{s}(ℝ^{N};X)} = (∑_{j=1}^{∞}(2^{js} ∥ϕ_{j} ∗ α_{j}∥_{L_{Φ}(ℝ^{N};X)})^{q})^{1/q}

≤ c (∑_{j=1}^{∞}(2^{js} ∥α_{j}∥_{L_{Φ}(ℝ^{N};X)})^{q})^{1/q}

≤ ∥α∥_{l_{q}^{s}(L_{Φ}(ℝ^{N};X))}.

Thus B_{Φq}^{s}(ℝ^{N}; X) is a retract of l_{q}^{s}(L_{Φ}(ℝ^{N}; X)).

Definition 3.4. Assume that s ∈ ℝ and let Φ be an Orlicz function satisfy the Δ_{2}-condition. We define the Sobolev-Orlicz spaces W_{Φ}^{s}(ℝ^{N}; X) by

W_{Φ}^{s}(ℝ^{N}; X) = {f ∈ S′ : J^{s}f ∈ L_{Φ}(ℝ^{N}; X)}.  (13)

Here J^{s} is the Bessel operator .

We equip the space W_{Φ}^{s}(ℝ^{N}; X) with the norm

∥f∥_{W_{Φ}^{s}(ℝ^{N};X)} = ∥J^{s}f∥_{L_{Φ}(ℝ^{N};X)}.

Definition 3.5. Let Ω ⊂ ℝ^{N} be an arbitrary domain.Further,let −∞ < s < ∞, 1 < p < ∞ and 1 ≤ q ≤ ∞. Then B_{pq}^{s}(Ω; X) is the restriction of B_{pq}^{s}(ℝ^{N}; X) to Ω,

∥f∥_{B_{pq}^{s}(Ω;X)} = inf_{g|_{Ω}=f} ∥g∥_{B_{pq}^{s}(ℝ^{N};X)}, g ∈ B_{pq}^{s}(ℝ^{N}; X).

And W_{p}^{s}(Ω; X) is the restriction of W_{p}^{s}(ℝ^{N}; X) to Ω,

∥f∥_{W_{p}^{s}(Ω;X)} = inf_{g|_{Ω}=f} ∥g∥_{W_{p}^{s}(ℝ^{N};X)}, g ∈ W_{p}^{s}(ℝ^{N}; X).

We shall now introduce two mappings S and R.The mapping S (coretraction) maps B_{pq}^{s}(Ω; X) to B_{pq}^{s}(ℝ^{N}; X).It is defined by

S(f) = lim_{j} fψ_{j}, f ∈ B_{pq}^{s}(Ω; X)

The mapping R (retraction) is given by

R(∑_{j} β_{j}) = ∑_{j} ψ_{j} ∗ f_{j},

where the mapping I (coretraction) maps B_{pq}^{s}(ℝ^{N}; X) to l_{q}^{s}(L_{p}(ℝ^{N}; X)) is defined by

I(g) = {g ∗ ϕ_{j}}_{j=1}^{∞} = (g_{j})_{j=1}^{∞}, g ∈ B_{pq}^{s}(ℝ^{N}; X),

and the mapping P (retraction) is given by

P (α) = ∑_{j} ψ_{j} ∗ g_{j}, α = (g_{j})_{j=1}^{∞} = ∑_{j} β_{j}.

Therefore, the mapping IS (coretraction) maps B_{pq}^{s}(Ω; X) to l_{q}^{s}(L_{p}(ℝ^{N}; X)) and the mapping RP is the corresponding retraction.

Theorem 3.6. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain. If −∞ < s < ∞, 1 < p < ∞ and 1 ≤ q ≤ ∞, then the restriction operator from B_{pq}^{s}(ℝ^{N}; X) onto B_{pq}^{s}(Ω; X) is a retraction. If r ∈ ℕ, then there exists a corresponding coretraction independent of 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < r.

Proof : See H.Trieble [T p. 311]

Remark 3.7. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, s ∈ ℝ, 1 ≤ q ≤ ∞ and Φ be an Orlicz function satisfy the Δ_{2}-condition. Then B_{Φq}^{s}(Ω; X) is a retract of l_{q}^{s}(L_{Φ}(Ω; X))

Theorem 3.8. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, Φ be an Orlicz function which satisfies, together with its conjugate Φ^{∗}, the Δ_{2}-condition. Then there exist p_{0} and p_{1}, with 1 < p_{0}, p_{1} < ∞, such that

⟨B_{p_{0}q}^{s}(Ω; X), B_{p_{1}q}^{s}(Ω; X)⟩_{ρ,p} = B_{Φq}^{s}(Ω; X),  (14)

where ρ is given by (10) .

Proof : Using remark (3.7), remark (2.6) and theorem (2.7), we have

⟨B_{p_{0}q}^{s}(Ω; X), B_{p_{1}q}^{s}(Ω; X)⟩_{ρ,p} = ⟨RP (l_{q}^{s}(L_{p_{0}}(Ω; X))), RP (l_{q}^{s}(L_{p_{1}}(Ω; X)))⟩_{ρ,p}

= RP (l_{q}^{s}(⟨L_{p_{0}}(ℝ^{N}; X), L_{p_{1}}(Ω; X)⟩_{ρ,p}))
= RP (l_{q}^{s}(L_{Φ}(Ω; X))) = B_{Φq}^{s}(Ω; X).

Theorem 3.9. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain , 1 < p, q < ∞, and s ∈ ℝ. Then we have, for Banach spaces X_{0}, X_{1}

⟨B_{pq}^{s}(X_{0}), B_{pq}^{s}(X_{1})⟩_{ρ,p} = B_{pq}^{s}(⟨X_{0}, X_{1}⟩_{ρ,p}).  (15)

Proof : Using remark (3.7) , remark (2.6) and theorem (2.5), we have

⟨B_{pq}^{s}(X_{0}), B_{pq}^{s}(X_{1})⟩_{ρ,p} = ⟨RP (l_{q}^{s}(L_{p}(X_{0}))), RP (l_{q}^{s}(L_{p}(X_{1})))⟩_{ρ,p})

= RP (l_{q}^{s}(⟨L_{p}(X_{0}), L_{p}(X_{1})⟩_{ρ,p})
= RP (l_{q}^{s}(L_{p}(⟨X_{0}, X_{1}⟩_{ρ,p})))

= B_{pq}^{s}(⟨X_{0}, X_{1}⟩_{ρ,p}).

Theorem 3.10. Let 1 < p, q, u_{0}, u_{1}, v < ∞ and s ∈ ℝ. Then we have

⟨B_{pq}^{s}(B_{u_{0}v}^{σ}(Ω; X)), B_{pq}^{s}(B_{u_{1}v}^{σ}(Ω; X))⟩_{ρ,q} = B_{pq}^{s}(B_{Φv}^{σ}(Ω; X)),  (16)

with,

ρ(τ) = τ^{\frac{u_{1}}{u_{1}−u_{0}}} Φ^{−1}(τ^{\frac{u_{0}u_{1}}{u_{0}−u_{1}}} ).

Proof : Using theorem (3.9) and theorem (3.8) ,we have

⟨B_{pq}^{s}(B_{u_{0}v}^{σ}(Ω; X)), B_{pq}^{s}(B_{u_{1}v}^{σ}(Ω; X))⟩_{ρ,q}

= B_{pq}^{s}(⟨B_{u_{0}v}^{σ}(Ω; X), B_{u_{1}v}^{σ}(Ω; X))⟩_{ρ,q}) = B_{pq}^{s}(B_{Φv}^{σ}(Ω; X)).

4. Schatten-Orlicz classes

Definition 4.1. Let Φ be an Orlicz function as before. We define for 1 ≤ q < ∞, the Lorentz-Orlicz spaces l_{Φq} as follows

l_{Φq} = \{x = (x_{n})_{n} : ∥x∥_{l_{Φq}} := inf (c : (∑_{n=1}^{∞} Φ(\frac{|x_{n}|}{c})^{q})^{1/q} ≤ 1) < ∞\} ,  (17)

and,

l_{Φ∞} = \{x = (x_{n})_{n} : ∥x∥_{l_{Φ∞}} := inf (c : sup_{n} Φ(\frac{|x_{n}|}{c}) ≤ 1) < ∞\} .  (18)

Definition 4.2. Let Φ be an Orlicz function . We define for 1 ≤ q ≤ ∞, the Schatten-Orlicz classes S_{Φq}^{s} as follows

S_{Φq}^{s}(X) := {T ∈ L(X) : (s_{n}(T))_{n} ∈ l_{Φq}}.  (19)

Theorem 4.3. Let 1 < p_{0}, p_{1} < ∞. Then we have

⟨S_{p_{0}q}^{s}(X), S_{p_{1}q}^{s}(X)⟩_{ρ,p} ⊂ S_{Φq}^{s}(X),  (20)

where ρ is given by (10).

Proof : Let T ∈ ⟨S_{p_{0}q}^{s}(X), S_{p_{1}q}^{s}(X)⟩_{ρ,p} ⊂ ⟨S_{p_{0}∞}^{s}(X), S_{p_{1}∞}^{s}(X)⟩_{ρ,p} with ∥T ∥_{⟨S_{p_{0}∞}^{s}(X),S_{p_{1}∞}^{s}(X)⟩_{ρ,p}} ≤ 1 and let (T_{n})_{n} be an admissible sequence for T such that

max_{k=0,1} sup_{J} ∥∑_{n∈J} r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}T_{n}∥_{L_{p}([0,1],S_{p_{k}∞}^{s}(X))} < 1.

Since the sequence (r̃_{n}(.)\frac{2^{kn}}{ρ(2^{n})}T_{n})_{n∈ℤ} is L_{p}([0, 1], S_{p_{k}∞}^{s})-summable , given ε > 0, there exist a finite subset J_{ε} ⊂ ℕ such that for all J′ ⊂ ℕ with J′ ∩ J_{ε} = ∅ we have

∥(∑_{n∈J′}(\frac{2^{kn}}{ρ(2^{n})}T_{n})^{2})^{1/2}∥_{S_{p_{k}∞}^{s}(X)} ≤ ε^{1/p_{k}},

Now setting

A_{k}(m) = max_{n} n^{1/p_{k}} ∥(∑_{n∈J′}(\frac{2^{kn}}{ρ(2^{n})} s_{m}(T_{n}))^{2})^{1/2}∥_{X} ≤ ε^{1/p_{k}} and

A(m) = max_{k=0,1} (\frac{A_{k}(m)}{2ε})^{p_{k}} .

From the fact, \frac{ρ(t)}{t} is non-increasing and lemma (2.4)

\frac{1}{2ε}∥ ∑_{n∈J′} s_{m}(T_{n})∥_{X} ≤ c\frac{A_{0}(m)}{2ε} ρ (\frac{A_{1}(m)}{A_{0}(m)})

≤ cA(m)^{\frac{1}{p_{0}}} ρ(A(m)^{\frac{1}{p_{1}} − \frac{1}{p_{0}}} ) ≤ cΦ^{−1}(A(m)).

Therefore,

Φ (\frac{1}{2cε}∥ ∑_{n∈J′} s_{m}(T_{n})∥_{X}) ≤ A(m)

≤ \frac{1}{2} ((\frac{A_{0}(m)}{ε})^{p_{0}} + (\frac{A_{1}(m)}{ε})^{p_{1}}) ≤ 1.

Therefore,∥ ∑_{n∈J′} T_{n}∥_{S_{Φ∞}(X)} ≤ 2cε, the sequence (T_{n})_{n} satisfies the Cauchy condition in S_{Φ∞}(X). Then, we have T ∈ S_{Φ∞}(X) ⇒ T ∈ S_{Φq}(X). Hence,

⟨S_{p_{0}q}^{s}(X), S_{p_{1}q}^{s}(X)⟩_{ρ,q} ⊂ S_{Φq}^{s}(X) with Φ^{−1}(t) = t^{\frac{1}{p_{0}}} ρ(t^{\frac{1}{p_{1}} − \frac{1}{p_{0}}} )

which equivalent to the relation (10).

5. Eigenvalue distribution of integral operators

Now, we derive our main results using the eigenvalue theorem for Besov spaces and the new interpolation results (interpolation of S_{pq}^{s}-spaces, and B_{pq}^{s}-spaces). We will prove that the eigenvalues of integral operators defined by kernels k ∈ B_{Φq}^{s}(B_{Ψv}^{σ}) belong to some Lorentz-Orlicz space l_{Φq}.

Theorem 5.1. (Eigenvalue theorem I) [K] Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, 1 ≤ p, q, u, v ≤ ∞ and s, σ > 0 with s + σ > N(\frac{1}{p} + \frac{1}{u} − 1). Define t by \frac{1}{t} := \frac{s+σ}{N} + \frac{1}{max(2,u′)}. Then the eigenvalues of any kernel k ∈ B_{pq}^{s}(Ω; B_{uv}^{σ}(Ω)) belong to the Lorentz sequence space l_{t,q} with

∥(λ_{n}(k))_{n∈ℕ}∥_{l_{t,q}} ≤ c∥k∥_{B_{pq}^{s}(B_{uv}^{σ})}.  (21)

The constant c depend only on the indices and the domain Ω.

Theorem 5.2 (K). Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, 1 ≤ p, q, u ≤ ∞ and s > 0 with s > N(\frac{1}{p} − \frac{1}{u′}). Define t by \frac{1}{t} := \frac{s}{N} + \frac{1}{max(2,u′)}. Then the eigenvalues sequence of any kernel k ∈ B_{pq}^{s}(Ω; L_{u}(Ω)) belong to the Lorentz sequence space l_{t,q} with

∥(λ_{n}(k))_{n∈ℕ}∥_{l_{t,q}} ≤ c_{p,q,u,s,Ω}∥k∥_{B_{pq}^{s}(L_{u})}.  (22)

Theorem 5.3 (K). Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain , 1 ≤ p, u, v ≤ ∞ and σ > 0 with σ > N(\frac{1}{u} − \frac{1}{max(p′,v′)}). Define t by \frac{1}{t} := \frac{σ}{N} + \frac{1}{max(2,u′)}. Then the eigenvalues of any kernel k ∈ L_{p}(Ω; B_{uv}^{σ}(Ω)) belong to the Lorentz sequence space l_{t,v} with

∥(λ_{n}(k))_{n∈ℕ}∥_{l_{t,v}} ≤ c_{p,u,v,σ,Ω}∥k∥_{L_{p}(B_{uv}^{σ})}.  (23)

Remark 5.4. (i) As shown by Besov [B] , the Sobolev spaces W_{u}^{s}(Ω) of integer order s ∈ ℕ satisfy the inclusion W_{u}^{s}(Ω) ⊂ B_{uv}^{s}(Ω) with v := max(2, u). Hence the eigenvalues of the kernel operator defined in L_{p}(Ω) by any kernel k ∈ L_{p}(Ω; W_{u}^{s}(Ω)) belong to l_{tv} for \frac{1}{t} := \frac{s}{N} + \frac{1}{max(2,u′)} provided that s > N(\frac{1}{u} − \frac{1}{max(p′,v′)}). (ii) For v ≥ p, r ∈ ℕ,we have

W_{p}^{r}(Ω; X_{θ,v}) ⊂ (W_{p}^{r}(Ω; X_{0}), W_{p}^{r}(Ω; X_{1}))_{θ,v}.

This yields that the eigenvalues of kernels k ∈ W_{p}^{r}(Ω; B_{uv}^{σ}(Ω)) belong to l_{t}, v, \frac{1}{t} := \frac{σ+r}{N} + \frac{1}{max(2,u′)}, provided that σ + r satisfies σ + r > N(\frac{1}{u} − \frac{1}{max(p′,v′)}). Using the above inclusion W_{p}^{s}(Ω) ⊂ B_{uv}^{s}(Ω) for v := max(u, 2), one finds:

If r, s ∈ ℕ with r + s > N(\frac{1}{u} − \frac{1}{max(p′, v′)}) and k ∈ W_{p}^{r}(Ω; W_{u}^{s}(Ω)),

the eigenvalues of T_{k} (e.g., in L_{p}(Ω) if p ≥ u′) belong to l_{t,v} for

\frac{1}{t} := \frac{r + s}{N} + \frac{1}{max(2, u′)} and v := max(2, u).

Theorem 5.5. (The eigenvalue theorem II.) Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, 1 ≤ p, q, u_{0}, u_{1}, v ≤ ∞ , u_{1} < u_{0} and s + σ > 0 with s + σ > N(\frac{1}{p} + \frac{1}{u_{1}} − 1). Define

Ψ^{−1}(τ) = τ^{\frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{min(2,u_{0})}} Φ^{−1}(τ).  (24)

Then the eigenvalues of any kernel k ∈ B_{pq}^{s}(Ω; B_{Φv}^{σ}(Ω)) belong to the Lorentz-Orlicz space l_{Ψq} with

∥(λ_{n}(k))_{n∈ℕ}∥_{l_{Ψ,q}} ≤ c∥k∥_{B_{pq}^{s}(B_{Φv}^{σ})}.  (25)

Proof : Using the eigenvalue theorem (5.1), we have, for s + σ > N(\frac{1}{p} + \frac{1}{u_{1}} − 1), i = 0, 1

B_{pq}^{s}(B_{u_{i}v}^{σ}) ⊂ S_{t_{i},q}^{s} with \frac{1}{t_{i}} := \frac{s + σ}{N} + \frac{1}{max(2, u′_{i})}.  (26)

By theorem (3.10), we have

⟨B_{pq}^{s}(B_{u_{0}v}^{σ}(X)), B_{pq}^{s}(B_{u_{1}v}^{σ}(X))⟩_{ρ,q} = B_{pq}^{s}(B_{Φv}^{s}(X)),  (27)

with,

ρ(τ) = τ^{\frac{u_{1}}{u_{1}−u_{0}}} Φ^{−1}(τ^{\frac{u_{0}u_{1}}{u_{0}−u_{1}}} ).  (28)

(i) The case u_{1} < u_{0} < 2 (2 < u′_{0} < u′_{1}):

Using theorem (4.3),for \frac{1}{t_{i}} := \frac{s + σ}{N} + \frac{1}{u′_{i}} , we have,

⟨S_{t_{0}q}^{s}(X), S_{t_{1}q}^{s}(X)⟩_{ρ,q} = S_{Ψq}^{s}(X),  (29)

with,

ρ(τ) = τ^{\frac{t_{0}}{t_{0}−t_{1}}} Ψ^{−1}(τ^{\frac{t_{0}t_{1}}{t_{1}−t_{0}}} ).  (30)

Let β = \frac{t_{0}t_{1}}{t_{1} − t_{0}} ⇒ \frac{1}{β} = \frac{1}{t_{0}} − \frac{1}{t_{1}} = \frac{u′_{1} − u′_{0}}{u′_{0}u′_{1}} ⇒ β = \frac{u′_{0}u′_{1}}{u′_{1} − u′_{0}} ,

and γ = \frac{t_{0}}{t_{0} − t_{1}} = −\frac{1}{t_{1}} β = (\frac{s + σ}{N} + \frac{1}{u′_{1}} )(\frac{u′_{0}u′_{1}}{u′_{0} − u′_{1}} ).

Consequently, we have

ρ(τ) = τ^{γ} Ψ^{−1}(τ^{β}).  (31)

From (27) , (28) , (29) , (31) and the fact, that if X_{k} ⊂ Y_{k}, (k = 0, 1) ⇒ ⟨X_{0}, X_{1}⟩_{ρ,p} ⊂ ⟨Y_{0}, Y_{1}⟩_{ρ,p}, we have

B_{pq}^{s}(B_{Φv}^{s}(X)) ⊂ S_{Ψq}^{s}(X) with Ψ^{−1}(τ^{β}) = τ^{\frac{u_{1}}{u_{1}−u_{0}} −γ} Φ^{−1}(τ^{\frac{u_{0}u_{1}}{u_{0}−u_{1}}} ).

But,β = \frac{u′_{0}u′_{1}}{u′_{1}−u′_{0}} = \frac{u_{0}u_{1}}{u_{0}−u_{1}} and

\frac{u_{1}}{u_{1} − u_{0}} − γ = \frac{−β}{u_{0}} + β(\frac{s + σ}{N} + \frac{1}{u′_{1}} ) = β(\frac{s + σ}{N} + \frac{1}{u′_{1}} − \frac{1}{u_{0}} ).

Then, Ψ^{−1}(τ^{β}) = τ^{β( \frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{u_{0}} )} Φ^{−1}(τ^{β}).

Take τ^{β} = ξ , we find

Ψ^{−1}(ξ) = ξ^{\frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{u_{0}}} Φ^{−1}(ξ).  (32)

(ii) The case u_{1} < 2 < u_{0} (u′_{0} < 2 < u′_{1}): Using the same method as above, we can find

Ψ^{−1}(ξ) = ξ^{\frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{2}} Φ^{−1}(ξ).  (33)

Combining (32) and (33), we have

Ψ^{−1}(τ) = τ^{\frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{min(2,u_{0})}} Φ^{−1}(τ).

Then, we readily get the desired results.

Theorem 5.6. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, 1 ≤ p, q, u_{0}, u_{1} ≤ ∞, u_{1} < u_{0} and s > 0 with s > N(\frac{1}{p} + \frac{1}{u_{1}} − 1). Define

Ψ^{−1}(τ) = τ^{\frac{s}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{min(2,u_{0})}} Φ^{−1}(τ).  (34)

Then the eigenvalues of any kernel k ∈ B_{pq}^{s}(Ω; L_{Φ}(Ω)) belong to the Lorentz-Orlicz space l_{Ψq} with

∥(λ_{n}(k))_{n∈ℕ}∥_{l_{Ψ,q}} ≤ c∥k∥_{B_{pq}^{s}(L_{Φ})}.  (35)

Proof : Replacing s + σ by s in theorem (5.5) and using the same method, we get directly the desired results

Theorem 5.7. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, 1 ≤ p, u_{0}, u_{1}, v ≤ ∞, u_{1} < u_{0} and σ > 0 with σ > N(\frac{1}{u_{1}} − \frac{1}{max(p′,v′)}). Define

Ψ^{−1}(τ) = τ^{\frac{σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{min(2,u_{0})}} Φ^{−1}(τ).  (36)

Then the eigenvalues of any kernel k ∈ L_{p}(Ω; B_{Φv}^{σ}(Ω)) belong to the Lorentz-Orlicz space l_{Ψv} with

∥(λ_{n}(k))_{n∈ℕ}∥_{l_{Ψ,v}} ≤ c∥k∥_{L_{p}(B_{Φv}^{σ})}.  (37)

Proof : Using theorem (5.3) and the proof of theorem (5.5) .

Remark 5.8. (i) As shown by Besov [B], The Sobolev spaces W_{u}^{σ} of integer order σ ∈ ℕ satisfy the inclusion W_{u}^{σ} ⊂ B_{uv}^{σ} with v := max(2, u). Hence, using theorem (5.5) together with remark (5.4), we can prove that, the eigenvalues of any kernel k ∈ W_{p}^{s}(Ω; W_{Φ}^{σ}(Ω)) belong to the Lorentz-Orlicz space l_{Ψ,v} with

Ψ^{−1}(τ) = τ^{\frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{min(2,u_{0})}} Φ^{−1}(τ),

provided that s + σ > N(\frac{1}{u_{1}} − \frac{1}{max(p′,v′)}). (ii) Also, we can prove that, k ∈ L_{p}(Ω; W_{Φ}^{σ}(Ω)) belong to the Lorentz-Orlicz space l_{Ψ,v} with

Ψ^{−1}(τ) = τ^{\frac{s+σ}{N} +_{\frac{1}{u′_{1}}} − \frac{1}{min(2,u_{0})}} Φ^{−1}(τ),

provided that s > N(\frac{1}{u_{1}} − \frac{1}{max(p′,v′)}).

Corollary 5.9. Let Ω ⊂ ℝ^{N} be a bounded C^{∞}-domain, 1 ≤ p, q, u_{0}, u_{1}, v ≤ ∞, u_{1} < u_{0} and s + σ > 0 with s + σ > N(\frac{1}{p} + \frac{1}{u_{1}} − 1). Define t by

\frac{1}{t} := \frac{s + σ}{N} + \frac{1}{u′_{1}} − \frac{1}{min(2, u_{0})} + \frac{1}{u} with \frac{1}{u} = \frac{1 − θ}{u_{0}} + \frac{θ}{u_{1}} ,

0 < θ < 1. Then the eigenvalues of any kernel k ∈ B_{pq}^{s}(Ω; B_{uv}^{σ}(Ω)) belong to the Lorentz-Orlicz space l_{tq} with

∥(λ_{j}(k))_{j∈ℕ}∥_{l_{tq}} ≤ c ∥k∥_{B_{pq}^{s}(B_{uv}^{σ})}.  (38)

Proof : Consider ρ(t) = t^{θ}.Then using the prove of theorem (5.5),we get the desired result.

Acknowledgment

The results presented in this paper are part of the dissertation of the second author. He would like to thank H. König for advice during the preparation of the dissertation

References

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[FG] D. L. Fernandez and J. B. Garcia, Interpolation of Orlicz-valued spaces, Stud. Math. 99 (1991) 23–39.

[GP] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Stud. Math. 60 (1977) 33–59.

[K] H. König, Eigenvalue distribution of compact operators, 1985.

[T] H. Triebel, Interpolation theory, Function spaces, Differential operators, 1994.