ON THE APPROXIMATION NUMBERS OF HARDY-TYPE OPERATORS

Desmond J. Harris

Introduction

While attending last year’s Edinburgh Workshop on elliptic PDE’s and related areas of harmonic analysis Michael Solomyak suggested that David Edmunds, Desmond Evans and I should try to develop the results of our paper [1] so that they might be more easily applied and their connections with related results of his and Birman in various papers, in particular in [2], be more readily established. I shall describe our so far limited success in this direction. It is necessary first to rehearse some of the definitions and theorems of [1].

1. Preliminaries (see [1])

We are concerned with the problem of estimating the approximation numbers a_{n}(T) of the weighted Hardy operator T from L^{p}(0, ∞) into itself given by

T f(x) = v(x) ∫_{0}^{x} f(t)u(t) dt,

where u ∈ L_{loc}^{p′}(0, ∞), v ∈ L^{p}(0, ∞), 1/p + 1/p′ = 1 and

a_{n}(T) = inf{∥T − P_{n}∥: P_{n} ∈ F_{n}},

where F_{n} is the set of bounded linear operators on L^{p}(0, ∞) of rank less then n.

1.1. Definitions. If I = (a, b) is an interval in ℝ^{+} = (0, ∞) we make the following definitions:

ℓ(I; f) = ∫_{I} ∫_{I} |v(x)v(y) ∫_{x}^{y} f(t)u(t) dt|^{p} dx dy, f ∈ L^{p}(I),

216 DESMOND J. HARRIS

L(I; u, v) = sup \{\frac{ℓ(I; f)}{μ(I)} : ∥f∥_{L^{p}(I)} ≤ 1\}^{1/p} , where dμ = v^{p}(x) dx,

A_{c} = sup_{a≤s≤c} ∥u∥_{L^{p′} (s,c)}∥v∥_{L^{p}(a,s)},

B_{c} = sup_{c≤s≤b} ∥u∥_{L^{p′} (c,s)}∥v∥_{L^{p}(s,b)},

J(I) = max(A_{c}, B_{c}), where μ(a, c) = \frac{1}{2}μ(I)

or J(I) = inf_{a≤c≤b} max(A_{c}, B_{c}),

K(I) = sup{∥F − F_{I}∥_{L^{p}(I,μ)} : ∥f∥_{L^{p}(I)} ≤ 1},

where F (x) = ∫_{0}^{x} f(t)u(t) dt and F_{I} = \frac{1}{μ(I)} ∫_{I} F (x) dμ(x).

1.2. Lemma. For 1 ≤ p < ∞, p ≠ 2,

\frac{1}{2}J(I) ≤ K(I) ≤ \{ \begin{matrix}2J(I) (p = 1) \\ 2p^{1/p}p^{′1/p′}J(I) (p > 1), \end{matrix}

and if p = 2, then

\frac{1}{\sqrt{2}}J(I) ≤ K(I) ≤ 2J(I).

Also

K(I) = L(I)/\sqrt{2} (p = 2),

\frac{1}{2}L(I) ≤ K(I) ≤ L(I) (p ≠ 2).

1.3. Definition. The sequence defined by

c_{0} = 0, c_{k+1} = inf{t: L(c_{k}, t) > ε}

is called the (ε, L)-sequence.

There are two possibilities:

(i) The sequence is finite of length N + 1 so that

0 = c_{0} < c_{1} < ⋯ < c_{N} < c_{N+1} = ∞

and

L(c_{k}, c_{k+1}) = ε (k = 0, 1, …, N − 1), L(c_{N}, c_{N+1}) ≤ ε. (ii) The sequence is infinite and then L(c_{k}, c_{k+1}) = ε for all k.

1.4. Theorem. (i) T is bounded if and only if L(0, ∞) < ∞.

(ii) If N + 1 is the length of the (ε, L)-sequence, then

a_{N+2}(T) ≤ ε, a_{N}(T) ≥ \frac{1}{4}ε if p ≠ 2, 1 ≤ p < ∞

and

a_{N+2}(T) ≤ \frac{1}{\sqrt{2}}ε, a_{N}(T) ≥ \frac{1}{\sqrt{2}}ε if p = 2.

2. A connection with ∫_{0}^{x} u(t)v(t) dt

We need two lemmas. Throughout we suppose that u ∈ L_{loc}^{p′}(0, ∞), v ∈ L^{p}(0, ∞).

2.1. Lemma. If the weights u_{0}, v_{0} are non-negative and constant on I, then L(I; u_{0}, v_{0}) = α_{p}|I|u_{0}v_{0}, where α_{p} = L((0, 1); 1, 1).

2.2. Lemma. If v(x) > 0 almost everywhere and u is non-negative, then L(I; u, v) = L(I^{∗}; w, 1), where μ(x) = ∫_{0}^{x} v^{p}, I^{∗} = (μ(a), μ(b)) and w = (u ∘ μ^{−1})/(v ∘ μ^{−1})^{p−1} ∈ L_{loc}^{p′}(I^{∗}) and ∫_{I^{∗}} w = ∫_{I} uv.

The proofs follow easily by direct substitution.

2.3. Definition. For an (ε, L)-sequence c_{0} < c_{1} < ⋯ < c_{N+1} = ∞ define N(x, ε) = max{k: c_{k+1} ≤ x}.

Then we have

2.4. Theorem. The equality

lim_{ε→+0} N(x, ε)ε = α_{p} ∫_{0}^{x} uv

holds provided u and v are continuous and strictly positive on [0, ∞) and 0 ≤ x < ∞.

The proof follows from the definition of the Riemann integral and properties of L, and Lemma 2.1, using Lemma 2.2 to transpose the problem to one in which u, v are replaced by w, 1, and ℝ^{+} by [0, μ(∞)).

2.5. Corollary. The same conclusion holds if u ∈ L_{loc}^{p′}(0, ∞), v ∈ L^{p}(0, ∞),

v > 0 almost everywhere, and ess inf_{0≤t≤x} \frac{u(t)}{v(t)^{p−1}} > 0.

This follows by an argument depending on the density of the continuous functions in L^{p}(0, x), L^{p′}(0, x) and on continuity on L^{p}(0, x) × L^{p′}(0, x) of the functionals involved.

218 DESMOND J. HARRIS

2.6. Corollary. If lim inf_{N→∞} Na_{N}(T) < ∞ then uv ∈ L^{1}(0, ∞). Proof. By Theorem 1.4, α_{p} ∫_{0}^{x} uv = lim_{ε→+0} εN(x, ε) ≤ lim inf_{ε→+0} εN(∞, ε) ≤

4 lim inf_{N→∞} Na_{N}(T). □

3. Restriction to the case of non-decreasing w Now suppose u/v^{p−1} is non-decreasing, i.e., w is non-decreasing on the

interval [0, μ(∞)). We have 3.1. Lemma. If w is non-decreasing then

sup_{a≤x≤b}{w(x)(b − x)} ≤ p^{1/p}p′^{1/p′} sup_{a≤y≤b} {(∫_{a}^{y} w^{p′})^{1/p′} (b − y)^{1/p}}

and

sup_{a≤y≤b} \{(∫_{a}^{y} w^{p′})^{1/p′} (b − y)^{1/p}\} ≤ (\frac{p}{p′} )^{1/p′} sup_{a≤x≤b}\{w(x)(b − x)\}.

Proof. Since w is non-decreasing,

w(x)(y − x)^{1/p′} ≤ (∫_{x}^{y} w^{p′})^{1/p′} ≤ (∫_{a}^{y} w^{p′})^{1/p′} if x ≤ y, and so

sup_{a≤y≤b} {(∫_{a}^{y} w^{p′})^{1/p′} (b − y)^{1/p}} ≥ w(x) sup_{x≤y≤b}{(y − x)^{1/p′}(b − y)^{1/p}}

= (\frac{1}{p})^{1/p} (\frac{1}{p′} )^{1/p′} w(x)(b − x).

Conversely,

(∫_{a}^{y} w^{p′})^{1/p′} ≤ [ sup_{a≤x≤b}\{w(x)(b − x)\}] [∫_{a}^{y} \frac{1}{(b − x)^{p′}} dx]^{1/p′} ,

and

∫_{a}^{y} \frac{1}{(b − x)^{p′}} dx ≤ \frac{1}{p′ − 1} \frac{1}{(b − y)^{p′−1}} .

Thus,

sup_{a≤y≤b} \{(∫_{a}^{y} w^{p′})^{1/p′} (b − y)^{1/p}\} ≤ (\frac{p}{p′} )^{1/p′} sup_{a≤x≤b}\{w(x)(b − x)\}.

□

3.2. Corollary. L(I; u, v) = O(∫_{I} uv) uniformly in I provided u/v^{p−1} is non-decreasing.

Proof. By Lemma 1.2, we have L(I; u, v) ≤ c(p)J(I; u, v). It suffices to use Lemma 3.1 and the fact that ∫_{I^{∗}} w = ∫_{I} uv. □

The proof of the next theorem is then a simple consequence of Theorem 2.4 and Corollary 2.6.

3.3. Theorem. If u/v^{p−1} is non-decreasing on [0, ∞) then

lim_{ε→+0} εN(∞, ε) = α_{p} ∫_{0}^{∞} uv

(with ∞ a possible value for each).

3.4. Corollary. If p ≠ 2, then

\frac{1}{4} ≤ lim inf_{N→∞} \frac{Na_{N}(T)}{α_{p} ∫_{0}^{∞} uv} ≤ lim sup_{N→∞} \frac{Na_{N}(T)}{α_{p} ∫_{0}^{∞} uv} ≤ 1

and if p = 2, then

lim_{N→∞} Na_{N}(T) = \frac{α_{p}}{\sqrt{2}} ∫_{0}^{∞} uv.

This follows directly from Theorem 1.4 (if correctly interpreted for the ∞ case). Note it is well-known from Sturm-Liouville theory that α_{2} = 1/π.

A simple example. Let u(x) = e^{Ax}, v(x) = e^{−Bx}, where A < B. Then for p = 2,

a_{N} ∼ \frac{1}{Nπ\sqrt{2}} ∫_{0}^{∞} e^{(A−B)x} dx = \frac{1}{N(B − A)π\sqrt{2}}

as N → ∞, and for p ≠ 2,

\frac{B − A}{4α_{p}} ≤ lim inf_{N→∞} \frac{a_{N}}{N} ≤ lim sup_{N→∞} \frac{a_{N}}{N} ≤ \frac{B − A}{α_{p}}

I should like to conclude by thanking the organizers of this workshop for their invitation here and for many past kindness.

References

[1] D. E. Edmunds, W. D. Evans and D. J. Harris, Approximation numbers of certain Volterra integral operators, J. London Math. Soc. 37, no. 2, 471–489.

[2] M. Birman and M. Solomyak, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk 32, no. 1, 17–84 ((Russian)); English translation in Russian Math. Surveys 32 (1977), no. 1.