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Growth properties of hyperplane integrals of Sobolev functions in a half space (Dedicated to Professor Masayuki Ito on the occasion of his sixtieth birthday)

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Shimomura, T. Osaka J. Math. 38 (2001), 759–773 GROWTH PROPERTIES OF HYPERPLANE INTEGRALS OF SOBOLEV FUNCTIONS IN A HALF SPACE Dedicated to Professor Masayuki Ito on the occasion of his sixtieth birthday TETSU SHIMOMURA (Received January 21, 2000) 1. Introduction Let D ⊂ R ( ≥ 2) denote the half space D = { = ( ′ ) ∈ R −1 × R1 : > 0} and set S = ∂D; we sometimes identify ′ ∈ R −1 with ( ′ 0) ∈ S. We define the hyperplane integral ( ) over S by ( ) = (∫ S | ( ′)| ′ )1/ for a measurable function on S and > 0. Set ( ′) = ( ′ )− −1∑ =0 ! [( ∂ ∂ ) ] ( ′ 0) for quasicontinuous Sobolev functions on D, where the vertical limits( ∂ ∂ ) ( ′ 0) = lim →0 ( ∂ ∂ ) ( ′ ) exist for almost every ′ = ( ′ 0) ∈ ∂D and 0 ≤ ≤ − 1 (see [8, Theorem 2.4, Chapter 8]). Our main aim in this note is to study the existence of limits of ( ) at = 0. More precisely, we show (in Theorem 3.1 below) that lim →0 −ω ( ) = 0 760 T. SHIMOMURA for some ω > 0. Consider the Dirichlet problem for polyharmonic equation ( ) = 0 with the boundary conditions( ∂ ∂ ) ( ′ 0) = ( ′) ( = 0 1 . . . − 1) We show (in Corollary 3.1 below) that if 1 < ≤ < ∞, / − ( − 1)/ < 1 and ∈ (D) is a solution of the Dirichlet problem with ( ′) = (∂/∂ ) ( ′ 0) for 0 ≤ ≤ − 1, then lim →0 / −( −1)/ − ( ) = 0 where ( ′) = ( ′ )−∑ −1 =0 ( / !) ( ′). To prove our results, we apply the integral representation in [6, 8]. For this pur- pose, we are concerned with -potentials defined by ( ) = ∫ ( − ) ( ) for functions on R satisfying weighted condition:∫ R | ( )| | |β <∞ In connection with our integral representation, ( ) is of the form λ| |− for a multi-index λ with length . Our basic fact is stated as follows (see Theorem 2.1 be- low): lim →0 / −( −1)/ − ( ) = 0 where ( ′) = ( ′ )−∑ −1 =0 ( / !)[(∂/∂ ) ]( ′). In the final section, we give growth estimates of higher differences of Sobolev functions. For related results, see Gardiner [2], Stoll [14, 15, 16] and Mizuta [5

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