We define Hardy spaces of pairs of conjugate temperatures on ${ℝ}_{+}^2$ using the equations introduced by Kochneff and Sagher. As in the holomorphic case, the Hilbert transform relates both components. We demonstrate that the boundary distributions of our Hardy spaces of conjugate temperatures coincide with the boundary distributions of Hardy spaces of holomorphic functions.

In this preliminary Note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation $\partial {}_{t}u=\mathrm{△}u+\frac{c}{\left|x^2\right|}u(0<c<\frac{{\left(n-2\right)}^2}{4};n\ge 3)$. A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence of the Schrödinger operator $Hu=-\mathrm{△}u-\frac{c}{{\left|x\right|}^2}u$ with the opposite of the weighted Laplacian ${\mathrm{△}}_{\lambda }v=\frac{1}{{\left|x\right|}^{\lambda }}\text{div}\left({\left|x\right|}^{\lambda }\nabla v\right)$ when $\lambda =2-n+2\sqrt{c_0-c}$.

An asymptotic analysis is given for the heat equation with mixed boundary conditions rapidly oscillating between Dirichlet and Neumann type. We try to present a general framework where deterministic homogenization methods can be applied to calculate the second term in the asymptotic expansion with respect to the small parameter characterizing the oscillations.

We study the homogenization of parabolic or hyperbolic equations like$${\rho }_{\epsilon }\frac{{\partial }^n{u}_{\epsilon }}{\partial t^n}-\mathrm{div}\left(a_{\epsilon }\nabla u_{\epsilon }\right)=f\text{in}\Omega \times (0,T)+\text{boundary}\text{conditions},n\in \{1,2\},$$when the coefficients ${\rho }_{\epsilon }$, $a_{\epsilon }$ (defined in $Ø$) take possibly high values on a $\epsilon$-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

We study the homogenization of parabolic or hyperbolic equations like $${\rho }_{\epsilon }\frac{{\partial }^n{u}_{\epsilon }}{\partial t^n}-\mathrm{div}\left(a_{\epsilon }\nabla u_{\epsilon }\right)=f\text{in}Ø\times (0,T)+\text{boundary}\text{conditions},n\in \{1,2\},$$ when the coefficients ${\rho }_{\epsilon }$, $a_{\epsilon }$ (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.