408 Index
distribution, 6
(x1 ± i0)λ, 12
δ(S), 19
1
S±i0
, 19
1
x1±i0
, 16
p.v.
1
S
, 19
p.v.
1
x1
, 15
x±,
λ
16
tempered, 34
Fourier transform of, 39
singular support of, 57, 217
support of, 20
wave front set of, 58
distributions
direct product of, 27
domain of dependence, 77, 243, 246
domain of influence, 77, 243, 245
double layer potential, 83, 103, 177
duality of Sobolev spaces, 54
eigenvalues, 301
eigenvectors, 301
eikonal equation, 344
electrical impedance problem, 171
elliptic ψdo on a manifold, 266
elliptic boundary conditions, 274
elliptic class of symbols Ell(T
∗(Rn)),
316
elliptic differential operator, 204
elliptic Fourier integral operator, 338
elliptic pseudodifferential systems, 259
elliptic regularity, 206
elliptic symbol, 206
ellipticity of the boundary conditions,
277
embedding, 208
equivalent norm for s = m + γ, 52, 117
existence of fundamental solution, 85
external product, 319
factorization of elliptic symbol, 266, 279
Faddeev’s fundamental solution, 85, 181
Faddeev’s scattering amplitude, 181
fiber, 264
finite rank operator, 210, 302
focal point, 371
formally adjoint operator, 118
Fourier integral operator (FIO), 330
Fourier inversion formula, 37
Fourier transform, 37
of convolutions, 45
of distributions, 39
Fourier-Laplace transform, 229
Fredholm alternative, 149, 209
Fredholm equation, 183
Fredholm operator, 208, 311
Fredholm property, 273
Friedrichs regularization, 260
function spaces
C(R1,
Hs−
1
2
(Rn−1)),
54
C(R1,
S
(Rn)),
72
C(BN ), 149
C(R+, 1 L2(Rn−1)), 79
Cm(∂Ω),
178
C0
m(Rn),
51
C0
∞(Rn),
2
Cα,N , 191
Cα,N
r
, 191
Cm−1,s, 237
L1[(0,
T ),
Hs(Rn)],
237
L∞,α, 148
S, 33
S , 34
Vm,s(Rn), 142
D, 4
D , 8
D (Ω), 8
D(Ω), 8
E , 22
H(s), 267
H(s)(Ω), 271
fundamental solution, 69, 72, 75
for the Cauchy problem, 74
arding inequality, 234
general form of a ψdo, 212
generalized scattering amplitude, 164
global FIO, 360
Green’s formula, 107, 132, 136, 169, 176
Hamiltonian system, 334
heat equation, 70
heat kernel, 13, 71, 230, 302, 310
heat trace of an operator, 305
Helmholtz equation, 81, 91, 157
Hilbert-Schmidt operator, 150, 183,
207, 302, 304
homotopy, 314
hyperbolic Cauchy problem, 342
hypoelliptic operator, 87
index of Fredholm operator, 311
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