Partial Gaussian bounds for degenerate differential operators II

2012/02/09 by ter Elst, A. F. M., Ouhabaz, E. M.
#35J70 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1202.2139

Abstract

Let A = - ∑ ∂_k c_kl ∂_l be a degenerate sectorial differential operator with complex bounded mesaurable coefficients. Let Ω⊂ \mathdsR^d be open and suppose that A is strongly elliptic on Ω. Further, let χ∈ C_{\rm b}^∞(\mathdsR^d) be such that an ε-neighbourhood of \supp χ is contained in Ω. Let ν∈ (0,1] and suppose that the c_kl|Ω ∈ C^0,ν(Ω). Then we prove (Hölder) Gaussian kernel bounds for the kernel of the operator u ↦ χ S_t (χ u), where S is the semigroup generated by -A. Moreover, if ν= 1 and the coefficients are real, then we prove Gaussian bounds for the kernel of the operator u ↦ χ S_t u and for the derivatives in the first variable. Finally we show boundedness on L_p(\mathdsR^d) of various Riesz transforms.

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