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Showing papers in "Arkiv för Matematik in 1976"


Journal ArticleDOI

222 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the Riesz potentials of Lp functions are valid for m = 1, 2 for the remaining integer values of m Theorem 4 of section 4.
Abstract: This paper is a direct outgrowth of a conversation the author had with Professor V. G. Maz'ya in the Spring of 1974 concerning the existence of certain Lp estimates for the restriction of Riesz potentials of Lp functions to small sets. The main open question concerned the validity of the estimate llI,,f[]v,~ 2 (re

115 citations




Journal ArticleDOI
TL;DR: In this paper, the authors give a necessary condition for an interpolation pair to have its interpolation spaces characterized by K-monotonicity, which is the strongest form of monotonicity which holds in such generality.
Abstract: For any interpolation pair (A 0 A 1), Peetre’sK-functional is defined by: $$K\left( {t,a;A_0 ,A_1 } \right) = \mathop {\inf }\limits_{a = a_0 + a_1 } \left( {\left\| {a_0 } \right\|_{A_0 } + t\left\| {a_1 } \right\|_{A_1 } } \right).$$ It is known that for several important interpolation pairs (A 0,A 1), all the interpolation spacesA of the pair can be characterised by the property ofK-monotonicity, that is, ifa∈A andK(t, b; A0, A1)≦K(t, a; A0, A1) for all positivet thenb∈A also. We give a necessary condition for an interpolation pair to have its interpolation spaces characterized byK-monotonicity. We describe a weaker form ofK-monotonicity which holds for all the interpolation spaces of any interpolation pair and show that in a certain sense it is the strongest form of monotonicity which holds in such generality. On the other hand there exist pairs whose interpolation spaces exhibit properties lying somewhere betweenK-monotonicity and weakK-monotonicity. Finally we give an alternative proof of a result of Gunnar Sparr, that all the interpolation spaces for (L v p , L w q ) areK-monotone.

80 citations


Journal ArticleDOI
Abstract: A series of Stieltjes is a (formal) power series f ( z ) = ~ ~ e.(-z)" where e.=fo t"de(t) for some real, bounded, nondecreasing function e(t) assuming infinitely many values on t~0. These functions were first studied by Stieltjes who proved that the moment problem on [0, ~[ associated with" {e.}~= 0 is determinate if and only if the corresponding continued fractions expansion off(z) converges except on the negative real axis. (Stieltjes [10].) This theorem is also of significance for the theory of Pad4 approximation (Pad4 [8]),

24 citations



Journal ArticleDOI
TL;DR: In this article, it was shown that for any positive integer r > 1, one can find a sequence of a-fields with respect to which the above lacunary exponentials become martingale differences.
Abstract: It has been recognised for a long time that the sequence {exp (irkO); k = O, 1, 2 .... with r an integer greater than one and 0~0<=2rc, is quite similar to a sequence of independent random variables. That is, many statements that are valid for sums o f independent variables a r e also true for sums of exponentials of the above type. This coincidence may be explained by the observation that the sequence, while not independent, is a martingale difference sequence. Our purpose in this note is to discuss this type Of martingale in the context o f the theory of H/'-spaces. In fact, we show that for any positive integer r > 1 one can find a sequence of a-fields with respect to which the above lacunary exponentials become martingale differences. Using this, we define HP-spaces in a manner analogous to what has been done in the classical case (cf. [2]). These HP-spaces are translation invariant subspaces of LI(T) that coincide with LP(T) for p >1. The most interesting case is when p ~ 1; here the spaces which we denote by H i are translation invariant subspaces of L I(T), distinct from the classical Hardy space HL The space H~ may be characterised as follows: fEH~ if and only if f a n d its \"conjugate\" y~ belong to L 1 (T). Here j~ is, of course, not the harmonic conjugate function; nevertheless, it is obtained from f by a Fourier multiplier taking the values + 1. The spaces H i and their associated conjugate functions are closely related to some results of Taibleson and Chao [3]; we indicate this in some detail in w 3. We also use these ideas to obtain some recent results on lacunary series. We discuss these applications in w 2. For background on martingale theory and HP-spaces, we cite [1] and the excellent exposition by Garsia [4].

18 citations


Journal ArticleDOI
TL;DR: In this paper, the Fourier-Borel transform of u E d'(C") is denoted t~ or flu, where flu denotes the real and the imaginary part of EC.
Abstract: 1. The notation is standard. N, Z, R, R +, C denote respectively the natural, integral, real, real positive, and complex numbers. Re ~ and Im ~ is the real and the imaginary part of ~EC". [1 is the usual norm in C". By g(t), tER + u {+ oo}, we denote the C = functions defined for Ix[

14 citations




Journal ArticleDOI
TL;DR: In this article, Lesmes gave sufficient conditions for a subalgebra AcCm(E) to be dense in (CmE, z~), when m = l, and E is a real separable Hilbert space.
Abstract: (1) for every xE U, there exis ts fEA such t h a t f ( x ) r (2) for every pair x, yEU, with xr there exists fEA such that f(x)r (3) for every xE U and vEE, with v ~ 0 , there exists fEA such that Df(x)v~O. In [1], Lesmes gave sufficient conditions for a subalgebra AcCm(E) to be dense in (Cm(E), z~), when m = l , and E is a real separable Hilbert space. In fact, he proved that (1), (2), (3) (with U=E) and



Journal ArticleDOI
TL;DR: In this paper, it was shown that there exists a neighborhood U of I such that Q (U ) c f ( I ) + + +B, (X) is an analytic path in a complex Banach space X satisfying f(O) = O.
Abstract: Lemma. Let f : I ~ X be a path in a complex Banach space X satisfying f(O) = O. Let ~ >0 and 0< r < 1. Suppose that E is a closed subset o f the boundary o f A of linear measure 0 which does not contain a point z o, lz01=l. Denote D = A n K(r, co). There exists a continuous function ~ : A ~ X , analytic on A and having the following properties (a) f b (A)c f ( I ) + B , (X) (b) I[q~(z)l[ < 8 ( z E A D ) (c) ]If(l)-q~(Zo)[[ < e (d) ~(z) = 0 (z~E). Proof. By the Mergelyan theorem for vector-valued functions [1] there exists a polynomial P: C-*-X such that IF f ( z ) -P(z)][



Journal ArticleDOI
TL;DR: In this paper, an eigenfunction expansion for the operator −°+V under assump-tions (1.2) −(1.5) was obtained, where V is the number of assumps.
Abstract: We obtain an eigenfunction expansion for the operator −°+V under assump-tions (1.2)–(1.5) given below.

Journal ArticleDOI
TL;DR: In this article, the authors give necessary and sufficient conditions for the weight functions i n order that the operators (0-1) are continuous on LZ(R) and introduce some hypotheses on Z, o, ~.
Abstract: with X, ~o, ~ fixed weight functions on ~+ = {~ER, 4=>0}. Our aim is to give necessary and sufficient conditions for the weight functions i n order that the operators (0-1) are continuous on LZ(R"). As a matter of fact, we shall restrict ourselves to the one-dimensional case, n = 1, and we shall introduce some hypotheses on Z, ~o, ~. In the first section we enunciate the results and we give some applications. Particularly we obtain for the classes of H6rmander soma on R X R X R the results in Calder6n--Vail lancourt [3], HOrmander [6] and also a result of Ching [4]. Another application refers to the classes of pseudo differential operators in Beals--Fefferman [1]. In the second section we give the proofs.


Journal ArticleDOI
TL;DR: In this paper, the e-entropy of a subset of a metric space relative to f2 is defined as the smallest integer for which the answer is positive, and the number of points that can approximate the set A in the sense that knowing them we can reproduce it to within an accuracy e.
Abstract: One approach to approximation theory is the following (see [15, Chapter 10]). I f (f2, Q) is a metric space, A is a subset of f2, and e>0 one asks whether there exist points zl, zz . . . . . z, in fl such tha t A ~U?=I S(z,, e). If N.~(A) is the smallest integer for which the answer is positive the points z~, z~ . . . . . zAr ~ approximate the set A in the sense that knowing them we can reproduce the set A to within an accuracy e. The quantity H~(A)=logzN,O(A) is called the e-entropy of A relative to f2, and one is then interested in its asymptotic growth as e tends to zero. This approach has been the subject of much activity (see [13], [14] and [19]). Since the covering of sets by spheres of equal radii can be quite inefficient, it is, for some purposes, preferable to consider covers by spheres of varying radii. Suppose that (r,) is a sequence Of positive real numbers which are decreasing with limit zero. We say that (r,) is majorizingfor A in f2 if there exists a sequence (zi) of points in I2 such that A c= 0 S(zi, rl) for each n, i = .