## A Gaussian measure analogue to Sobolev's inequality |

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Banach lattice Banach space bounded operator Calderon choose classical Sobolev inequality Closed Graph Theorem complete with respect Complex Method concave continuously embedded contraction convergence in measure converges a.e. converges to f convex Definition denote derivatives Dual Space element equivalent norms f and g f(it finite measure space fn(x follows that f function f function-type g ln|g given I I I implies f Interpolation of Banach interpolation pair L2 ln L2 ln+L Lebesgue measure Lebesgue space let f linear ln|f measurable functions measurable set N-function norm dominates norm of exp(-tHQ operator from L2 operator HQ original space Orlicz space positive real Proof resp results of Chapter satisfies the A2-condition self-adjoint operator sequence show that g simple functions spaces of functions subadditivity tends to zero Theorem 12 Three Lines Theorem topological vector space Young's complement Young's inequality