# Algebraic topology: a first course by Marvin J. Greenberg

By Marvin J. Greenberg

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Zm are homogeneous of degree 1, and since it is also true that any homogeneous diﬀerential operator of degree d can be written as ηα (u) Z α , where ηα is a homogeneous function of degree |α| − d, the above reasoning also implies that in the deﬁnition of the class FµA one can use these Zi ’s instead of the Yi ’s. The case when P is a diﬀerential operator of local degree d can be proved in the same way observing that if P = fα (u) Dα then the Taylor expansion of fα (u) Dα ∼ cαβ uβ Dα is the formal sum of homogeneous diﬀerential operator of degree d.

Proof. 18) with R1 = cR. 14). 5. FRACTIONAL INTEGRALS 25 We now apply to the fundamental solution of HA the previous results, to get an upper Gaussian bound for hA , which holds when d (x, y) is bounded away from zero. 10. There exists a constant τ such that for any R > ε > 0 we have hA (t, x, y) ≤ c1 (ε, R) e−c/t for every t ∈ (0, τ ) , x, y ∈ Rn with ε ≤ d (x, y) ≤ R. Proof. For any ﬁxed y ∈ Rn , let C = Cε,R = {(t, x) : 0 ≤ t ≤ 1, ε < d (x, y) < R} . Let us cover C with a ﬁnite number of balls Bj such that the balls Bj∗ concentric with Bj and with radius four times the radius of Bj are at positive distance from the origin, let C = Bj∗ .

K ∈ FµA and P is a diﬀerential operator of local degree d, then ∂tp P (k) ∈ Fµ−d−2p A γ 3) Given γ > 0 we can ﬁnd µ so that Fµ ⊆ C (R × G) . 4) If k ∈ FµA1 , h ∈ FµA2 , µ1 , µ2 > 0 and (k ∗ h) (t, u) = k t − s, u ◦ v −1 h (s, v) dvds then (k ∗ h) ∈ Fµ1 +µ2 . Proof. 1). To show 2) observe that any homogeneous diﬀerential operator of degree d can be written as ηα (u) Y α where ηα is a homogeneous function of degree |α| − d. 40 PART I: OPERATORS WITH CONSTANT COEFFICIENTS We have |∂tp P (k)| = ηα (u) ∂tp Y α k α u |α|−d c (p, α, µ) t(µ−2−2p−|α|)/2 gA (Ct, u) α α u √ t |α|−d c (p, α, µ) t(µ−2−2p−d)/2 gA (ct, u) .