An Infinitesimal Approach to Stochastic Analysis by H. Jerome Keisler

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62-63]. This shows that ue -+ u, as e ~ 0. Finally, if we replace ¢ - u by u in the last series of 18 2. Sobolev spaces inequalities, we get ][ue]lL~(~) <_ CHUHL~(~) and thus ue 6 L~(~). 4 has the following corollary, which is proved in the same way as when w = 1 (see R. A. Adams [9, pp. 52-53]). 5. Suppose that f~ is an open subset of R N, and let ~' be an open subset of l~ such that dist(~', ~c) > 0. Suppose that u 6 w m ' P ( ~ ) , where m > 1, 1 < p < co, a n d w 6 Ap. 4, then as ~ -~ O, u~ --+ u in w m ' p ( ~ ' ) .

181-192], which works for domains satisfying a "minimal smoothness condition" and for 1 _~p < c~, does not easily generalize to the weighted case. 20 2. e. 2) IITflIL~(R~) ~ CIIflIL~(R~), with C depending only on o]w. IIwlIL~(R~), IIVwlIL~(R~), N, p, and the Ap constant P r o o f . It is readily seen that the kernel K satisfies A IK(x)l ~ ixl---~ for every x E R N \ {0}, and AlYl IK(x) - K ( x - Y)I < -IxINq-1 - ' if x, y E R N and Ixl > 2[yl (for the proof of the second inequality, see Stein [99, pp.

13. Let w E Ap, 1 < p < co, and let m >_ 1 be an integer. Suppose that ~ C R N is open and satisfies the following cone condition: there is an open covering { uj } M1 o] Of~ and corresponding finite, congruent cones M (Cj}j= 1 such that if x • Uj N f l ]or some j, t h e n x + Cj C 12. Then there exists a bounded, linear extension operator E : W~,P(~) -+ W~'P(RN). P(R N) <_ Cllullw:,p(, ). 5) The constant C depends only on 12, m, N, p, and the Ap constant of w. P r o o f . First, let u • C °O( ~ ) N W ~ ,p (f}).

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