The Gaussian Cotype of Operators from C(K)

S.J. Montgomery-Smith

Abstract We show that the canonical embedding C(K) → L_{Φ}(μ) has Gaussian cotype p, where μ is a Radon probabilty measure on K, and Φ is an Orlicz function equivalent to t^{p}(log t)^{\frac{p}{2}} for large t.

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In [6], I showed that the Gaussian cotype 2 constant of the canonical embedding l_{∞}^{N} → L_{2,1}^{N} is bounded by log log N. Talagrand [9] showed that this embedding does not have uniformly bounded cotype 2 constant. In fact, a careful study of his proof yields that the cotype 2 constant is bounded below by \sqrt{log log N}. In this paper, we will show that this is the correct value for the Gaussian cotype 2 constant of this operator. However, we will show this via a different result, which we will give presently. First, let us define our terms.

We will write Φ_{p} for an Orlicz function such that Φ_{p}(t) ≈ t^{p}(log t)^{\frac{p}{2}} for large t.

For any bounded linear operator T : X → Y, where X and Y are Banach spaces, and any 2 ≤ p < ∞, we say that T has Gaussian cotype p if there is a number C < ∞ such that for all sequences x_{1}, x_{2}, … ∈ X we have

E ∥∑_{s=1}^{∞} γ_{s}x_{s}∥ ≥ C^{−1} (∑_{s=1}^{∞} ∥T x_{s}∥^{p})^{\frac{1}{p}} .

(Here, as elsewhere, γ_{1}, γ_{2}, … denote independent N(0, 1) Gaussian random variables.) We call the least value of C the Gaussian cotype p constant of T, and denote it by β^{(p)}(T).

Throughout this paper, we shall use the letter c to denote a positive finite constant, whose value may change with each occurence. We shall write A ≈ B to mean A ≤ c B and B ≤ c A.

Theorem 1. Let μ be a Radon probability measure on a compact Hausdorff topological space K, and let 2 ≤ p < ∞. Then the canonical embedding C(K) → L_{Φ_{p}}(μ) has Gaussian cotype p.

Finding the Gaussian cotype p constant of an operator from C(K) involves finding lower bounds for the quantity E ∥∑_{s=1}^{∞} γ_{s}x_{s}∥_{∞}, where x_{1}, x_{2}, … ∈ C(K). In fact, since we really only need to consider finite sequences x_{1}, x_{2}, …, x_{S} ∈ C(K), in order to prove Theorem 1, it is sufficient to show that the Gaussian cotype p constant of the canonical embedding C(K) → L_{Φ_{p}}(μ) is uniformly bounded over all finite K. Now we see that we are trying to find lower bounds for the supremum of the finite Gaussian process, sup_{ω∈K} |Γ_{ω}|, where Γ_{ω} = ∑_{s=1}^{∞} γ_{s}x_{s}(ω). Hence we can apply the following result due to Talagrand [8].

Theorem 2. Let ( Γ_{ω} : ω ∈ K) be a finite Gaussian process.

i) Let

V_{1} = E ( sup_{ω∈K} |Γ_{ω}|) .

ii) Let V_{2} be the infimum of

(sup_{t≥1} \sqrt{1 + log t} (E |Y_{t}|^{2})^{\frac{1}{2}} ) ( sup_{ω∈K} ∑_{t=1}^{∞} |α_{t}(ω)|)

over all Gaussian processes (Y_{t})_{t=1}^{∞} and over all sequences (α_{t})_{t=1}^{∞} of functions on K such that Γ_{ω} = ∑_{t=1}^{∞} α_{t}(ω)Y_{t}.

Then V_{1} ≈ V_{2}.

We can rewrite this corollary in the following way. First, let us define the following spaces (here we are assuming K is finite).

G = { (x_{s} ∈ C(K))_{s=1}^{∞} : ∥(x_{s})∥_{G} = ∥∑_{s=1}^{∞} γ_{s}x_{s}∥_{∞} < ∞} ,
C(K, l_{1}) = { (α_{t} ∈ C(K))_{t=1}^{∞} : ∥(α_{t})∥_{C(K,l_{1})} = ∥∑_{t=1}^{∞} |α_{t}|∥_{∞} < ∞} ,
Y = \{ (y_{t} ∈ l_{2})_{t=1}^{∞} : ∥(y_{t})∥_{Y} = sup_{t≥1} \sqrt{1 + log t} ∥y_{t}∥_{2} < ∞\} .

Let m : C(K, l_{1}) × Y → G be the bilinear map m((α_{t}), (y_{t})) = (x_{s}), where

x_{s} = ∑_{t=1}^{∞} y_{t}(s)α_{t}.

Corollary 3. The map m has the following two properties:

i) m is bounded; ii) m is open, that is, if ∥(x_{s})∥_{G} ≤ 1, then there are ∥(α_{t})∥_{C(K,l_{1})} ≤ c and ∥(y_{t})∥_{Y} ≤ c

such that m(((α_{t}), (y_{t})) = (x_{s}).

Proof: This is just restating Theorem 2, setting Γ_{ω} = ∑_{s=1}^{∞} γ_{s}x_{s}(ω), and Y_{t} = ∑_{s=1}^{∞} γ_{s}y_{t}(s).

From this we obtain the following corollary, for which we first give a definition.

Definition. If 2 ≤ p < ∞, and T : C(K) → Y is a bounded linear map, where K is a finite Hausdorff space, and Y is a Banach space, then we set

H^{(p)}(T) = sup \{(∑_{s=1}^{∞} ∥T x_{s}∥^{p})^{\frac{1}{p}} \} ,

where the supremum is over all x_{s} = ∑_{t=1}^{∞} y_{t}(s)α_{t}, with α_{1}, α_{2}, … pairwise disjoint elements of the unit ball of C(K), and ∥(y_{t})∥_{2} ≤ \frac{1}{\sqrt{1+log t}} for each t ≥ 1.

Corollary 4. For any 2 ≤ p < ∞, and for any bounded linear operator T : C(K) → Y, where K is a finite Hausdorff space, and Y is a Banach space, we have

H^{(p)}(T) ≈ β^{(p)}(T).

Proof: This follows straight away from Corollary 3 and the following lemma.

Lemma 5. Let B be the set of (α_{t}) ∈ C(K, l_{1}) such that the α_{t} are pairwise disjoint elements of the unit ball of C(K). Then the closed convex hull of B is the unit ball of C(K, l_{1}).

Proof: See [5], Lemma 4 or [3], Proposition 14.4.

Now we are almost in a position to prove Theorem 1; we just need the following properties of L_{Φ_{p}}(μ).

Lemma 6. If μ is a Radon probability measure on a compact Hausdorff space K, then i) for any Borel subset I of K, we have ∥χ_{I}∥_{Φ_{p}} ≈ (μ(I))^{\frac{1}{p}} \sqrt{log \frac{1}{μ(I)}}; ii) the space L_{Φ_{p}} satisfies an upper p estimate.

Proof of Theorem 1: We want to show that H^{(p)}(C(K) → L_{Φ_{p}}(μ)) ≤ c, where μ is a probability measure on a finite Hausdorff space K. So consider (x_{s}), (α_{t}) and (y_{t}) as given in the definition of H^{(p)}(T). Then we need to show that

∑_{s=1}^{∞} ∥x_{s}∥_{Φ_{p}}^{p} ≤ c.

First note, by Lemma 6, that

∥x_{s}∥_{Φ_{p}}^{p} ≤ c ∑_{t=1}^{∞} y_{t}(s)^{p} ∥a_{t}∥_{Φ_{p}}^{p}

≤ c ∑_{t=1}^{∞} y_{t}(s)^{p}μ(I_{t}) (log \frac{1}{μ(I_{t})})^{\frac{p}{2}} ,

where I_{t} is the support of α_{t}. Hence

∑_{s=1}^{∞} ∥x_{s}∥_{Φ_{p}}^{p} ≤ c ∑_{t=1}^{∞} ∑_{s=1}^{∞} y_{t}(s)^{p}μ(I_{t}) (log \frac{1}{μ(I_{t})})^{\frac{p}{2}}

≤ c ∑_{t=1}^{∞} \frac{1}{(1 + log t)^{\frac{p}{2}}} μ(I_{t}) (log \frac{1}{μ(I_{t})})^{\frac{p}{2}} ,

since ∥y_{t}∥_{p} ≤ ∥y_{t}∥_{2} ≤ \frac{1}{\sqrt{1+log t}}. But now, splitting the sum into the two cases μ(I_{t}) ≥ \frac{1}{t^{2}} or μ(I_{t}) < \frac{1}{t^{2}} , we deduce that this sum is bounded by some universal constant.

Concluding Remarks We first note that there is a nice way to calculate the Orlicz norms ∥ · ∥_{Φ_{p}} provided by the following result of Bennett and Rudnick.

Theorem 7. If 1 ≤ p < ∞ and a ∈ R, then the Orlicz probability norm given by the function Θ(t) ≈ t^{p}(log t)^{a} (t large) is equivalent to the norm

∥x∥ = (∫_{0}^{1}(1 + log \frac{1}{t})^{a}x^{∗}(t)^{p} dt)^{\frac{1}{p}} ,

where x^{∗} is the non-increasing rearrangement of |x|.

Proof: See [1], Theorem D.

Thus we can now deduce the following result.

Theorem 8. The Gaussian cotype 2 constant of the canonical embedding l_{∞}^{N} → L_{2,1}^{N} is bounded by \sqrt{log log N}.

Proof: Let K = \{1, 2, …, N\}, and let μ be the measure μ(A) = \frac{|A|}{N}. Now notice that if x ∈ l_{∞}^{N} = C(K), then x^{∗}(t) is constant over 0 ≤ t ≤ \frac{1}{N}, and hence

∥x∥_{L_{2,1}^{N}} = \frac{1}{2} ∫_{0}^{1} \frac{x^{∗}(t)}{\sqrt{t}} dt

= \frac{x^{∗}(1/N)}{\sqrt{N}} + \frac{1}{2} ∫ 1_{\frac{1}{N}} \frac{x^{∗}(t)}{\sqrt{t}} dt

≤ (∫0^{\frac{1}{N}} (1 + log \frac{1}{t})x^{∗}(t)^{2} dt)^{\frac{1}{2}} + \frac{1}{2} (∫ 1_{\frac{1}{N}} \frac{1}{t(1 + log \frac{1}{t})} dt)^{\frac{1}{2}} (∫ 1_{\frac{1}{N}}(1 + log \frac{1}{t})x^{∗}(t)^{2} dt)^{\frac{1}{2}}

≤ c \sqrt{log log N} ∥x∥_{Φ_{2}} .

This is sufficient to prove the result.

An obvious question is the following.

Problem 9. Is there a rearrangement invariant norm ∥ · ∥_{X} on [0, 1] which is strictly larger than ∥ · ∥_{Φ_{p}}, but for which the canonical embedding C(K) → X(μ) has Gaussian cotype p?

For p > 2, the answer is yes. The embedding C(K) → L_{p,1}(μ) has cotype p (this follows from results in [2]). Hence X = L_{Φ_{p}} ∩ L_{p,1} equipped with the norm ∥x∥ = max{∥x∥_{Φ_{p}} , ∥x∥_{p,1}} provides the counterexample.

For p = 2, the answer is no. Talagrand [10] has recently shown that if C[0, 1] → X has Gaussian cotype 2, then ∥ · ∥_{X} is bounded by a constant times ∥ · ∥_{Φ_{2}}.

Another problem is also suggested by Theorem 1.

Problem 10. If T : C(K) → X is a linear map with Gaussian cotpye 2, does it follow that there is a Radon probability measure μ on K such that ∥T x∥ ≤ c ∥x∥_{L_{Φ_{2}} (μ)} for x ∈ C(K)?

Talagrand [10] has recently shown that this not the case.

Acknowledgements Much of the contents of this paper also appear in my Ph.D. thesis [7], which I studied at Cambridge University under the supervision of Dr. D.J.H. Garling, to whom I would like to express my thanks. I would also like to acknowledge simplifications communicated to me by M. Talagrand, including the statement of Corollary 4.

References

1. C. Bennett and K. Rudnick, On Lorentz–Zygmund spaces, Dissert. Math. 175 (1980), 1–72.

2. J. Creekmore, Type and cotype in Lorentz L_{p,q} spaces, Indag. Math. 43 (1981), 145–152.

3. G.J.O. Jameson, Summing and Nuclear Norms in Banach Space Theory, London Math. Soc., Student Texts 8.

4. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I—Sequence Spaces, Springer-Verlag.

5. B. Maurey, Type et cotype dans les espaces munis de structures locales inconditionelles, Seminaire Maurey-Schwartz 1973–74, Exposés 24–25.

6. S.J. Montgomery-Smith, On the cotype of operators from l_{∞}^{n}, preprint.

7. S.J. Montgomery-Smith, The Cotype of Operators from C(K), Ph.D. thesis, Cambridge University, August 1988.

8. M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99–149.

9. M. Talagrand, The canonical injection from C([0, 1]) into L_{2,1} is not of cotype 2, Contemporary Mathematics, Volume 85 (1989), 513–521.

10. M. Talagrand, Private communication.

S.J. Montgomery-Smith, Department of Mathematics, University of Missouri at Columbia, Columbia, Missouri 65211, U.S.A.