| Preface |
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vii | |
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1 Fourier analysis and Laplace operators on IR" |
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1 | (44) |
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1 | (14) |
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1.1.1 Some definitions from measure theory |
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1 | (1) |
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1.1.2 Polar coordinates in Rn |
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2 | (2) |
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1.1.3 The spaces of C∞c(Rn) and S(Rn) |
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4 | (8) |
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1.1.4 The space of tempered distribution S(Rn)* |
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12 | (3) |
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1.2 Convolution units and the group algebra |
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15 | (5) |
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1.2.1 Convolution of functions |
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15 | (5) |
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1.3 Fourier transform oftempered distributions |
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20 | (1) |
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1.4 Lp spaces and its duals |
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21 | (17) |
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1.4.1 Pointwise convergence |
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28 | (3) |
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1.4.2 Convolution of finite measures |
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31 | (5) |
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1.4.3 Convolution with tempered distributions |
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36 | (2) |
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1.5 Solving partial differential equations |
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38 | (2) |
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1.6 The convolution kernels of the functions of Laplacian |
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40 | (5) |
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2 The model domain, the sub-Laplacian operator and Cauchy-Szego kernels |
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45 | (22) |
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2.1 Background of the problem |
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45 | (13) |
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2.1.1 Cauchy-Riemann geometry |
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45 | (2) |
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2.1.2 Tangential Cauchy-Riemann operator |
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47 | (6) |
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53 | (3) |
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2.1.4 The nonisotropic Heisenberg group |
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56 | (2) |
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2.2 The model domain in Cn+1 |
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58 | (3) |
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2.3 The Cauchy-Szego kernel on |
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61 | (6) |
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3 The fundamental solution for the operator Δλ: k = 1 |
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67 | (36) |
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3.1 The fundamental solution for Δλ: k = 1 and n = 1 |
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67 | (7) |
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3.2 Relative fundamental solution and the Cauchy-Szego kernel |
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74 | (9) |
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83 | (13) |
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3.3.1 Fundamental solution with singularity at the origin |
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84 | (9) |
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3.3.2 Fundamental solution with singularity at an arbitrary point y |
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93 | (3) |
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3.4 Connection with the operator Δλ |
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96 | (7) |
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4 Fundamental solution for the operator Δλ:k = 2 and n = 1 |
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103 | (38) |
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103 | (4) |
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4.2 Calculation of the fundamental solution |
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107 | (11) |
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4.3 The special case λ = 0 |
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118 | (23) |
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133 | (2) |
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135 | (3) |
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138 | (3) |
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5 Fundamental solution for the operator Δ0: k = 2 |
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141 | (34) |
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5.1 Adapted polar coordinates |
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143 | (1) |
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5.2 Adapted spherical harmonics |
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144 | (2) |
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5.3 Partial fundamental solution |
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146 | (5) |
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5.4 Complete fundamental solution |
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151 | (4) |
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5.5 The case k = 2 and n = 2 |
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155 | (1) |
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5.6 Computation of the sum over odd integers |
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156 | (3) |
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5.7 Computation of the sum over even integers |
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159 | (9) |
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5.8 Computation of the Fourier integral (in τ) of K(+) and k(-) |
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168 | (7) |
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6 Green's function of the operator AA for general n and k |
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175 | (22) |
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177 | (3) |
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6.2 Derivation of the Green's function Kλ: singular part |
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180 | (1) |
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6.3 Derivation of the Green's function Kλ: regular part |
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181 | (6) |
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6.4 Analytic properties of Gn,k and Fλ |
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187 | (4) |
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6.5 Derivation of the Green's function: computation of cn,k |
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191 | (6) |
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7 A geometric formula for the fundamental solution |
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197 | (48) |
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197 | (1) |
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7.2 Hamiltonian formalism |
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197 | (3) |
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7.2.1 Newtonian mechanics |
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197 | (1) |
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7.2.2 Lagrangian formalism |
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198 | (2) |
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7.2.3 Hamiltonian formalism |
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200 | (1) |
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200 | (3) |
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203 | (3) |
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7.5 Hamiltonian formalism on H1 |
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206 | (5) |
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7.6 Lagrangian formalism on H1 |
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211 | (15) |
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7.6.1 Connectivity by geodesies |
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213 | (13) |
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7.7 Fundamental solution in terms of geometric invariants |
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226 | (11) |
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7.8 Fundamental solution Kλ for Δλ when |Re(λ)| > 1 |
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237 | (5) |
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7.9 The case |Re(λ)|≤1 with λ ε Σ |
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242 | (3) |
| Bibliography |
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245 | (4) |
| Index |
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249 | |
