A Remark on Eigenfunction Estimates by Heat Flow

A Remark on Nelson's Best Hypercontractive Estimates

By using a combinatorial estimate we provide a new proof of Nelson's best hypercontractive estimates from L2 to L*. Let G be the differential operator \ -A; +x4on L2(R,tt~x/2e-x2dx). 2 dx2 dx Hypercontractive estimates on e~'G have played a key role in constructive quantum field theory; see e.g. [6]. In [5] Nelson proved the estimate (1) ll'"'%<l|/||, if (2) e<V(<?l)/(pI) where ||-|| is the LP(...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

A Remark on Cavitational Flow.

pp. 76-83. Cf. also R. Brauer, Bull. Am. Math. Soc., 51, 749-755 (1945). For calling our attention to the first reference, we are indebted to Dr. Ernst Snapper; for the third, to Dr. H. N. Shapiro. 4 Cipolla, Periodico. di Mat., 22, 36-41 (1907). For every m in (5a) equal to zero then N reduces to the number of sets of roots of (A), a fortnula known to V. A. Lebesgue, for n = 1, and this was em...

On Strichartz and Eigenfunction Estimates for Low Regularity Metrics

A bstract . We produce, for dimensions n ≥ 3, examples of wave operators for which the Strichartz estimates fail. The examples include both Lipschitz and C1,α metrics, for each 0 < α < 1, where by the latter we mean that the gradient satisfies a Hölder condition of order α. We thus conclude that, on the scale of Hölder regularity, an assumption of at least 2 bounded derivatives for the metric (...

Remark about Heat Diffusion on Periodic Spaces

Let M be a complete Riemannian manifold with a free cocompact Z-action. Let k(t, m1, m2) be the heat kernel on M . We compute the asymptotics of k(t, m1, m2) in the limit in which t → ∞ and d(m1, m2) ∼ √ t. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on R coming from the Hodge inner product on H(M/Z;R).