The Gaussian Cotype of Operators from C(K)

We show that the canonical embedding C(K)→ LΦ(μ) has Gaussian cotype p, where μ is a Radon probabilty measure on K, and Φ is an Orlicz function equivalent to t(log t) p 2 for large t. * * * * * * In [6], I showed that the Gaussian cotype 2 constant of the canonical embedding l ∞ → L2,1 is bounded by log logN . 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 √ log logN . 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 Φp for an Orlicz function such that Φp(t) ≈ t(log t) 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 x1, x2, . . . ∈ X we have