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2 Basic properties of the spaces Lp
3 Linear functionals on Lp
4 Compactness of spaces
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adjoint analogous arbitrary function averaged functions boundary value problem bounded domain bounded function boundedness Cauchy problem characteristic cone coefficients consequently const constant construct continuous derivatives continuous functions continuously differentiable corresponding defined denote depend derivatives of arbitrary derivatives of order Dirichlet problem du/dt dx'n dxnn element equal to zero equicontinuity equivalent estimate exists finite number follows formula func given Holder inequality hyperplane Imbedding Theorem initial data integral lemma linear functional minimizing sequence Minkowski inequality n-dimensional space n<lp Neumann problem norm obtain Obviously plane polar coordinates polynomial polynomial of degree proof radius respect right hand side satisfying the conditions Sn-s solution space Wf spherical projection operator sufficiently smooth Suppose surface theorem is proved tion triangle inequality unit sphere vanish variables vector virtue wave equation whole space xu x2 xux2