A Note on the Almost Everywhere Convergence to Initial Data for Some Evolution Equations

Abstract

The weighted Lebesgue spaces of initial data for which almost everywhere convergence of the heat equation holds was only very recently characterized. In this note we show that the same weighted space of initial data is optimal for the heat-diffusion parabolic equations involving the harmonic oscillator and the Ornstein-Uhlenbeck operator.2010 Mathematics Subject Classification. Primary: 35K15, 35K05. Secondary: 42B37, 42B25, 42B35.

Pointwise convergence to initial data of heat and Laplace equations

“…Theorem 1.1, in the setting considered in this section, is a direct consequence of the next three propositions. Our argument is more direct than that in [1], and also valid in greater generality.…”

“…We finally remark that, although this method gives no explicit expression for the weight u, we are able to show that it is "almost" in the same D p class as v; namely, for every ε > 0 we can choose a weight u such that u 1−ε ∈ D p (this is always the case in the Poisson setting, and also in the heat setting if a is sufficient small, or T * = ∞; see Remark 3.6). This result is new, even in the special cases already considered in the literature [7,1].…”

“…This case was studied earlier in [1], although from a slightly different perspective. Namely, in that paper the authors seek minimal conditions on a function f so that there exists some ε > 0 for which e −tL f (x) is well-defined up to time ε and e −tL f (x) → f (x) for a.e.…”

“…and search for minimal assumptions in f so that u(t, x) = e −tL f (x) solves the equation in the full band t ∈ (0, T), and secondly, it satisfies e −tL f (x) → f (x) a.e. This more precise approach is also slightly more general than [1], since their class of admissible initial data f will coincide with the union of our classes as T > 0 varies.…”

“…For completeness we have stated the results for both the heat and Poisson equations, although in the heat setting some of the results are already known; see [1]. Our main contribution concerns then the Poisson equation.…”