By Michael E. Taylor
For the previous 25 years the idea of pseudodifferential operators has performed a tremendous position in lots of intriguing and deep investigations into linear PDE. during the last decade, this instrument has additionally started to yield fascinating leads to nonlinear PDE. This publication is dedicated to a precis and reconsideration of a few used of pseudodifferential operator suggestions in nonlinear PDE. The publication could be of curiosity to graduate scholars, teachers, and researchers drawn to partial differential equations, nonlinear research in classical mathematical physics and differential geometry, and in harmonic research.
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Additional info for Pseudodifferential operators and nonlinear PDEs
51) if q > n, since q = n + a =⇒ 1 − q 1 a a = 1− =⇒ s > q 1 + . 53). 49) complementary to the result of DeTurk and Kazdan [DK] that, if g is C 2 and R m is C k+α , in harmonic coordinates, then g is C k+2+α . It is clear that further generalizations can be established. 51) holds, then f ∈ H σ,p , σ ≥ −1 =⇒ u ∈ H σ+2,p . I can be brought to bear on some of the material in [AC]. Further related results arise in [[AK2LT]]. 35) in the case when the coefficients ajk (x) are scalar. A proof is given in Appendix C.
24) ϕj (ξ)2 1= j=0 where ϕj ∈ C ∞ , ϕ0 (ξ) is supported on |ξ| ≤ 1, ϕ1 (ξ) is supported on 1/2 ≤ |ξ| ≤ 2, and ϕj (ξ) = ϕ1 (21−j ξ) for j ≥ 2. 26) Φ(f ) = ϕ0 (D)f, ϕ1 (D)f, ϕ2 (D)f, . . This is clearly an isometry, though of course it is not surjective. 27) Φ∗ (g0 , g1 , g2 , . . 28) on L2 (Rn ). 29) Φ(ξ) = (ϕ0 (ξ), ϕ1 (ξ), ϕ2 (ξ), . . ). 22) is satisfied by both Φ(ξ) and Φ∗ (ξ). 30) Φ∗ : Lp (Rn , 2 2 ) ) −→ Lp (Rn ). 31) u Lp ≈ Φu Lp (Rn , 2) 2 ), . 32) Cp u Lp |ϕj (D)u|2 ≤ j=0 1/2 Lp ≤ Cp u Lp , for 1 < p < ∞.
9) m OP S1,0 : X s+m −→ X s . 7) have this property provided p ∈ (1, ∞). 9) fails for the spaces C s if s is an integer. 10) X s = C∗s (Rn ), Σ = (0, ∞), which coincide with C s is s is not an integer, but differ from C s if s is an integer. Some important properties of Sobolev spaces and Zygmund spaces are discussed in Appendix A. 11) pj (x, ξ) p(x, ξ) ∼ j≥0 in terms homogeneous of degree m − j in ξ (for |ξ| ≥ 1), in the sense that the m−N difference between p(x, ξ) and the sum over j < N belongs to XS1,0 .