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By Charles Bradfield Morrey Jr.

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Extra info for Partial Differential Equations (Proceedings of Symposia in Pure Mathematics, vol. 4)

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Con- stants will be denoted by C. One or more subindices appended to C will indicate quantities on which it depends. The Fourier transform of a function or a distribution f will be denoted by f We shall use the same notation for points in E. and its dual, where the Fourier transforms of functions on E. are defined. Bessel potentials. The Bessel potential of order z (z being any complex number) of a temperate distribution f, denoted by Jzf is defined by (Jzf )-. _ (1 + 4a2IxI2)-z/2f This is meaningful for all complex z.

F f (x + Ay)dA A2 Now S(x + Ay) >_ S(x) - r, so that IIf'(x)yll 5 A(S - r)-ar-1. Set r = S/(a + 1). We obtain (8 - r)-ar-1 = 8-a-1(1 + a)(1 + l/a)a 5 e(1 + a)8-a-1. It is evident how this lemma enables us to pass from estimates of ul, , UN-1 to estimates of their derivatives. The next step in applying the recursion formula (4) depends on a generalization of Bernstein's inequality (see [1]) to Banach spaces. In the most important case of finite or infinite dimensional complex Euclidean space, or Hilbert space, sharper forms were already obtained by Kellogg [8].

They are Hilbert spaces), then there exists a canonical isomorphism {S, T} of [V, W, ] onto the self-pairing [W, W, (w1, w2)] with T = I and S an isometric isomorphism of V onto W. The following theorem is immediate. THEOREM 3. Let {S, T} bean isomorphism of [V, W ] into [V', W',

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