By B. Ja. Levin

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G. 5 Estimates for linear semigroups φ (y) dy 1 2π W (t) φ = W (t) φ = 1 4πt , 2 |x−y|≤βt and for n = 3 35 β 2 t2 − |x − y| |x−y|=βt φ (y) dωy . 58) Lp,b for all 1 ≤ p ≤ ∞ and b ≥ 0, k = 0, 1, where κ = 0 for the one dimensional case n = 1 and κ = 1 for the case of space dimensions n = 2, 3. 30. 33. 57) takes place. Then the estimates are true provided that the righthand sides are ﬁnite + Ct ≤ Ct− δ ( r − p )+ δ t n b |·| ∂tk G (t) φ Lp 1 1 −n δ (r−p) t −k i∇ 1 1 kκ b −k i∇ −α 2t φ Lr,b kκ i∇ +e φ Lr kκ φ Lp,b and b |·| ≤ Ct G (t) φ − ϑt− δ G0 (·) t− δ n b−a 1 −n δ (1− p )+ δ −α 2t +e φ 1 φ L1,a +t Lp 1 1 b −n δ ( r − p )+ δ − γδ t φ Lr,b Lp,b for all t > 0, where ϑ = Rn ≤ r ≤ p ≤ ∞, 0 ≤ b ≤ 2p φ (x) dx, max 1, 2+p min (γ, a) , k = 0, 1, here κ = 0 for the one dimensional case n = 1 and κ = 1 for the case of space dimensions n = 2, 3.

66) Lr 2p for all t > 0, where max 1, 2+p ≤ r ≤ p ≤ ∞, 0 ≤ b ≤ a. 66) the estimates of the lemma follow. 33 is proved. 2 Estimates in the L2 - theory By the Sobolev Imbedding Theorem the Lp norms can be estimated by derivatives in L2 norms and weighted L1,a norms can be estimates by weighted L2,b norms with b > a + n2 . And the L2 - theory is convenient for passing to Fourier transform representations. 67) for all ξ ∈ Rn , where α > 0, δ > 0, µ ≥ 0. , n, with some N ≥ n + 2. We now prove L2 estimates for the Green operator G (t).

Also we deﬁne ˆ = t− δ G0 (t) φ = F ξ→x e−tL0 (ξ) φ(ξ) G0 (x − y) t− δ φ (y) dy, 1 1 Rn where the kernel G0 (x) = F ξ→x e−L0 (ξ) . Denote the mean value θ = (2π) −n 2 φ (x) dx. 37. 73). Then the estimate is true ρ |∇| (G (t) φ − θG0 (t)) + Ct n − ρ+1 δ −δ ( q1 − 12 ) L2 · i∇ ≤ Ct− −µ φ ρ+γ δ 1 1 −n δ (q−2) −αt Lq + Ce i∇ |∇| ρ −µ i∇ φ −µ Lq φ L2 for all t ≥ 1, where 1 ≤ q ≤ 2, ρ ≥ 0, provided that the right-hand side is ﬁnite. 5 Estimates for linear semigroups 41 Proof. 67) we have ξ S (t, ξ) + e−tL0 (ξ) µ L∞ (|ξ|≥1) ≤ Ce−αt for all t ≥ 1.