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Some Relations Among Orlicz Spaces

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Some Relations Among Orlicz Spaces RICE UNIVERSITY SOME RELATIONS AMONG ORLICZ SPACES by Max August Jodeit, Jr. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS Thesis Director's signature: Houston, Texas May, 1965 ABSTRACT In this paper it is shown that for the study of Orlicz spaces the condition that a Young's function A be convex can be replaced by the more general (and more convenient) condition that A(x)/x be non-deereasing. Some properties of the lattice of Orlicz spaces ordered by inclusion are given. R. O'Neil has shown that the condition is sufficient in order that GO (f*g)(x) - f f(y)g(x-y)dy belong to L^, whenever f € L^, g € Lg. The condition is also sufficient when the convolution is formed over the in¬ tegers or (0,2TT]. It is proven here that the condition is also necessary; all triplets A, B, C of Y-functions for which the condition holds are determined. ACKNOWLEDGMENT The author wishes to thank Professor R. O'Neil for un¬ derstanding, insight, great patience, and valuable suggestions while this thesis was being written. The research contained in this thesis was supported in part by the Air Force office of Scientific Research. Thanks are tendered to Mrs. Nancy Singleton and to Miss Janet Gordon for their skillful typing and helpful suggestions. And to my wife Jane go thanks and gratitude. INTRODUCTION In this paper we study the partially ordered (by inclusion) set of Orlicz spaces over a measure space (X,S,n). Background results and definitions are given in Section 1. Young's functions (among which are N-functions) are equi¬ valent if they always determine the same Orlicz space. If A is a Young's function the condition A(kx) ^ B(x) ^ A(Kx) is that of equivalence expressed in terms of functions. This re¬ lation holds for many functions B, not all of which are convex. In Section 2 we introduce the Y-functions, which are among these, and show that they are, for our purposes, entirely equivalent to Young's functions. Each Y-function A satisfies the condition that A(x)/x be increasing. This is our substitute for convexity. In particular, Y-functions are convex at 0 (a function is convex at p if the chord from the point p to the graph lies on or above the graph). We find the dual notion of concavity at 0 to be useful, and discuss inverse Y-functions. Given a Y-function A we can find a convex function C equivalent to it. Indeed we may choose C so that C(x)^A(x)SC(2x). For a measurable function f we define ||f||^ as in Section 1, and in a similar fashion the quantity |f|^, using A in place of C. We then have ||f||c^|f |A^2||f|(c. A does not determine a metric, however, since the triangle inequality breaks down. How¬ ever since Y-functions are non-decreasing and 0 at 0 the 2 function A determines the metric p(f,g) - inf {K>0: f A( | f-g |/K) dpi =5 K) X (see [5;p.l09]) which turns out to be equivalent to the norm- metric given by C, so that A determines a uniform topology for !,£ equivalent to that given by C. The use of Y-functions instead of Young's functions in many cases simplifies proofs. In some cases involving natural constructions convexity at 0 is preserved but convexity is not. An example of the first kind is the function (used in [l;Th.9]) defined for xgl by Q(x) » sup M(y)N(x/y) , lSy^x where M,N are convex. It is not easy to show that Q is convex (for xsl), but if 0sl Q(0x)/0x = sup M(y)N(0x(y) 5 sup M(y)N(x(y) 5 Q(x)/x, l^y^0x l^yS0x so that Q is a Y-function. The pointwise minimum of two Y-functions is a Y-function, but that of two Young's functions is not necessarily convex. We derive results on the structure of the set of Y-func¬ tions and of equivalence classes of such functions, and interpret these in terms of Orlicz spaces. These form a distributive sublattice of the lattice of subspaces of measurable functions (modulo equality almost everywhere). This lattice is not CT-complete. A natural complementary ordering on classes of Y-functions induces an order relation among Orlicz spaces in 3 which pairs of spaces are comparable. Section 3 is based on properties of the operation of forming the Young's complement of a Y-function : convexity at 0 is again a satisfactory substitute for convexity. Complementation solves the problem: given the Orlicz space L^, to find B such that f £ L^, g 6 Lg => |J* fg d|a|<«>. X The mapping A -* A induces a dual automorphism on the lattice of equivalence classes of Y-functions, and yields the result that the complementary ordering is isomorphic to the original one. A second dual automorphism relates the behavior of a Y- function for large values and for small values of the argument. Further information on the structure of the lattice follows. N-functions may be naturally introduced at this point. What might be called the "interior" of the lattice consists of N-functions. In Section 4 we consider the problem: to express, in terms of Y-functions, the relation among Orlicz spaces L^, Lg,!^ over a topological group given by f € L^, g € => f * g € LQ. We assume that the group is locally compact and unimodular. The basic result in the solution is given by R. O'Neil [6;Th.2.5] the relation _1 :L 1 A (X)B’ (X) ^ xC“ (x) is sufficient. It turns out to be necessary as well; we treat 4 the problem using Y-functions, and the associated problem of finding a third function, given two of A,B,C, for which the relation holds. In each case, if there is such a function, a best possible one can be found. We also consider the cases in which only large values, or only small values, are relevant. 5 (1.1) Definltion: The function A:[0,°°) -♦ [0,°°] is a Young's function if A(0) a 0, and A is left-continuous, convex, and non-decreasing. The two trivial Young's func¬ tions are those which on (0,°°) are respectively identically 0, identically +®. (1.2) Definition; The Orlicz space L^(X,S,|j) determined by A over (X,S,n) consists of those (equivalence classes of) measurable functions for which the quantity ||f||^ = inf {K > 0: J* A( I f (x) | /K)d|_i <; 1} (inf 0 = +») (1.3) X is finite. || • ||^, where finite, is called the L^-norm. The functions x^, lsp<» are the Young's functions which determine the spaces Lp(X,S,n). L°°(X,S,|_i) is determined by the func¬ tion ( 0 0<;x<;l >^(x) = J (1.4) I CD 1<X We use the following results, which may be found in [4] . 1° Two Orlicz spaces L^, Lg are comparable under inclusion if there is a positive constant k such that A(x) ^ B(kx), x s 0. In this case 2 Lg. We write A £ B for this relation. In case the underlying measure space has a a-finite, atom-free subset of infinite measure the condition A s B is necessary in order that 2 Lg . [4; Thm.5 p 52] 6 2° The Young's complement, £, of a Young's function A is defined by £(x) = sup(xy - A(y)) (1.5) y A,A satisfy Young's inequality xy <; A(x) + X(y), x,y > 0 , (1.6) from which is derived a pairing of L^, by means of the bilinear functional (defined on L^xL^) (f,g) -» 5 fg dpi . X Holder's inequality in the form ;x Ifgldn * 2||f||Allg|lK is immediate. (see [4;p 47] and [3;§14]) The map A -♦ reverses the order in 1° and "K = A. If A satisfies the Ap-condition A(2x) s KA(x), x ^ 0, K > 0 a constant, (1.7) then is the continuous dual of L^. In certain cases the condition is necessary. [4;pp 58-60] 3° If the measure space (X,S,n) is totally finite the values of Y-functions for small values of their arguments play no role; if (X,S,|_i) is purely atomic, and its atoms all have measure ^ 6 > 0 for some 6, the values of the Y-functions for large values of their arguments play no role. To see this, consider the following: Let Eu = {x € X:|f(x)| > u}, 7 u > 0. Then J A(|f(x) |/K)d(i(x) = J* A ( | f (x) | /K)d(j (x) + J* A(|f (x) |/K)dM(x). X E, X~E u u The second integral, in the presence of total finiteness, is dominated by |j(X)A(u/k). If all the atoms of a purely atomic (X,S,n) have measure ^ 6 > 0 the first integral, when represented as a sum, can be made to vanish by taking K sufficiently large. 8 (2.1) Definition: A function A:[0,®) -* [0,®] is a Y-function if A is left-continuous, convex at 0, and A(0) =0. A Y-function is non-trivial if it is not identi¬ cally 0 or identically ® on (0,®). For reference we have (2.2) Lemma: The function A defined on [0,®) is a Y-function if and only if 1° A(x) I 0, all x i 0; 2° A(x)/x is non-decreasing; 3° A(0) = 0; 4° A is left-continuous.
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