Lorentz-Zygmund Spaces and Interpolation of Weak Type Operators Citation Rudnick, Karl Hansell (1976) Lorentz-Zygmund Spaces and Interpolation of Weak Type Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7TM3-QK38. https://resolver.caltech.edu/CaltechTHESIS:02222017-144637159 Abstract The Lorentz-Zygmund spaces L pa (log L) α are a class of function spaces containing as special cases the classical Lebesgue spaces L p , the Lorentz spaces L pa and the Zygmund spaces L p (log L) α . It is shown here that the Lorentz-Zygmund spaces provide the correct framework for the interpolation theory of weak type operators. The interpolation principles established here unify many classical results in harmonic analysis. In particular, there are applications to the Fourier transform, the Hardy-Littlewood maximal operator, the Hilbert transform, and the Weyl fractional integrals. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Bennett, Colin Thesis Committee: Unknown, Unknown Defense Date: 20 May 1976 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Record Number: CaltechTHESIS:02222017-144637159 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02222017-144637159 DOI: 10.7907/7TM3-QK38 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10061 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 22 Feb 2017 23:40 Last Modified: 23 Aug 2024 22:57 Thesis Files Preview PDF - Final Version See Usage Policy. 21MB Repository Staff Only: item control page

LORENTZ-ZYGMUND SPACES AND INTERPOLATION OF WEAK TYPE OPERATORS Thesis by Karl Rudnick In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1976 (Submitted May 20, 1976) ii To Jill (and Thor) iii ACKNOWLEOOEMENTS I am deeply gratef'ul to my advisor, Dr. Colin Bennett, for his encouragement, advice and friendship. His tremendous enthusiasm and brilliant insight have been an inspiration to me. I would also like to thank the students and faculty of the Ma.thematics Department for many stimulating mathematical discussions. I am indebted to the California Institute of Technology for financial support over the past four years in the form of Fellowships and Teaching Assistantships. Finally, special thanks is due to Lorraine R. Smith for her expert typing of the manuscript. iv ABSTRACT The Lorentz-Zygmund spaces Lpa(log L)a are a class of function spaces containing as special cases the classical Lebesgue spaces LP, the Lorentz spaces Lpa and the Zygmund spaces Lp(log L)a. It is shown here that the Lorentz-Zygmund spaces provide the correct framework for the interpolation theory of weak type operators. The interpolation principles established here unify many classical results in harmonic analysis. In particular, there are applications to the Fourier transform, the Hardy-Littlewood maximal operator, the Hilbert transform, and the Weyl fractional integrals. v TABLE OF CONTENTS page Chapter I. ·1• Introduction Introduction • • Chapter II. .............. Inequalities and Preliminaries 2. Generalized Hardy Inequalities •• 17 3. Decreasing Rearrangements of Functions • 27 4. Averaging Operators 29 5. 30 Lorentz Spaces Lpa •• Chapter III. The Lorentz-Zygmund Spaces 6. The Lorentz-Zygmund Spaces Lpa(log L)a 32 7. Lorentz-Zygmund Spaces on the Unit Circle 35 8. Lorentz-Zygmund Spaces on the Integers . 37 9. The Spaces Lpa(log L)a + Lqb(log L)A and Lpa(log L)a n Lqb(log L)~ • 44 o. Inclusion Relations 54 11. The Auxiliary Lorentz-Zygmund Spaces • 59 12. The Embedding Theorem 1 Chapter rf. • • • 69 The Interpolation Theory 13. Operators of Weak Type (p,q;r,s) • 14. Proof of the Main Results References . .. . ... 75 81 88 1 CHAPrER I INTRODUCTION 1. Introduction The principle of operator interpolation has extensive applications in harmonic analysis. To illustrate this principle let us consider a special case of the classical interpolation theorem of Riesz-Thorin [43, Chapter XII]. Let T be a linear operator defined on the Lebesgue 1 space L (f), the class of Lebesgue integrable functions on the unit circle ~. SUppose T has the following properties: and T 1 co co -+ 1 ; equivalently, Tis strong type (1,1) and strong type (co,co), respectively (cf. (1.4)). Since the LP spaces on the circle~ satisf'y the inclusions 1 < p < co, it is then natural to ask whether T is bounded on the LP spaces, 1 < p < co. The Riesz-Thorin theorem provides the affirmative answer: 1 ( ) The notation T : X-+ Y, where X and Y are (quasi) normed spaces, signifies that T is a continuous map of X into Y. 2 1 < p < co. (1• 1 ) Thus, the Riesz-Thorin theorem takes the strong type hypotheses on the 1 co "endpoint spaces" L and L and "interpolates" to establish the intermediate result (1. 1). We shall see that, for the applications, the conditions strong type (1, 1) and (co, co ) are much too stringent. Marcinkiewicz (28] was able to relax these conditions by introducing the notion of weak type (p,q) (cf . (1.5)). Under the weaker assumptions that the operator Tis of weak types (1,1) and (oo,co), Marcinkiewicz was still able to establish the desired interpolation result (1.1 ). Calderon [9] and Hunt [21] have shown that the Lorentz spaces Lpa form the natural setting for Marcinkiewicz's interpolation theorem. Of course, the LP spaces are not the only function spaces 00 1 intermediate between L and L for which we may want to interpolate operators of weak types (1,1) and (oo,co). For instance, let L log L denote the class of all functions on the unit circle T for which is finite. The space L log L is related to the LP spaces by the following inclusions: 1 LP c L log L c L , < p ::: co. 1 Hence, L log L is very "close" to the "endpoint space" L • linear operator of weak types (1,1) and result due to Zygmund [43, p. 119] : If Tis a (oo,oo), we have the following 3 1 T : L log L ~ L • (1 .2) The classical theory contains many results of the types described above. It was shown in [6] that the natural setting for all of these results is a class of function spaces rf>a(log L)a called the LorentzZygmund spaces. They contain as special cases both the Lorentz spaces Lpa (take a = 0) and the Zygmund spaces Lp(log L)a (take a = p). Furthermore, the interpolation theorems for these spaces produce easily the various classical estimates. These theorems were established in [6] in the context of the unit circle f. The purpose of this dissertation is to extend the results of [6] to arbitrary measure spaces. This more general setting requires a more complex technical machinery. However, we show that the simple structure of the results themselves remains intact. The following operators arise natura.lly in classical harmonic analysis: the Fourier transform 7; the Hardy-Littlewood maximaJ. operator M; the Hilbert transform H (conjugate-function operator); and the fractional integrals IA, 0 <A< 1. Each of these operators has a definition on functions on the unit circle 'r, the integers ~ , or euclidean space Rn (cf. [36, 42, 43]). There are many classical results concerning the mapping properties of these operators on the Lebesgue spaces LP, the Zygmund spaces Lp(log L)a and the Lorentz spaces Lpa. Each of these mapping properties is intrinsica.l.ly interesting but, in their existing form, they are apparently unrelated. However, in the setting of the Lorentz-Zygmund spaces, all of these classical results are unified by a natural, comprehensive interpolation 4 theory. Let (X,µ) be any measure space and let f * denote the decreasing rearrangement of a measurable function f on X (cf. (3.2)). 0 < p, a For S oo, the Lorentz space Lpa(X) =Lpa consists of all (classes of) measurable functions f for which the quasinorm (~ [tl/pf*(t)]a dt/t) l/a 0 0 < a < oo, , ( 1 • 3) a = oo, is finite. A quasilinear measure space ~J,v) operator T mapping measurable functions on a (X,µ) into measurable functions on a measure space is strong ty;pe (p,q) if 0 < p,q s oo, ( 1 • 4) and is weak type (p,q) if T 0 < p < oo, T p = oo, 0 <q 0 < q s oo, (1.5.i) s oo, ( 1 • 5. ii) For p ~ 1, the notion of weak type (p,q) is indeed weaker than the notion of strong type (p,q) (cf. Section 5), ool The definition (1.3) shows that the Lorentz space L Hence, the definition (1.5 .i ) is of no interest when p = oo, = [o}. The definition (1.5.ii) is somewhat artificially made toaccornmodatethis situation. The definitions (1 . 4) and (1.5.ii) show that ~eak type (oo,oo) is equivalent to strong type (oo,oo). Thus, the condition weak 5 type (oo,oo) is much too restrictive as we shall see later. The two theorems cited below give the known weak type and strong type mapping properties for the aforementioned operators. THEOREM 1. 1 (a) : The Fourier transform :f is strong (hence weak) types (1,oo) and (2,2). (b) (Hardy-Littlewood [ 15]). The Hardy-Littlewood maximal operator M is weak types (1,1) and (oo,oo). (c) (Zygmund (41 ]). The fractional integral operator IA, 0 <A< 1, 1 1 is weak types ( 1, ( 1 - X ) and ( X- , oo) • r (d) (Kolmogorov [23]). THEOREM 1.2 (a) The Hilbert transform His weak type (1,1). (Hausdorff-Young [19, 43, p. 101]) : The Fourier transform :if has the property: 1 < p < 2, (b) (Hardy-Littlewood [15, 42, p. 32]). 1/p + 1/q = 1. The Hardy-Littlewood maximal operator M has the property: 1 < p < oo. (c) (Hardy-Littlewood (14, 43, p. 142]). The fractioneJ. integral operator IX' 0 < X < 1, has the property: 1 < p < q <co, (d) (M. Riesz [34, 42, p. 254]). property: 1/p - 1/q = X. The Hilbert transform H has the 6 1<p<oo. Theorems 1.1.d and 1.2.d show one might expect (by "extrapolation") some form of a weak type estimate for the Hilbert transform . 00 H at the "endpoint" oo • However, H is not bounded on L ; in fact, H is not even bounded on characteristic f'unctions (cf. [37]). (Actually H 00 maps L into the larger class of functions of bounded mean oscillation, BMO (cf. [12]).) Hence, His not weak type (oo,oo)(= strong type (oo,oo)). This difficulty is overcome by introducing the notion of weak ty;pe (p,q;r,s) (p < r, and q ~ s; see Section 13 for the definition and details). The notion of weak type (p,q;r,s) was first introduced in [4] when p = q and r = s, and in general was first presented in [6]. If r <co, a quasilinear operator T is of weak type (p,q;r,s) only if T is weak types (p,q) and (r,s). when r = oo. if and The crucial difference occurs The following theorem summarizes the weak type estimates for the operators :F, M, H and I\ relative to the notion of weak type (p,q;r,s). THEOREM 1. 3 ( [ 6, Pa.rt IV]): (a) The Fourier transform:! is weak type (1,oo;2,2). (b) The Hardy-Littlewood maximal operator M is weak type (1, 1 ;oo,oo). (c) The fractional integral operator IA.' O <A.< 1, is weak type ( 1, ( 1 - (d) A. ) -1 ; A. -1 , oo ) • The Hilbert transform His weak type (1,1;00,00), 7 Theorem 1.3.d shows that we now have weak type estimates at both "endpoints" 1 and oo for the Hilbert transform H. This fact, in conjunction with the interpolation theorems cited below, gives a direct verification of the strong type mapping properties for H (cf. Theorem 1.2 . d) in addition to establishing a variety of other mapping properties. Suppose O < p, a ~ oo, -oo <a < oo. The Lorentz-Zygmund space Lpa(log L)a defined on any measure space (X,µ) is the set of measurable functions for which the quasino.rm 0 < a < oo, ( 1 • 6) sup t l/p(l + Ilog ti )a f*(t), a = oo, O<t<ix> is finite. When a = o, the space Lpa(log L)O reduces to the Lorentz space Lpa. When the underlying measure space is the circle l' and a = p, the Lorentz-Zygmund space Lpp(log L)a reduces to the Zygmund space LP(log L)a (cf. [6, Section 10]). With respect to these Lorentz-Zygmund spaces and operators of weak type (p,q;r,s), we shall prove the following generalization of the fundamental Marcinkiewicz interpolation theorem. THEOREM A: Let 0 < p < r ~ oo and 0 < q,s ~ oo, with q ~ s. SUppose T suppose 0 < e < 1 is a quasilinear operator of weak type (p,q;r,s). and let 1 e 1 - a -=- + - p r u 1 e 1 - e -=-+-v q s (1. 7) 8 Let 0 < a < oo and - oo < a < oo • Then Using Theorem A and the weak type estimates of Theorem 1 • 3, we now have a simple proof of Theorem 1 .2. Next we choose e' such that 0 < &' < e < 1, and we define u' and v' so that u', v' and e' are related as in (1.7). ShOWS that (for 0 < a I :::: 00 1 - oo Then Theorem A also < CX 1 < oo) and The next theorem deals with the limiting case of these results where we let e ~ 1 and e' ~ O (that is, u = p, v = q, u' = r and v' = s). THEOREM B: Let O < p < r :::= oo, and O < q, s :::= oo, with q I s. is a quasilinear operator of weak type 1 :::= a ~ b :::= oo, (a) (p,q;r,s). 1 :::= c :::= d :::= oo and - oo <a, ~' y, suppose T Suppose o < oo. Then, if a+ 1/a = 8 + 1/b > O and y + 1/c = 5 + 1/d > o, we have pa T : L a+1 re y+1 qb 8 sd o (log L) + L (log L) ~ L (log L) + L (log L) ; and (b) if a + 1/a = a + 1/b < 0 and y + 1/c = 0 + 1/d < o, we have 9 The sums appearing in Theorem B are defined as follows. If p < q, the space Lpa(log L)a + Lqb(log L)e is generated by the quasinorm (1.10) and if p > q, the generating quasinorm is (1.11) rPJ + ( j [t 1I * P(1+log tPf (t)]adt/t )1/a • 1 These spaces are the usual algebraic sums of the spaces Lpa(log L)a and Lqb(log L)~ in all instances except when one of the spaces is trivial (cf. Section 9). The function space generated by ( 1 • 10) when p > q, or by (1 • 11 ) when p < q is just the usual set theoretic intersection Lpa(log L)a n Lqb(log L)R (cf. Section 9). In the case where the underlying measure space is the unit circle f (or any finite measure space), the sums and intersections in Theorem B reduce to a single Lorentz-Zygmund space (cf. Section 7). When the underlying measure space is the integers .?l,the sums and intersections now reduce to a Lorentz-Zygmund sequence space t pa(log t)a (cf. Section 9), consisting of sequences (c } for which n the quasinorm 10 * is finite (here [c } 00 n n=1 is the decreasing rearrangement of the sequence In view of the above remarks, when T maps function spaces on ~ or 7l into function spaces on I' or 7l , the statement of Theorem B is simplified. Restating Theorem B in these special instances, we have the following theorems. THEOREM B1 (Measure spaces ~, I') O < q < s S oo. SUppose O < p < r S oo and Then and THEOREM B2 (Measure spaces I', 7l ) : Suppose 0 < p < r < oo and 0 < s < q S oo. Then and (b) T. : Lra(log L)a+l ~ tsb(log t)~, if a+ 1/a =A + 1/b < o. Similar restatements of Theorem B can be obtained by other combinations of the measure spaces ~ and 7l • We may now present some classical results in harmonic analysis which, after reformulation in terms of Lorentz-Zygmund spaces, are direct consequences of Theorems B1 or B2. To apply Theorem B1 or B2 11 we restrict our attention to operators defined on the circle r. THEOREM 1.4 (Hardy-Littlewood (15, 43, pp. 158-159]): The Hardy- Littlewood maximal operator M has the property: M : L(log L) -t L1 • THEOREM 1.5 (a) (Zygmund [39, 41, 42, 43)): For the conjugate-function operator H, we have H: L(log L) -t t 1 • (b) If Ir! < 1 a.e., then for some positive constants y and C independent of f. (c) H also has the property: a> o. THEOREM 1 • 6: If 0 < 'A < 1, the Weyl fractional integral. operator ; has the following properties: (a) (Zygmund [40]). IA : L(log L)l-'A -t Ll/(l-\) • (b) (Zygmund [ 43, pp. 158-159]). If l!fll l /\ ~ 1, then 1 ~nexp(v1Ixfl 1 /(l-\)) ~ c < ~, 0 . where y and C are positive constants independent of f. 12 ( c) (O'Neil (31]) . Let p = (1-\)-1. Then L(log L)a ~ KP(log+K)p(a- 1 ), (i) I\ '( ii) I\ : L(log L)a ~ Lra and THEOREM 1.7 :J(f) = (c 1 -1 , O<a<1. (Hardy-Littlewood [16,17], Zygmund [40]): Let 00 be the sequence of Fourier coefficients of f with int respect to the orthonormal system e , n = o, :!:_ 1 , :!:_ 2, • • • Let n n=- oo (c:1n: 1 denote the decreasing rearrangement of (en 1n:- 00 • Let f E +,(log L)a, a > o. (a) For some constants A and B independent of f, we have a a 00 j n- 1 (log nf' - 1c: :'.': ~ ~"JrJ (log+Jfl )a +Ba • n=1 (b) If O < a ::'.: 1, then 00 \ L n=1 We have already remarked that the spaces Lpa and Lp(log L)a are the Lorentz-Zygmund spaces Lpa(log L)O and LPP(log L)a, respectively. The following theorem, proved in (6, Section 10], shows that the other classes of functio ns mentioned in the four previous theorems are also Lorentz-Zygmund spaces . THEOREM 1 , 8: (a) 0000 If a > o, the Lorentz-Zygmund space L (log L)-a is the Zygmund space consisting of those functions f (on ~) for which 13 for some positive constant \ = \(f). (b) If O <a, p < oo, the Lorentz-Zygmund space Lpl (log Lf' coincides with O'Neil's space Kp(log+K)ap. (The space Kp(log+K)a consists of all f for which where Df is the distribution .function off (cf. (3.1))). 00 (c) For sequences {cn }· n=-oo of complex numbers and a > o, we have that (i) ~l (c n } n=-oo 00 00 \' ·..- n n=l -1 a * } ool a (log n) en< oo = t (log t) ·, and (ii) if O <a~ 1, we have Applying Theorem 1.8 to reformulate Theorems 1.4 through 1.7, these theorems now reflect that the various operators M, H, IX and .7 are continuous transformations of Lorentz-Zygmund spaces. THEOREM 1 •4 ' (Hardy-Littlewood ) : THEOREM 1 • 5 ' (Zygmund ) : 14 (b) H 0000 L (log L) 0 0000 -+ L (log L) - 1 • a> o. ( c) 1 Let p = (1-A.r • THEOREM 1.6' (Zygmund, O'Neil): (a) IA 11 1-A pp 0 L (log L) -+ L (log L) • IA LA (i) IA L11 (log L)a-+ LP 1 (log L)a-l, (ii) IA : L11 (log L)a .... Lp,l/a(log 1) 0, (b) (c) -1 -1 A (log L)O-+ L0000 (log L)A-l. a;:::1, and O<a<l. THEOREM 1. 7 1 (Hardy-Littlewood, Zygmund): (a) :; : L11 (log L)a-+ t 001 (log t)a-l, a> o. (b) 00 :I: L11 (log L)a-+ t 'l/a(log t) 0, O<a<l. It is clear that the weak type estimates of Theorem 1.3 in conjunction with the interpolation Theorems Bl and B2 give simple proofs of Theorems 1. 4' through 1.7'. Let us call the number a+ 1/a an index of smoothness for the Lorentz-Zygmund space Lpa(log L)a (or tpa(log t)a). The common feature of all the results in Theorems 1.4' through 1.7' is that the index of smoothness of the domain is always greater than the index of smoothness of the indicated image space. This is the simple essence of Theorem B. We conclude this section with a discussion of the Hilbert transform H on the real line ~ and the fractional integrals IA. (Riesz potentials) on euclidean space ~n. These results are implicit in the 15 ; work of Calderon and Zygmund [10, 11] and are also discussed in the work of Torchinsky [38] and Koizumi (22]. THEOREM 1.9: For the Hilbert transform Hon the real line, we have that (1.12) for a > O and 1 < p < oo. THEOREM 1. 10: Let 0 < X < 1 and 1 < p < q < oo, with 1/p - 1/q = X. I Then the fractional integral operator IA (Riesz potential) has the property: (1.13) The space L(log L)a +LP appearing in (1.12) has the quasinorm (cf. ( 1 • 10) Ir1 (1-log t)af* (t)dt + Vo (f * [f (t)]Pdt)1/p • 1 Hence, Theorem 1.9 is really just a combination of the local result in Theorem 1.5.c with the strong type result of Theorem 1.2.d for the Hilbert transform H. Similarly, we see that the result (1.13) for IA is a combination of the local result in Theorem 1.6.a with the strong type result in Theorem 1.2.c. Both Theorems 1.9 and 1.10 are instances of the following theorem, which is the limiting case of the results (1.8) and (1.9) where we let either e ~ 1 ore'~ o. THEOREM C: Let 0 < p < r '.'.:: oo, and 0 < q, s :=: oo, with q ~ s. T is a quasilinear operator of weak type (p,q;r,s). Suppose Suppose 16 1::: a::: b::: oo, o < c::: oo, -oo <a,~,v < oo and o < 9 < 1. 1 e 1-e -= - + u p r (a) 1 9 Let 1-0 -=-+-. v q s If a + 1/a = ~ + 1/b > o, we have (i) T : Lpa(log L)a+l + Luc(log L)V ~ Lqb(log L)~ + Lvc(log L)V, and (b) If a + 1/a = q + 1/b < o, we have (i) T : rf'a(log L)a+l n Luc(log L)V ~ Lqb(log L)~ n Lvc(log L)V, and 17 CHAPI'ER II INEXtUALITIES AND PRELIMINARIES 2. Generalized Hardy inequalities The inequalities established in this section form the foundation of the subsequent development and will be appealed to frequently. We begin with two modest technical lemmas. LEMMA 2.1: Let"-,µ and a be positive real numbers. Then, we have (2. 1 ) Proof: Clearly, and Adding the two inequalities and taking ath powers, we have the righthand inequality in (2.1). Replacing A by Al/a and~ by ~l/a in the inequality just established, then taking ath roots, we obtain This is precisely the left-hand inequality in (2.1) with a replaced by 1/a. L~ 2.2: Let 11 > o, a real. Then there is N = N(a,A) > 1 such that (a) 8 t- (N + Ilog ti )0 is decreasing for t E (o,~); 18 (b) t~(N + jlog ti )a is increasing for Proof: t E (O,oo). An examination of the derivative shows that N = 1 + jaj/~ will suffice. The following two lemmas were established in [6, Lemmas 6.1 and 6.2] . LEMMA 2.3: suppose 0 <a ~ oo, and 0 < v < oo. decreasing f'unction on (O,oo). " Let cp be a nonnegative Then, for each O < t < oo, we have (rt" 2 sup s cp ( s) ~ c J [ s cp( s) ]ads/ s)1/a ( ) O O<s<t (2.2) sup s "cp (s) ~ c ([ [s "cp(s) ]ads/s)1/a, t2s<ixi t/2 (2. 3) and where c is a constant independent of cp and t. LEMMA 2.4 : Let 0 < " < oo and let q> be a nonnegative decreasing function on (O,oo) . (a) Then, for 0 < t < oo, if 0 <a~ 1, then rt s \)cp (s)ds/s ~ c (rt [s \)cp(s)]ads/s)1/a, j (2.4) j 0 0 and (2 ) Throughout this paper, when a= oo, an integral is to be interpreted as ess sup v(t). c<t<d (~[t(t)]adt/t)l/a c 19 r" ( s cp(s)ds/s ~ c .JrlX' [s " cp(s)] ads/s)1/a • t t/2 (b) (2. 5) If 1 ~a~ oo, then (Jrt0 [s"cp(s)]ads/s)1/a ~ cIt0s"cp(s)ds/s, (r " and (2. 6) ) [s cp(s)]ads/s 1 /a ~ cjr:xi s "cp(s)ds/s. t t/2 (2.7) The next theorem is a variant of the classical Hardy inequalities (cf. [21, p. 256]). For measurable functions on (0,1), it was first proved in [6, Theorem 6.4]. suppose A. > o, THEOREM 2.5: 1 ~a ~ oo and nonnegative measurable function on (O,oo). -oo < o: <co. Let V be a Then the following four inequalities hold: (a) (2. 8) (b) (2. 9) ~ c ( c) ( r (J01[tA.+l (1-log t)o:;(t)Jadt/t)1 /a ; t 1/a [t-A.(1+log t)o:J +(s)ds]adt/t) 1 1 (2. 10) 20 I :P" A. I i [t (1+log t) (d) \., 1 ar°° .J t a )1/a '1t(s)ds) dt/t (2. 11 ) ("oo \1/a \ : '.: C\J (t ' +1 (l+log tP Ht) ]adt/t) 1 • 1 Furthermore, suppose 0 < a < 1 and 1~ ( t ) = t u.-1 q> ( t ) , where µ. > 0 and~ is a nonnegative decreasing function. Then (2.8) and (2.9) remain valid while (2.10) and (2.11) are replaced by 1 ['°° A. a ,.t ) 1I a :\.; [t- (1+log t) J w(s)ds]adt/t 1 1 (CI) (2.10 1 ) rP> 1/a :::: c(J [t-\+l (1+log t)a1Jl(t/2)]adt/t) ; 1 r"° A. rPl 1/a ( j (t (1+log t)aj $(s)ds]adt/t) (d I) t 1 (2.11 1 ) The constant c depends only on A., a, a (andµ when O <a< 1). Proof: The proof given here for (2. 10) and (2.11) is a modification of that given for the classical Hardy inequalities in [21, p. 256). The inequalities (2.8) and (2.9) are the content of Theorem 6.4 in (6) and so we omit their proofs. First, we note that when N ~ 1, 1 + !log ti < N + Ilog ti ::'.: N(1+llog ti). (2.12) Hence, it will suffice to prove the inequalities (2. 10) and (2. 11) with (1 + Ilog ti) replaced by (N + !log ti). Let 1:::: a:::: oo. To prove (2.10) choose y so that 21 1 -X < 'V < 1. (2. 13) We next write, for t > 1, rt 1 .i 1 rt if(s)ds = j [syv(s)][s 1 1- Y]ds/s • Now we apply HBlder's inequality (with respect to the measure ds/s) to the expression on the right to obtain the inequality rt )1/a 1 y(rt : v(s)ds <ct - ! [s"'v(s)]ads/s • "'1 - (2. 14) "1 Inequality (2.14), plus a change in the order of integration yields (2.15) ~ c~[syt(s)Ja(~t-(X+Y-l )a(N+log t)aadt/t)ds/s. s 1 Let N = N(a,1/2(X+Y-1)) as in Lemma 2.2.a so that t-(\+V-l)/2 (N+log t)a decreases fort E [1,oo). largest when t = s. Thus, on [s,oo), t-(X+V-l)/2 (N+log t)a is Letting I denote the right-hand side of (2.15), we have I~ cJ rs"'t(s)·s-(X+y-l )/2 (N+log s)aJa(J°t-(X+V-l)a/2dt/t)ds/s. 00 1 s The choice of V, (2.13), allows us to evaluate the integral. on (s, 00 ) , giving (3 ) The constant c may change from line to line. 22 2c j -A+1 a a I ~ a().+y-l) j [s (N+log s) +(s)] ds/s. (2. 1 6) 1 The estimates (2.15) and (2.16) produce (2. 10) (modulo (2.12)) for 1 < a < oo. For the case a= oo, (2.14) reads By Lenuna. 2.2.a, decreases. t -A we choose N = N(a,). + v - 1 ) so that t 1-v->..( N+log t )a Let 1 < t < oo, then rt ( ) a 1 -V-A a 'V (N+log t) JlW s ds ~ct (N+log t) sups '(s) 1<s<t < c sup s-X+l(N+log s)0 w(s). - 1<s<t Replacing the supremum over (1,t) by the supremum over (1,oo), and taking the supremum of the left-hand side over 1 < t < oo, we have (2,10) for this case, too. Now suppose O <a< 1. and~ is decreasing . We then have •(s) = sµ-l~(s) whereµ> O In addition to (2.13), we choose y so that v > 1 - µ. • (2. 17) Now we write, for 1 < t < oo, By (2 . 17), µ. + y - 1 > o, so we may apply (2.5) (with t = 1) to obtain 23 This is the analogue of the crucial inequality (2.14). We then have the estimate Enlarging the range of integration (1,oo) to (1/2,oo) and then interchanging the order of integration, we obtain, Proceeding as before, we have the estimate J < cjrx' - 1/2 [s -\+1 (N+ Ilogs I )a t(s)] a ds I s. Changing variables (let s = t/2), we have 1 which is (2.10 ) (modulo (2.12)). The inequalities (2.11) and (2.11 ')are proved quite similarly. We choose V now so that 1<y<1+\. (2. 18) For 1 ~a~ oo, the analogue of the crucial inequality (2.14) is (2. 19) 24 By (2.18), 1 - V + X > o, so we choose N by Lemma 2.2.b so that t(l-V+X)/2 (N+log t)a increases. Applying (2.19), we have the estimate ~ c~[tl-v+X(N+log t)aJa(~[svt(s)]ads/s)at/t t 1 = cjrx' [s vit(s)] a( jrs [t 1 -v+\ (N+log t)a ]adt/t) ds/s 1 1 rP> \+1 a a [s (N+log s) 'lt(s)] ds/s, ~ cj 1 which is the desired inequality (2.11). As before, the case a= oo is easy so we omit it. When 0 < a < 1, we write r ·Hs)ds = [ s l -Yf sµ+v- 1cp(s) ]ds/s ~ t 1-vfsµ+V-\p(s )ds/s. t t t The inequality (2.5) now gives the slightly different version of the crucial inequality (2.19): Proceeding as before, we obtain (2.11 1 ). This completes the proof. The following theorem is a generalization of the classical.. Hardy inequalities (they may be obtained by taking a= o). The proof is entirely similar to the proof of Theorem 2.5 so we omit it. 25 THEOREM 2.6: Let X > o, 0 <a< oo and -oo <a< oo. If either l < a < oo and t is a nonnegative measurable function on ( o, oo) ; (a) or (b) 0 < a < 1 and V( t) = tµ.-\p( t), where µ. > 0 and cp is a nonnegative, measurable decreasing function; then (i) and (ii) (2.21) ~ c(~rtX+ 1 (1+jlog ti )at(t)]adt/t) 1 /a. 0 The following variants of Hardy's inequalities correspond to the limiting cases X = 0 in Theorem 2.5. In contrast to Theorem 2.5, the order of the logarithmic term now increases by a factor of one from left to right. The inequalities (2.22) and (2.24) were established in [1, Theorem 6.2]. THEOREM 2. 7: suppose 1 <a< oo and a+ 1/a. f O. measurable function on (O,oo). (a) If a + 1/a > o, then Let it be a nonnegative 26 (i) (J:rc1-1og tlaJ:v(s)ds]adt/t) 1/a (2 .22) 1 ~ c(J [t(1-log t)a+l'(t)]adt/t) 1/a, 0 (ii) 1 (f,[(1+log t)o:(w(s)ds]adt/t) /a (2. 23) ~ c(~[t(1+log t)a+l~(t)]adt/t) 1 /a. 1 (b) If a + 1/a < o, then (2. 24) 1 ~ c(J [t(1-log t)a+l*(t)]adt/t) 1/a, 0 (iv) (~[(1+log t)o:Jtv(s)ds]adt/t) 1/a 1 1 (2.25) ~ c(~[t(1+log t)a+l•(t)]adt/t) 1/a. 1 Proof: We shall prove (2.23). Let a' satisf'y 1/a + 1/a' = 1. Since a + 1/a > O, we have a + 1 > 1/a 1 • Thus, we may choose V so that 1/a' < y <a+ 1. (2.26) We now write 27 Then, we apply H8lder's inequality and (2.26) to obtain (2.27) For 1 ~ a < oo, we can now make the estimate fr (1+log t)ar'41(s)ds]adt/t 't 1 ~cf [(1+log t)a-y+l/a' Ja(f [s(1+log s)Yv(s)]ads/s)dt/t t 1 = c~[s(l+log s)Yt(s)Ja(Js(1+log t)aa-ya+a- 1dt/t)ds/s 1 1 r [s(1+log s)a+1 W{s)] ads / s, ~ c 1 (the last inequality is available by (2.26)). (2.23) for 1 < a < oo. The above estimate gives The case a = oo is easier and so we omit it. The inequality (2.25) is proved in much the same way. Since a + 1/a < o, we now choose y so that a+ 1 <y<1/a'. The crucial inequality (2.27) is replaced by rt j w(s)ds ~ c(1+log t)-'V+ 1 and we proceed as before. 3. i/a '(Jt [s(l+log. s)y'(s)]ads/s)1/a , 1 This completes the proof. Decreasing Rearrangements of Functions Let (l,µ) be any measure space. For each complex-valued measurable function f on l we have its distribution function 28 Df ( y) = µ. [x : If ( x) I > y}, 0 < y < oo. The function Df is decreasing and right continuous. (3.1) Hence, it has a right continuous inverse f * (t) = inf[y : Df(y) ~ t}, 0 < t We call f * the decreasing rearrangement of f. < oo. (3.2) The mapping f ~ f * is not subadditive, but we do have (f+g) * (t) ~ f * (t/2) + g* (t/2), 0 < t < oo. (3.3) 0 < t < oo, (3.4) 0 < t < oo. (3.5) However, for the averaged rearrangement f ** (t) = t _,It * f (s)ds, 0 the mapping f ~ f ** is subadditive. That is, (f+g) ** (t) ~ f ** (t) + g** (t), More generally, if~ is a nonnegative decreasing function on (O,ro), then It * ,.t * <p(s)(f+g) (s)ds ~ J cp(s)f (s)ds + 0 0 It * q>(s)g (s)ds, 0 < t < oo. (3.6) 0 Another useful fact is: given two measurable functions f and g and a measurable subset E of X, we have rI lfgldµ < r(E) f * (t)g* (t)dt. VE - 0 (3. 7) Discussion and proofs of these results may be found, for instance, in the work of Luxemburg (27], Lorentz (24], and Hunt (21]. 29 4. Averaging Operators For 0 < p ~ oo, the averaging operators A , B , C and D are p defined as follows: p p p Let f be a measurable function on ('I,µ) and f * For O < t < oo, define its decreasing rearrangement. (A f * )(t) = t -1/pJt s 1/pf * (s)ds/s; (4. 1 ) 0 p (4.2) (Cf* )(t) = t -1/p sup s 1/pf * (s) p O<s<t (4. 3) 1 (Dpf * )(t) = t-l/p sup s /Pr*(s) . t<s<Jx> (4.4) , As seen in the work of Calderon [9] and Boyd [7], the operators A and p B play an important role in the theory of weak type interpolation. p The operators C and D first appear in [6] and play an equally p p prominent role (cf. Section 13). Each of the functions Apf*, Bp f *, Cp f * and Dp f * is decreasing and right continuous; hence, each is equal to its own decreasing rearrangement. Note that A1f * = f ** and so A1 is subadditive by (3.5). The relation (3.6) implies A (f'+g) p * <- Ap f * + Ap g*, The following relations among the operators AP, BP, CP and DP will be useful later. 30 LEMMA 4. 1: Let O < p ~ oo. Then f * (t) -< (cf * )(t) <- c(Ap f * )(t), p (4. 6) f * (t) -< (Dp f * )(t) -< c(Bp f * )(t/2), (4.7) and for all t > O. Proof: The constant c is independent of f and t. The left-hand inequalities in (4.6) and (4.7) are obvious. The right-hand inequalities in (4.6) and (4.7) are precisely the inequalities (2.2) and (2.3), respectively (letting a = 1 and v = 1/p). 5. Lorentz Spaces Lpa This section is devoted to a brief discussion of the Lorentz spaces Lpa a Lpa(l) (recall (1.3)). (A complete discussion of the Lorentz spaces may be found, for instance, in [21 ].) The Lorentz spaces are complete linear spaces, but the functional l\ •1\pa defined by ( 1 • 3) is in general ~ a quasinorm. L 11 The spaces and Lpa, 1 < p ~ oo, 1 ~a~ oo are (equivalent to) Banach spaces. For 1 < p ~ oo, 1 ~ a ~ oo, the equivalent norm is obtained by replacing f* by f**(cf. (3.4)) in (1.3). The quasinorm !\fll in the case 0 <a< p ~ oo (cf. (25]). actually is a norm pa The Lorentz spaces Lpa generalize the classical Lebesgue spaces LP since Lpp = LP. We have the following inclusion relations among Lorentz spaces with the same primary indices p: O < a < b < oo • (5. 1) 31 In particu1ar, we have (5.2) and (5.3) Note that if T is a quasilinear operator such that then (5.2) and (5.3) show that Thus, for p ~ 1, the notion of weak type (p,q) is indeed weaker than the notion of strong type (p,q) (recall (1.4) and (1.5)). Lorentz spaces Lpa with different primary indices are related in special circumstances. For example, if µ(X) < 001 we have o < p < q ~ oo, O < a, b ~ oo. (5.4) If µ(E) > 1 for every set E of positive measure (which happens when X = 7l , the integers) , then o < p < q < oo, O < a, b ~ oo or q = b = co. (5. 5) REMARK 5. 1: Throughout this thesis, the notation X ~ Y, where X and Y are quasinormed linear spaces, will signify continuous embedding. 32 CHAPrER III THE LORENTZ-ZYGMUND SPACES 6. The Lorentz-Zygmund Spaces Lpa(log L)a: Let 0 < p, a ::: co and -co < a: < co. The Lorentz-Zygmund space Lpa(log L)a: on (I,µ) consists of all (classes of) measurable functions f for which the quasinorm ( 6. 1 ) is finite . The inequalities (3.3) and (2.1) show that llf\lpa;a: is indeed a quasinorm. As for the Lorentz spaces Lpa(cf. [21 ]), the space Lpa(log L)a: is complete with respect to the quasinorm 11 · ll ,..., . pa ;u. When a: = o, the Lorentz - Zygmund space Lpa(log L)O is just the Lorentz space Lpa. When a = p and the underJ.~"ing measure space is the unit circle ~, the Lorentz-Zygmund space Lpp(log L)a: is the Zygmund space Lp(log L)a: consisting of all functions f for which is finite. In view of these last remarks, we will make the following abbreviations: 33 pp a p a 11 a a L (log L) = L (log L) ; L (log L) = L(log L) ; 11 1 L (log L) = L log L. The next theorem describes the action of the averaging operators Ap , Bp , Cp and Dp (cf. Section 4) on the Lorentz-Zygmund spaces Lpa(log L)a. In the context of this section, Theorem 6.1 will aid in classif'ying which Lorentz-Zygmund spaces are in fact Banach spaces. Specifically, Theorem 6.1 shows that the operators A and C act on p p Lqa(log L)a as the identity provided p < q; if p > q, B and D act p p on Lqa(log L)a as the identity. THEOREM 6. 1: Let O < q, a ::: co and -oo <a < co. Then we have (6.3) provided 0 < p < q. The result (6.3) is also valid for Cp • Similarly, we have (6.4) provided 0 < q < p ::: co. The result (6.4) is also valid for D • p (4 ) Since A f* and r* are defined on the interval (O,co) it is implicit in p(6.3) and (6.4) that the measure space underlying the quasinorm ll·llpa;a is the interval (O,co). The symbol ",,,.'' denotes equivalence; i.e. there are positive constants c 1 and c2 independent of f such that c 1 \lf*llqa;a ::: llAPf*llqa;a ::: c2 llf*11qa;a • 34 Proof: Suppose 0 < p < q. By the relations (4.6) and the definition (6.1 ), we clearly have and To obtain the converse inequality for A , we apply the Hardy inequality p 1 (2.20) with\= 1/p - 1/q > 0 and *(t) = t /p-lf*(t) (in which case Theorem 2.6 applies for all 0 <a~ oo). To obtain the reverse inequality for Cp' we use the result for AP and the relation (4.6). The proof of (6.4) is similar, now utilizing the relation (4.7) · and the Hardy inequality (2.21). COROLLARY 6.2: Let 1 < p ~ oo, 1 < a < oo and -co <ex < oo. Then the functional (6.5) defines an equivalent norm on the Lorentz-Zygmund space Lpa(log L)a. Hence, for these choices of p,a and a, Lpa(log L)a is (equivalent to) a Banach space. Proof: We have that A1r* = f** (cf. (3.4)) and so the equivalence ( 6. 3) shows that because p > 1. The mapping f ~ f ** is subadditive (cf. (3.5)); hence, 35 by Minkowsk.i's inequality, !lfll:;a is indeed a norm. At times the expression (1+llog ti )a is awkward in computations. We have that The equivalence (6.6) is valid because 1 + !log ti and !log ti are asymptotically the same at 0 and oo, and the condition a + 1/a > 0 assures that the integral in (6.6) converges at t 7. = 1. Lorentz-Zygm.und Spaces on the Unit Circle In this section let (l,µ) be the unit circle ~ with normalized Lebesgue measure dt/2n. THEOREM 7.1: In addition to Corollary 6.2 we have: 11 The Lorentz-Zygmund space L (log L)a on the unit circle r is a Banach space whenever a ~ o. Proof: In this situation (6.1) becomes !lfll 11 ·a = ' ,(1-log t)af* (t)dt. I0 (7. 1 ) Since a~ o, the function (1-log t)a is decreasing and this is necessary and sufficient that the quasinorm (7.1) defines a norm (cf. [25)). This completes the proof. As for the Lorentz spaces Lpa on a finite measure space (cf. (5.4)),we have the following inclusion relations when the primary indices differ. THEOREM 7. 2 : Let O < p < q :'.: oo, O < a , b :'.: oo and -oo < a , B < oo. Then for the Lorentz-Zygmund spaces on~ (or any finite measure space), (7.2) Proof: Choose r so that p < r < q. (7.3) It will suffice to establish that (7.4) We first make the estimate: 1 1 1 0 8 1 = (J:rt /rr*(t)]a[t /P- /r(l-log t) ] dt/t) /a 1 ::: sup t 1 /rf*(t)(J [tl/p-l/r(1-log t)a]adt/t) 1/a . O<t<1 0 The supremum is, by (6. 1 ), the qu.asinorm llrllroo;O • The last integral is finite by (7.3). This proves the first inequality in (7.4). To verif'y the second inequality in (7.4) we apply (2.2) (with v = 1/r, ~ = f * ) to get (7.5) If ~ ~ o, (7. 3) implies O<t<1. 37 Inserting this inequality into (7.5) we have the second inequality in (7.4) for q > O. If B < o, choose E > O so that 1/q < 1/q + E < 1/r. Notice that t-E(1-log t)~ ~ c > o, where c is a constant independent of t. O < t < 1, Hence, This inequality, in conjunction with (7.5), gives the second inequality in (7.4) when B < o. An extensive discussion of the Lorentz-Zygmund spaces Lpa(log L)a on the unit circle r may be found in [6]. 8. Lorentz-Zygmund Spaces on the Integers We open this section with a couple of results valid for any measure space (l,µ). It is of interest to know when the spaces Lpa(log L)a are trivial. LEMMA 8.1: Suppose p = oo. If either (i) o <a< oo and a+ 1/a ~ o, or (ii) a = oo and a > o, then Looa (log L)a = rtO } • Proof: and rPJ 1 If 0 < a < co, then (l+jlog ti )aadt/t is finite if and only if a+ 1/a < 0. "o If a = co, then which can be finite only if a :S O. The next technical lemma is very important to the rest of the development. LEMMA 8.2: Suppose O < p < r < q :S oo, 0 < a, b :S oo and -co < a , p < co . Then for any measurable function f on l, ( 8. 1 ) Proof: We rewrite the left-most expression in (8.1) as 39 This is clearly dominated by The last integral is finite since r < q, proving the first inequality in (8. 1 ) • To prove the second inequality in (8.1), we fix t, 1 ~ t < oo, and write t 11r = tl/r 1 r(t /r_1/2r) It s 11rds/s < cIt s 11rds/s . 1/2 - 1/2 Therefore, since f * is decreasing, we have t lr* f (t) <cft I - .J 1/2 s lr* If (s)ds/s < c - r If 1/2 s lr* (s)as/s. (8.2 ) We rewrite the last integral as JrPl [ s 1/p ( 1+I log s I )a f * ( s) )[ s 1/r-1/p ( 1+I log s I ) -a ]ds/ s 1/2 When 1 ~a~ oo, we apply HBlder's inequality to get (by (8 . 2)) The right-hand side of (8.3) is dominated by 1 " cr( J [s 1 Ipf* (s)] a ds/s )1 Ia + (j~ [s 1 IP(l+log s)o:f * (s)] a ds/s)1 Ia -J:. . 1 I. _ 1/2 1 40 This proves the second inequality in (8. 1) when ~a~ oo. When 0 <a< 11 we apply (2.5) to (8.2) to give (8.4) Since p < r it is easy (cf. (7.6)) to show that s 1/r ~ cs 1/P(1+log s)a, 1 < s < oo. (8.5) The inequalities (8.4) and (8.5), together with (2.1), imply the second inequality in (8. 1) when O < a < 1. The proof is now complete. For the remainder of this section, the underlying measure space (I,µ.) for the Lorentz-Zygmund spaces will be the integers 7l . In addition to Corollary 6.2 we have: THEOREM 8.3: 11 The Lorentz-Zygmund space L (log L)a on the integers 7l is a Banach space whenever a ~ O. Proof: Let r1 j (1-log s)ads, O < t < 1, 0 (8.6 ) C?( t) = < t < oo. If f is a function on 7l , then f * (t) = f * (n-), n-1 ~t <n, n = 1, 2, • • . . (8.7) Therefore, by (6 . 1 ), we have . ) rx> * l!r\1 11 . = f * (1-) (r1 j (1-log t)adt + I (1+log t)af (t)dt. ,a o ' 1 41 Hence, by (8.6), we have The definition (8.6) and the fact a~ 0 show that~ is a decreasing function on (O,oo), Hence, the functional l\fll inequality (cf. (3.6)). 11 ,a satisfies the triangle ' We can now establish inclusion relations analogous to (5.5) for the spaces Lpa(log L)a(:;z ) with distinct primary indices. THEOREM 8.4: Let O < p < q ~ oo, suppose Lqb(log L)~ f o < a, b ~ oo, -oo <a, ~ < oo and [o} (cf. Lemma 8.1). Then, for the Lorentz- Zygmund spaces on :iZ, (8.8) Proof ·. Let f valid. Let us first suppose that either E q < oo; Lpa(log L)a. or q = oo, Sinee (...,N,µ ) = " '"' ' there1 at.ion (8 . 7) is . 0 < b < oo and A+1/b < 0 holds. By the inequalities (2.1) and (8.1) and the relation (8.7), we have that ~ c{f*(1-{(f [t 1 /q(1-log t)a]bdt/t)l/b + - 0 (f (8.10) ta/pat/t) 1/4 1 1aJ 42 The coefficient off * (1-) in (8.10) is a constant independent off by (8.9). Hence, we have llrllqb ;a '."'. c{(([t 1/p (1-log t )''r* (t) ]adt/t) 1/a ( 8. 11 ) + (~[tl/p(l+log t)af*(t)]adt/t)l/a}. 1 For O <a< oo, apply (2.1) to give (8.8). If a= oo, we use the following analogue of (2.1) to produce (8.8): 1/2( sup h(t) + sup h(t)) < sup h(t) < sup h(t) + sup h(t), 1<t<oo - O<t<oo - O<t<l O<t<l l<t<oo (8. 12) where h is any nonnegative f'unction on (O,oo). The only other case when Lqb(log L) 8 f (o} is when q = b = oo and S < O. The proof is similar so we omit it. For functions on the integers, i.e. sequences, it is often desirable (and necessary to accommodate the classical theory) to attempt to express (6.1) in terms of series. definition: We make the following If (en} is a sequence of complex numbers, let (c:}n:l denote its nonnegative decreasing rearrangement (if c c* = c*(n-)). n n = c(n), then The Lorentz-Zygmund sequence space tpa(log t)a for 0 < p , a ~ oo, -oo < a < oo is the set of sequences for which 00 \' 1/p (l+log n) a en] * an -1) 1/a ( l [n n=1 (8.13) When a = oo, this is to be interpreted in the obvious way as sup n1/P(l+log n)ac* • l<n<ixi n (5) 43 is finite. The following theorem shows that Lpa(log L)a = tpa(log t )a (provided Lpa(log L)a is non-trivial) when the underlying measure space is the integers ~ • THEOREM 8.5: SUppose O < p, a~ oo, -oo <a< oo and Lpa(log L)a ~ [o}. Then, (8. 14) Proof: We will only consider the case O < p ,a< oo and a:::: O. other cases are proved similarly. The First, we observe that 00 \ [n ·1-' 1/p a *a -1 (1 +log n) c ] n n n=1 (X) = c *1 + \'l n=2 r ( 8. 15) [n 1/p (1+log n)a c * (t)] a dt/n . n-1 Next, we notice there are constants k 1 and ~ such that k [t 1 1/p (l+log t) a ]a t -1 ~ [n 1/p( l+log n) a ]a n -1 ~ ~[t n = 2, 3, .... 1/p (8. 16) a a -1 (1+log t) ] t , for n-1 ~ t ~ n, is finite. Using this, (8.15) and the second inequality in (8.16 ), we have Since p < oo, the integral 44 00 00 :'5: k If 1 0 rt /P(1+1og t) c*(t)]adt/t n=1 n-1 n=1 The reverse inequality follows from the first inequality in (8.16 ) . REMARK 8.6: The spaces L a(log L) , when (i) O <a< oo and a+1/a ~ O 00 0 or (ii) a= oo and a> Oare all trivial (cf. Lemma. 8.1). However, the corresponding spaces t ooa( log t)a generated by the quasinorm 00 *a -1)1/a \ L [(1+log n)ex en] n ( \ n=1 are clearly non-trivial. For the Lorentz-Zygmund sequence spaces tpa(log t)0 we have the inclusions (8.8) with no restrictions on the para.meters q,b and R . THEOREM 8. 7: Suppose 0 < p < q :'5: oo, 0 < a , b :'5: oo and -oo < a , f3 < oo • Then (8.17) Proof: The inclusion (8.17) is precisely the content of Lemma 8.2 when the underlying measure space is the integers :,:z • Suppose O < p , q :'5: oo, O < a , b :'5: oo and -oo < a , El < oo. The Lorentz-Zygmund space Lpa(log L)a + Lqb(log L)R consists of all functions for which the quasinorm (1 . 10) is finite when p < q; when 45 pa a qb ·Q p > q, L (log L) + L (log L) · is generated by the quasinorm (1. 11 ) . First, we note that pa a qb R qb A pa a L (log L) + L (log L) = L (log L) + L (log L) • To justify the notation "+", we will show that Lpa(log L)a + Lqb(log L) 8 is just the usual algebraic sum of the Lorentz-Zygmund spaces Lpa(log L)a and Lqb(log L)~; the only exception occurs when either space is trivial (cf. Lennna. 8.1). The space Lpa(log L)a n Lqb(log L)B is generated by (1.11) when p < q, and is generated by (1.10) when p > q. For all values of the parameters we will show that Lpa(log LP n Lqb(log L)A is the usual set-theoretic intersection of the Lorentz-Zygmund spaces Lpa(log L)a and Lqb(log L) 8 • The quasinorms (1.10) and (1.11) are rather unwieldy, so we shall adopt the following notation convention: If h is a nonnegative measurable function on ( O,oo) and O < p, a ~ oo, -oo <a < oo, we define 1 I . h = pa,cr (J [t 1/p(1-log t)°h(t))adt/t) 1/a, ( 9. 1 ) 0 and J 1 .~h = (~[t /p(1+log t)°h(t))adt/t) 1/a. 1 pa,....,, We thus have (cf. (1 . 10) and (1.11 )) (9.2 ) 46 (9.3) I qb;~ f *+J qb;~ f * +J and I pa;a f * f * p<q ' ' p >q·, pa;a ' ( 9. 4) I pa;af * + J qb ;nof*, p > q. If (X1 , II· 1\ 1 ) and (~, ll · 11 2 ) are two quasinormed spaces continuously embedded in a larger topological vector space, we define two new quasinormed spaces x1 +~and x1 n x2 (cf. [8]). f E For each x1 + ~' the quasinorm of f is given by (9.5) where the infimum is taken o~er all decompositions f = f 1 + f 2 , f 1 E X 1 and f 2 E ~· For f E X 1 n ~' the quasinorm is given by (9.6) THEOREM 9. 1 : Let 0 < p < q :5 oo, 0 < a, b :5 oo and the space Lqb(log L)~ f -oo < a, ~ < co. {o}, then the quasinorm defined by (9.5) on the algebraic sum Lpa(log L)a + Lqb(log L)~ is equivalent to the quasinorm (9.3) . If Hence, the space defined by (9.3) is the usual algebraic sum provided neither space is trivial. 47 Proof: (i) Let us first assume O < q < oo so that Lqb(log L)A f tol. We pick fin the algebraic sum Lpa(log L)a + Lqb(log L)R. We may assume that f is real-valued. as follows: f(x) - f * (1 ), Define functions f 1 and f if f(x) ~ f * (1 ), r 1 (x) = f(x) + f * (1), o, 2 (9.7) otherwise; (9.8) With this choice of f 1 and f 2 we have f * (t) - f * (1), 0 < t < 1, f1* (t) = (9.9) o, 1 ~ t f * (1 ), O<t<1 < oo; and f2* (t) = (9.10) f * (t), 1 < t < oo. We thus have for x E l and 0 < t < oo, f(x) = f1(x) + f2(x); * + f2(t). * f * (t) = f1(t) ( 9. 11 ) We now must show (9.11) is a proper decomposition, i.e., E Lpa(log L)a and f E Lqb(log L)~. To this end, we let f = g + h 2 1 8 where g E Lpa(log L)a and h E Lqb(log L) . By (3.3), we have f f * (t) ~ g* (t/2) + h * (t/2). (9.12) 48 The description (9.9) shows that (cf. (6.1) and (9.1)) (9.1 3) This and ( 9. 12) show that ~ c(Ipa;a(g * (t/2)) + Ipa;a(h *(t/2)). Now Ipa;a(g*(t/2)) is finite since g E Lpa(log L)a. Also, I pa;..... ~(h * (t/2)) is finite because I pa ;a -<Iqb ; 8, (cf.(7.2)) and h E Lqb(log L)S. Hence, \lf 1llpa;a is finite and so f 1 E Lpa(log LP. The description (9.10) shows that (cf. (6.1), (9.2)) The first integral is finite since O < q <co. By (9.12), the second integral is dominated by (cf. (9. 2)) By Lemma. 8. 2, we have which is finite because g E Lpa(log L)a. 'llle expression Jqb;S(h*(t/2)) is clearly finite since h € Lqb(log L)~. qb estimates show that f 2 E L A (log L) , as we wished. These 49 To show that (9.3) dominates (9.5), we observe that (9.13) and (9. 14) give II f + \I 1 pa;o: II f 2 II qb;~ <- c {I pa;o:f * + Jqb;~ f * (9.15) 1 1 1 9 + f*(1)(J [t /q(1-log t) ]bdt/t) /b}. 0 We have because each of the integrals is finite. This last expression is dominated by since f * decreases. This estimate and (9.15) show IIf 1 II pa;o: + II f 2 II qb;e <- c (I pa;o:f * + Jqb;8 f *) • Ta.king the infimum over all decompositions f = f 1 + t 2 , we have that (9. 3) dominates (9.5). To prove the reverse inequality, let f f 1 E Lpa.(log L)o; and f 2 E Lqb(log = f 1 + f 2 where L)e. The inequality (3.3) shows that Applying Minkowski' s inequality (or (2. 1 ) if O < a < 1 ) and substituting t/2 ~ t, we obtain 50 ~ c{(J 112 Ipa·af* ' 1 t)af~(t)]adt/t)l/a + (J1/ 2 [t 1/P(1-log t)af;(t))adt/t) 1/a}. [t /p(1-log 0 0 < + I - c (I pa;af * 1 pa;af 2*) • Hence, it follows from (7.2) that I pa;af * < c(Ipa;af * + I f *) 1 qb;a 2 • (9.16) Estimating Jqb;af* similarly, only appealing to (8.1) instead of (7.2), we obtain J < c {I pa;a f * + J f + I f + J f 1 pa;a 1* qb;a 2* qb;A 2*} qb;B * f (9.17) Adding the inequalities (9.16) and (9. 17), then using (2.1 )(or (8.12)), we have the estimate where c is a constant independent of f, f 1 and f • Taking the 2 in:t'imum over all decompositions f = f 1 + f , we have that (9.5) 2 dominates (9.3). Hence, when O < q < oo, the quasinorms (9.3) and (9.5) are equivalent. (ii) The other instances where Lqb(log L)~ -/= {o} may be treated similarly so their proofs are omitted. THEOREM 9.2: suppose O < p < q ~ oo, O < a, b ~ oo and -oo < a, 8 < oo. Then the quasinorm defined by (9.6) on Lpa(log L)a n Lqb(log L)a is equivalent to the quasinorm (9.4). Hence, the "intersection" defined 51 by (9.4) is the usual set-theorectic intersection. Proof: Clearly, by (6.1), (9.1) a.nd (9.2), we have that proving that (9.6) dominates (9.4). To prove the reverse inequality, (2.1) (or (8.12) if a= oo) gives f pa;a -<cI 1111 ( pa;af * +Jpa;af*.) Applying (7.2) to Ipa;af *, we obtain (9.18) Similarly, and by Lemma 8. 2, we have llrl\q b·Q + J pa,a • r*). , t, ~ c(Iq b·of* ,o (9. 19) The estimates (9.18) a.nd (9.19) show that (9.4) dominates (9.6), completing the proof. REMARK 9.3: From now on, when we speak of sums or intersections of Lorentz-Zygmund spaces, we refer to the quasinorms (9.3) or (9.4). Theorem 9. 2 shows that this is consistent for the intersections. Theorem 9.1 shows that the sum given by (9.3) is the usual algebraic sum provided neither of the spaces is trivial. The sum given by the 52 qua.sinorm (9.3) is of greater interest for us. 1 001 algebraic sum of the spaces L and L 1 001 space L + L 1 is just L • However, the given by (9.3) consists of the larger class of functions for which I,f *(t)dt + rf * (t)dt/t 0 is finite. For example, the (9.20) ... , This space is of interest for the Hilbert transform H. For example, it is well known that Hf(x) is defined a.e. for f E LP, 1 ::Sp< oo (cf. [36, 42]). It is shown in [6] that Hf(x) is 1 001 actua.lly defined a.e. for functions f in the larger space L + L given by (9.20). THEOREM 9.4: Let O < p < q ::S oo, O < a, b,c,d ::S oo and -oo <a, P., v, o < oo. Then Proof: The (qua.si)norm of a function f in Lpb(log L)~ + Lqd(log L) 0 is (cf. (9.3)) * + Jq d·6f *• I pib•Sf ,. ' (9.22) The norm of a function fin Lpa(log L)a n Lqc(log L)V is (cf. (9.4)) I qc;vf * + Jpa;af *• (9.23) The inequalities (7.2) and (8.1) show that (9.23) dominates (9.22). This completes the proof. 53 The next theorem characterizes the sums and intersections of Lorentz-Zygmund spaces when the umer~ measure space is the unit circle f (or a:rry finite measure space) or the integers 7l. THEOREM 9.5: (i) Let o < p < q ~ oo, o < a,b ~ oo and -oo < cx,e < oo. If the underlying measure space is !.', then (a) Lpa(log L)cx + Lqb(log L) 8 = Lpa(log L)cx; (b) Lpa(log L)cx n Lqb(log L)a = Lqb(log L)A. If the underlying measure space is 7l , then (cf. ( 8. 13)) (ii) (c) Lpa.(log L)cx + Lqb(log L)a = tqb(log t) 6 ; (d) if, in addition, Lqb(log L)~ f to}, Lpa(log L)cx n Lqb(log L)R = tpa.(log t)cx. Proof: (a). (b). By (9.3) it is clear that By Theorem 9.2, the intersection in part (b) is the usual set-theoretic intersection, and by (7.2). (c). If Lqb(log L)~ f to}, Theorem 9.1 shows that the sum in (c) is the usual algebraic sum of the two spaces. 7l , the inclusion ( 8. 8) shows that So, for the integers 54 L pa a qb B qb B (log L) + L (log L) ' = L (log L) • The statement (c) now follows in this case by (8.14). If q = oo (which encompasses the situation Lqb(log L)e = [o}), then (9.3) becomes (cf. (8.7)) By the inequalities (8.16), this quasinorm is equivalent to 00 (I [(1+log n)~f*(n-)]bn- ) /b, 1 1 n=1 which is the quasinorm (8.13) (d). 00 8 on t b(log t) • Theorem 8.4 applies, so that Since Lpa(log L)a F {o} (p<j:io), the statement (d) follows by (8.14). 10. Inclusion Relations Theorems 7.2 and 8.4 exhibit inclusion relations for the spaces Lpa(log L)a when the primary indices are distinct and the underlying measure space is the unit circle [' or the integers ?Z. It is also of interest to study the inclusion relations when the primary indices are the same. The following theorem generalizes the facts that the spaces Lpa(log L)a decrease with increasing a, while the spaces Lpa increase 55 with increasing a. THEOREM 10.l ([6, Theorem 9.3] ( 6 )): and -oo <a,~ < oo. Suppose O < p ~ oo, O < a,b ~ oo Then ( 1 o. 1 ) whenever either of the following holds: (i ) a < b (ii) a >b REMARKS 10.2: and a ~ B, or and a + 1/a > B + 1/b. (i) If a > b, the condition a+ 1/a > B + 1/b in Theorem 10.1 cannot in general be relaxed to a+ 1/a ~ ~ + 1/b. Hence, the spaces Lpa(log L)a are not ordered along the "diagonals" where a + 1/a is constant . (ii) We wish to establish similar inclusion relations for the sums and intersections defined in Section 9. Theorem 1o.1 shows exactly what to expect (provided none of the spaces involved in the sums are trivial) • We shall need two technical lennnas; the first lemma shows that the integrals Ipa;af * and Jpa;af* (cf. (9.1) and (9.2)) satisfy a (6 ) This theorem was proved in [6] for the case where the underlying measure space is finite. However, the proof remains valid for any underlying measure space . convexity property. LEMMA 10.3: Let O < p ~ oo, 0 <a< b < oo and -co< ex,~ <co. Then we have I Pib; Q"" r*)a/b(I )1-a/b , f * <(I pa ;ex pco; y (10.2) * _< (J f*)a/b(J )1-a/b , pa;ex pco;y (10.3) b8-ao: ( 1o.4) and Jp~;o . f 'U 0 where V = b-a Proof: ( = B, if b = 00). The case b = oo is obvious, so we assume b < oo. We note that [t 1/P(1-log t)~f*(t))b = [t 1/P(1-log tPf*(t)]a[t 1/P(1-log t)Yf*(t))b-a. Hence, by (9.1 ), Ipb;Rf* 1 ~ [t 1/P(1-log t)exf*(t))adt/t) 1/b( sup [t 1/P(1-log t)Vf*(t))b-a)l/b (J 0 = (I pa;ex O<t~ f*)a/b(I f*)1-a/b pco;Y • The inequality (10.3) is proved similarly. LEMMA 10.4: Let 0 < p ~ oo, O <a <co and -oo <ex < oo. the inequality Then we have 57 1 · decreasing · on (1 ,oo ) We choose N so tha t s - (N+1 og s )aa is Pr oof : (cf. Lemma 2.2). Then, Jtsa/p(1+log s)aads/s ~ cJtsa/p(N+log s)aads/s 1 1 ~ ct- 1 (N+log t)aaJtsa/pds 1 We add cr{.ta to both sides and take a th roots to prove ( 1o.5). THEOREM 10.5: Let 0 < p < q ~ oo, 0 < a,b,c,d ~ oo and -oo < a, R,y,5 < oo. Then ( 1o. 6) and Lpa(log L)a n Lqc(log L)y ~ Lpb(log L)e n Lqd(log L) 5, (10.7) whenever one of the following conditions holds: (i) a<banda>e ., c ~ d and V ~ &, . c > d and y + 1/c > & + 1/d, (ii) a<banda>e (iii) a>banda+ 1/a > B + 1/b; c ~ d and y ~ 51 (iv) a>banda+ 1/a > ~ + 1/b; c > d and V + 1/c > &+ 1/d. Proof : ( 1o.8) By the Remark (10 .2.ii), the inclusion (10.7) is a direct consequence of (10. 1) and Theorem 9.2. The inclusion (10.6) follows from (10.1) and Theorem 9.1 provided Lqb(log L)R ! {o}. It will therefore suffice to prove only (10.6) when q = oo (cf. Lenlma. 8.1 ). We first assume that (10.8.i) holds. Since the space 00 Lpa(log L)a + L c(log L)V clearly decreases with increasing a and V, we may assume a = R and y = 5. We therefore need to show * I pib ;u rvf * + J ~ . :f * < k (I rvf * + J vvu.;y pa;u ooc;y. :f ) • ( 1o.9) The inclusion (10.1) (for a finite measure space) shows that I pb;af * <- kIpa;urvf *• (10.10) If c = d, the inequality (10.10) implies (10.9), so assume c < d. utilizing (10.3), we obtain (10.11) To estimate Jmxi;~ *, we use (10.5) and the fact that f * decreases to get (l+log t)Vf * (t) ~ k (f * (1Jc + It [(1+log s)Vf*(s)) cds/s)1/c , <t < 00 • 1 Replacing the range of integration (1,t) by the range (1,oo) and taking the supremum over 1 ~ t < oo, we have (10.12) The estimates (10.10), (10.11) and (10.12) show that if the right-hand side of (10.9) is finite, then the lef't-hand side of (10.9) is also finite. This shows that the (set-theoretic) inclusion (10.6) holds. An appeal to the closed graph theorem shows that the inequality (10. 9) al.so holds, i.e., the inclusion map is continuous. 59 Now assume the condition (10.8.ii) holds. We wish to show that * * Ipb; 0nf * + J 00d ;u~f* < - k(I pa;af + Jooc;vf ). The inequality (10.10) still. remains valid. (10.1 3 ) We assume that c < oo, and write Applying HBlders inequality with the conjugate exponents c/d and c/(c-d), we obtain c-d Jqd;&f* ~ (Jqc;'1*)(~(1+log t]~dt/t) cd , wh ere 'rl •r = cd( 5-v) c- d ( 1o. 14) The conditions c > d and y + 1/c > ~ + 1/d ( cf. (10.8.ii)) imply Tl< -1, and so the integral in (10.14) is finite and independent off. Thus, the estimates (10.10) and (10.14) imply the required result (10.13). The other parts of the theorem are proved in the srune way. The details are omitted. As noted in Remark 10.2.i, we still. fail to have inclusions along the "diagonals" a + 1/a = constant and/or y + 1/c = constant. 11 . The Auxiliary Lorentz-Zygmund Spaces The averaging operators Ap , Bp , Cp and DP of Section 4 are basic to the interpolation theory as we will. show in Section 13. To 60 study the action of these operators on the Lorentz-Zygmund spaces we shall now define auxiliary Lorentz-Zygmund spaces. Let O < p < q ~ oo, o < a, b ~ oo, and -oo <a,~ < oo. The auxiliary Lorentz-Zygmund spaces iPa(log :l)a + ~b(log L)~ , ;t.;Pa(log J!)a n i1b(log .l)a, '7f'a(log '!'11)a + ?#b(log '771)!3 and ,,pa(log 'lrl)a n ??flb(log 'lrl)e are generated by the following quasinorms (cf. (9.1) and (9.2)): !If!!~(log :l)a + ./}b (log .£,) A = I pa;a (Ap f * ) + Jqb ;~ (Bq f * ), Q (11.1) a + 1/a > o, ~ + 1/b > O; (11.2) a + 1/a < o, R + 1/b < O; 1\fll'Tf8. (log 'lrl)a + ...ab A H(- (log 'm) (11.3) = I pa;a (Cp f * ) + Jqb;B (Dq f * ) ' 1 1 a + a > O, e + b > O; llf\\...Da a b e wr (log 'lrl) n ~ (log 'm) = I q b·Q(D f * ) + J .~(C f * ), ,.., q pa,"" p (11.4) a + a1 < O, A + b1 < O. When p > q we make the obvious definition so that 61 To avoid unnecessary conf'usion, we have defined these auxiliary spaces for only the values of the parameters a, b, a and ~ that we will ultimately be interested in. By Lemma 4. 1 we have the following inclusions: and iP8'(log .£)an :f.lb(log £) 8 c:: 'nPa(log '!J?)a n mtb(log '71)~. (11.6) The next theorem gives important special cases when the auxiliary Lorentz-Zygmund spaces reduce to ordinary Lorentz-Zygmund spaces. THEOREM 11. 1: Suppose O < p,q < oo with p :/: q. Then, if a+1, ~+1 > o, and if a,R > o ( 11. 8) If a + 1, ~ + 1 < 0 and if a,B < o, (11.10) (7) We implicitly assume a+1/a > o, B+1/b > O so that the spaces are defined by (11.1) and (11.3). These implicit assumptions will be made throughout the remainder of the thesis without comment. Let p < q. Proof: To prove (11.7), we apply the definitions (9.1), (9.2), (4.1) and (4.2) and interchange the order of integration to show that I (A f * ) + J P1 ;a P Q q 1 ;p (B f * ) q = j 1s 1/ Pf* (s) (J1 (1-log t)adt/t) ds/s + r' s 0 r s 1/ qf * (s) (jrs (1+log t) ~ dt/t ) ds/s 1 1 r1 1/p (1-log s)a+1 f * (s)ds/s + rx> 1/q ~+1 f * (s)ds/s - I s I s (1+log s) vo = I V1 P1 ;a+ 1r * + Jq 1 ;p+ 1r * . Q Thus, (11.7) is proved by the above equivalence and the definitions of the spaces involved (cf. (11.1) and (9.3)). To prove (11.9), note that the (quasi)norm of a function fin iP1 (log £)an il1 (log £)A is (cf. (11.2)) s:(l-log tl~(~sl/qr*(s)ds/s)dt/t (11.11) interchanging the order of integration and noting that A~er the (quasi)norm -1 I ~+1 1 1q 1 ;~+ 1 f (11.11) becomes * + Ia+1 1-1 JP ;a+lf* 1 1 + la+11- 1 s 1/pf*(s)ds/s + l~+11- 1 ,f's 1 /qf*(s)ds/s. J 0 1 (11.12) The norm off in LP 1 (log L)a+ 1 n Lq 1 (log L)A+ 1 is (cf. (9.4)) I q1;R+1 f * + J p1;a+1 f * (11.13) The quasinorm (11.12) clearly dominates the quasinorm (11.13). To 1 1 verif'y the reverse inequality, we apply (7.2) to s /Pf*(s)ds/s l ..io and we apply (8.1) to 11sl/qf*(s)ds/s. J 1 This proves (11.9). The remainder of the theorem is proved similarly. The next theorem is the auxiliary space analogue of Theorem 10.5. Notice that now the spaces are ordered along the "diagonals" a + 1/a = constant and/or y + 1/c = constant (cf. Remark 10.2.i). THEOREM 11.2: Suppose O < p < q ~ oo, 0 < a,b,c,d ~ oo and < a,~,v,& < oo. -oo Then (11.14) and ,e'a(log £.)an .tlc(log £.)V <_:: ;;tl'b(log .l)~ n ild(log £.) 5, (11.15) whenever one of the following conditions holds: (i) a+ 1/a > ~ + 1/b y + 1/c > 8 + 1/d, a+ 1/a > 6 + 1/b y + 1/c = 8 + 1/d, (ii) c ~ d, (iii) a+ 1/a = R + 1/b, a< b v + 1/c > 5 + 1/d, (iv) a + 1/a = A + 1/b, a< b V + 1/c = 5 + 1/d, c < d. Proof: Let us first prove (11. 14). (11.16) 'lllen all the quantities a+ 1/a, A + 1/b, V + 1/c and 8 + 1/d are positive. First, we assume 64 a> band c > d. This is a subcase of (11.16.i), so a+ 1/b >A + 1/b and y + 1/c > 5 + 1/d. We need to prove that (cf. (11.1 )) * * I pb ; a(A f * ) + J d ~(Bf * ) < k(I I-' p q ;u q pa;a (Ap f ) + Jqc; v(B T q f )) Statements (10.10) and (10.14) are valid are valid. (11.17) Hence, and J d ~ (B f q ;u q * ) <- kJqc;1v(Bq f *) • (11.19) These inequalities (11.18) and (11.19) imply the required result (11.17). We next assume that a > b and c < d. This is a subcase of (11.16.i) or (11.16.ii), so a+ 1/a > B + 1/b and y + 1/c 2'. 5 + 1/d. Since the spaces ~a(log -;!_,)a + :flc(log .e.) 5 decrease with increasing y, we may assume y + 1/c = 5 + 1/d, c < d. (11.18) still holds. J In this case the inequality By (10.3) and (10.4) we have (B f*) < (J (B f*) )c/d(J (B f*) ) 1-c/d qd;8 q qc;y q qoo;v+1/c q • Therefore, to prove (11.19) (and hence, (11.17)) in this case it will. suffice to show * Jqoo; y+ 1I C (Bq f * ) < - kJq C ; Y(Bq f ). ( 11 .20) Since y + 1/c > o, we may replace (1+log t)v+l/c by (log t)v+ 1/c since the two f'unction are asymptotic as t ~ oo. We write t (log t)yc+l = (Vc+l)l (log u)vcdu/u. .., 1 (11.21) Then, for 1 ~ t < oo, we have (log t)vc+l(~s 1 /qf*(s)ds/s)c t = (Vc+1)Jt[logvu~s 1 /qf*(s)ds/s]cdu/u 1 t < (vc+1)Jt[logvu~s 1 /qf*(s)ds/s]cdu/u. 1 u Taking cth roots and passing to the supremum (over 1 ~ t < oo) we obtain the required result (11.20) (cf. (9.2) and (4.2)). Third, we assume a ~ b and c > d. This is either a subcase of (11.16.i) or (11.16.iii) so that V + 1/c > 5 + 1/d and o: + 1/a ~ e + 1/b. We may assume o: + 1/a = B + 1/b and a < b. this case (11.19) still holds. In To prove (11.18) (and hence, (11.17)), Lenuna 10.3 shows it will suffice to show that ( 11 .22) As in the previous case, we write 1 (log 1/t)o:a+l = (o:a+1)J (log u)o:adu/u, (11.23) t and proceed as before to prove the required result (11.22). Finally, we assume a ~ b and c ~ d. V + 1/c = 5 + 1/d and o: + 1/a = ~ + 1/b. (11.22). We may suppose a < b, c < d, We then have (11.20) and This completes the proof (via Lemma. 10.3). The inclusion (11.15) is proved in a similar fashion. 66 There is a similar theorem for the '!!I. spaces. Its proof is fashioned after the proof of Theorem 11.2 so we omit it. THEOREM 11.3: suppose O < p < q ~ oo, O < a,b,c,d ~ oo and < a,a,v,5 < oo. -oo Then (11.24) and 71f'a(log 7J?)a n ??fIC(log 'f!l.)V ~ 71f'b(log 'fnJ~ n '7fld(log ?J7.) 5, (11.25) whenever one of the conditions (11.16) hold. The next theorem gives some special inclusion relations for the spaces .J'_., 711 and L. THEOREM 11.4: suppose o < p < q ~ oo, o < a,b ~ oo and -oo <a,~< oo. If a + 1/a > O and ~ + 1/b > o, then (11.26) and (11.27) If a + 1 /a < O and ~ + 1 /b < o, then (11.28) and (11.29) 67 Proof: We choose y to satisfy O < y + 1 < min(a + 1/a, R + 1/b). Applying (11.14) and (11.7), we have 'W-1 ql v+1 = LP1 (log L)T' + L (log L)r • But, since y + 1 > o, the last space is clearly contained in Lpl + Lq 1 • This proves (11.26). The inclusion (11.27) follows in the same way from (11.24) and (11.8). The inclusions (11.28) and (11.29) follow similarly by choosing 'V so that ma.x(a + 1/a, ~ + 1/b) < V + 1 < O and applying (11.15) and (11.9) (to prove (11.28)) or applying (11.25) and (11 . 10) (to prove (11.29)). As seen in Theorems 11.1 - 11.4, the quasinorms (11.1) - (11.4) are quite convenient for technical purposes. The next theorem gives an equivalent reformulation of these quasinorms. As we shall see in Chapter Dl, these new expressions for the quasinorms on the auxiliary Lorentz-Zygmund spaces are the most natural for interpolation purposes. THEOREM 11. 5: Let O < p < q ~ oo, O < a, b < oo and -oo <a, R < '°• Then (11.30) 68 11f!I At>a ;C" a (log .£) .nb 8 n ;C (log ..J!,) · ,.., 11 (A p a lif!l....:Pa ~r (log '71) + B )f*11 q (11. 31) Lpa(log L) a b n Lq (log L) R b 9 + 'l7fl. (log 'l'fl) ,.., lice + n )fjj p q (11.32) Lpa(log L)a + Lqb(log L)R ( 11. 33) Proof: In each of the statements above it is clear from the definitions of the spaces involved (cf. (9.3), (9.4), (11.1) - (11.4)) that the right-hand side dominates the left. In order to prove the reverse inequality, let us concentrate on (11.30). implicitly that a + 1/a > 0 and ~ + 1/b > o. We have The right-hand side of (11.30) is (cf. (9.3)) I * * pa;a(AP + Bq )f + Jqb ; 9. (Ap + Bq )f , which is dominated by (cf. (2.1) and (11.1)) We let A= 1/p - 1/q > 0 and make the estimate (cf. (9.1) and (4.2)): I ·a(B f*) ~ pa, q 1 (J0 [tA(1-log t)a]adt/t) 1 /a~sl/qf*(s)ds/s '1 Noting the first integral is finite and applying the Hardy inequality (2.9), we obtain I pa;o: (Bq f *) <- c (Jq 1 ; 0f * + Ipa;o:f*) • (11.35) * similarly (using the Hardy inequality (2.10)), Estimating Jb.c.(A f) q , l·' p we get (11.36) * dominate f * (cf. Lemma 4.1), the estimates (11.35) Since Ap f * +Bf q and (11.36) show that Now, the inclusion (11.26) shows that The estimates (11.34) and (11.37) establish the reverse inequality for (11.30). The remaining equivalences are proved similarly using the variants of Hardy's inequalities given in Theorem 2.5. The details are omitted. 12. The Embedding Theorem The auxiliary Lorentz-Zygmund spaces introduced in Section 11 are related to the Lorentz-Zygmund spaces of Section 9 by the following basic embedding theorem. 70 Suppose 0 < p < q ~ oo, 1 ~ a ~ b ~ oo and -oo <a,~ < oo. THEOREM 12. 1: If a + 1/a > O and B + 1/b > o, then Lpa(log L)a+l + Lqb(log L)~+l C::~a(log ..t)a + .f!b(log ..!..)R ':: Lpa(log L)a+l/a + Lqb(log L)~+l/b ':: '7Pa(log 'm)a + 'J?flb(log 'm)B (12.1) ~ Lpa(log L)a + Lqb(log L)a • If a + 1/a < 0 and a + 1/b < O, then Lpa(log L)a+l n Lqb(log L)B+l ':: :ePa(log £)a n :tlb(log ..t)a ':: Lpa(log L)a+l/a n Lqb(log L)a+lfb ~ 'Tf8'(log 'm)a n 'J?flb(log ?71) 8 (12.2) ~ Lpa(log L)a n Lqb(log Proof: 1. tl . The last inclusion in both (12.1) and (12.2) is a consequence of Lemma. 4.1 and the definitions of the spaces involved ((9.3), (9.4), (11.3), (11.4)). 2. To prove the first inclusion in (12.1) we note that the norm of a function fin ~a(log ..!..)a+ :tlb(log ..t..)R is (cf. (11.1), (9.1), (9.2)) + (r (12.3) [(1+log t)·sf' j s 1/qf * (s)ds/s]bdt/t )1/b • t 1 Since a + 1/a > O and B + 1/b > o, we may apply the Hardy inequalities (2.22) and (2.23) to show that (12.3) is dominated by I pa;a+ 1 f * + Jqb·o * ,f"+ 1 f , 71 which is the norm of a function f in Lpa.(log L)a+l + Lqb(log L)~+l (cf. (9.3)). To prove the first inclusion in (12.2), it suffices show that (cf. (9.4), (11.4)) ( 12.4) Applying Minkowski's inequality, we obtain l The integral l (1-log t)~bdt/t is finite since A + 1/b < o. We now "o apply the Hardy inequality (2.24) to show that (12.5) Estimating Jpa;a(Apf* ) similarly (the Hardy inequality (2.25) now applies), we see that (12.6) Theorem 9.4 shows that * + Jq 1 ; of* <- c(Iqb ; 6+,f* + J pa;a+ lf* ). I P1 ; of (12. 7) The estimates (12.5), (12.6) and (12.7) produce the required result (12.4). 72 3. To establish the second inclusion in (12.1), we shall show that (12.8) Suppose O <a< oo. The identity (11.23) and the remark (6.6) show that Next, interchange the order of integration to obtain An application of the inequality (2.6) to the inner integral shows that * I pa;a+1 I a f * < - c I pa;a (Apf ) • ( 12 9) • When a= oo, the inequality (12.9) still holds (cf. (4.6)). Using the identity (11.21) and proceeding as above (using (2.7) instead of (2.6)), we obtain (12.10) The estimates (12.9) and (12.10) imply the desired result (12.8). For the second inclusion in (12.2) we proceed as above, employing the identities (1-log t)Bb+l = l~b+11Jt(1-log u)Bbdu/u; 0 (1+log t)aa+l = laa+11 ~(1+log u)aadu/u, t 73 which hold since a + 1/a < 0 and B + 1/b < O. 4. To prove the third inclusion in (12.1), it is required to show If If ) * * - c(Ipa;a+1 a * + Jqb;B+1 b * • r pa;a (cf p ) + Jqb;B (Dq f ) < We assume that 1 ~ a,b < oo. (12 • 11) The inequality (2.2) shows that We may now apply the Hardy inequality (2.22) to prove that * rpa;a (cp f * ) < - erpa;a+1 I a f • (12.12) To estimate J qb·S(D f * ), apply (2.3) to obtain ,. q utilizing the Hardy inequality (2.23) and changing variables (t/2 -+ t), we see that The estimates (12.12) and (12.13) combine to produce the required result (12.11). (The cases a= oo, b = oo are easy and thus omitted). To prove the third inclusion in (12.2) we proceed as above to obtain 74 \lt!l?fa(log 'mfi· n ?#b (log '71) A < cClltll I I + lltll Lp1 + Lq1 ). Lpa(log L)a+1 a n Lqb(log L)R+1 b An appeal to Theorem 9. 4 completes the proof. 75 CHAP.rER IV THE INTERPOLATION THEORY 13. Operators of Weak Type (p,q;r,s). Suppose 0 < p < r ~ oo and 0 < q,s ~ oo with q '/= s. Let 1/il = 1/p - 1/r, 1/£ = 1/q - 1/s, m = 11/£. (13.1) The quantity m represents the slope of the line segment a joining the points (1 /p, 1 /q) and (1 /r, 1 /s ). For each f € Lpl + Lr 1, let m (w(a)f*)(t) = t-l/qJt u 1 /Pf*(u)du/u 0 (13.2) + t-l/s~ u 1/rf*(u)du/u, tm DEFINITION 13.1: O<t<oo. Let T be a quasilinear operator mapping measurable functions on (l,µ) into measurable functions on ('lj,~). 0 < p < r ~ oo and 0 < q,s ~ oo with q ~ s. Suppose The operator T is of weak type (p,q;r,s) if T is defined on LP 1 + Lrl and the inequality (Tf) * (t) ~ c(w(a)f* )(t), is satisfied for all f € 1 o < t < oo, ( 13.3) 1 LP + Lr , where c is a constant independent of f. The next theorem reformul.ates the notion of weak type (p,q;r,s) in terms of the averaging operators Ap , BP, Cp and Dp of Section 4. We first need a lemma. LEMMA 13.2: suppose O < p < r ~ oo and f E LP 1 + Lr 1• Then the function increases on (O,oo~ and the function decreases on (O,oo). Proof: The definitions (4.1 ), (4.2) and (13.1) show that Let O < s < t < oo. Then g(s) - g(t) = - t u 1/pf * (u)du/u + s 1/11Jtu 1/rf * (u)du/u s s S The last term is less than or equal to zero since 11 > O (cf. (13.1)). On the other hand, we have s 1111Jtu1 /rf*(u)du/u ~ Jtu 1 /~l/rf*(u)du/u s s = t u1/pf* (u)du/u • J s Thus, g(s) ~ g(t) as required. similar. The proof that h is decreasing is 77 THEOREM 13. 3: suppose O < p < r ~ oo and O < q, s ~ oo with q f s. Th.en a quasilinear operator T is of weak type (p,q;r,s) if and only if the following inequality holds: (a) for q < s, (b) forq>s, where c is a constant independent of f. Proof (a): Since q < s, we have that m > 0 (cf. (13.1)). From Definition 13.1, Tis of weak type (p,q;r,s) if and only if 0 < t < oo, (13.6) where C = -1/q + m/p = -1/s + m/r. (13.7) It is required to show that (13.4) and (13.6) are equivalent. suppose (13.4) holds. By Lemma. 4.1, the left-hand side of (13.4) is greater than or equal to 2te/q(Tf)*(te). a~er the change of variable t £ ~ t, reduces to (13.6). Conversely, suppose (13.6) holds. then (13.6) and (13.7) show that The resulting inequality, Lett > o. If 0 < u ~ t e 1 78 Lemma. 13.2 and the fact m > 0 show that the right-hand side is largest when u = te. Taking the supremum over 0 < u ~ te, we obtain (13.8) Now fix u so that te ~ u < oo. Then (13.6) and (13.7) show that u 1 /s (Tf) * (u) ~cum/r ([~ + Br)f* )(um). Lemma. 13.2 and m > O imply that the right-hand side is largest when u = te. Passing to the supremum over te ~ u < oo, we have (13.9) Now the identities (13.10) show that (13.11) The estimates (13.8) and (13.11) imply the required inequality (13.4). The proof of part (b) proceeds similarly; only now m < o, and so Lemma. 13.2 implies that um/p([A + B )f*)(um) decreases while P r um/r([~ + Br]f*)(um) increases. This accounts for the interchange of q and s. The proof is now complete. The operator w(a) defined by (13.2) is closely related to Calder~n's S(cr) operator introduced in (9, p. 288). w(a) = S(a) except in the important case r = oo where In fact, 79 (w(cr)f * )(t) (s(cr)f* )(t) Thus, in all cases we have s(cr) ~ w(cr) (13.12) The following theorem is due to Calderon [9, Theorem 8]. THEOREM 13.4: (Calderon): suppose o < p < r ~ oo and o < q,s ~ oo with q ~ s. Let T be a quasilinear operator of weak types (p,q) and (r,s). Then (Tf) * (t) ~ cs(cr)f * (t), for all f € pl L o < t < oo, r1 + L if r < oo, or for all f € (13.13) pl L oo + L if r = oo, where c is a constant independent of f. The next theorem shows when an operator of weak type (p,q;r,s) is simultaneously of weak types (p,q) and (r,s). THEOREM 13.5: suppose O < p < r ~ oo, O < q,s ~co, q ~ s, and let T be a quasilinear operator. (a) If T is of weak types (p,q) and (r,s), then T is of weak type (p,q;r,s). (b) If T is of weak type (p,q;r,s), then T is of weak type (p,q). If, in addition, r < oo, then T is also of weak type (r,s). Proof: Part (a) is a direct consequence of (13.13) and (13.12). To prove part (b) we are first required to show that 80 (1 3 .ll~) i.e., Tis of weak type (p,q) (cf.(1.5.i)). Assume that q < s. Then the inequalities (13.8) and (13.9) remain valid. 1 f € LP • Let t ~ oo in (13.8) to obtain 1 l!Tfllqoo <- c(llf\l p 1 + t-\o lim t f u /rf*(u)du/u). t~ Using (13.1) and the fact f € suppose (13.15) 1 LP , we have The required result (13.14) now follows from (13.15). 1 If, in addition, r < oo, let f E Lr • Lett~ 0 in (13.9) to obtain The definition (13.1) and the fact f E Lrl show that These estimates show that T : Lrl ~ Lsoo, i.e., T is weak type (r,s). The case q > s is treated similarly so we omit the details. This completes the proof. 81 14. Proof of the Main Results We may now prove Theorems A,B and C which were stated in the Introduction. Proof of THEOREM A: Recall that 0 < p < r ~ oo, 0 < q,s ~ oo with q-/: sand Tis of weak type (p,q;r,s). For 0 < 0 < 1, u and v are defined by 9 ·- + u p 1-@ r v 1 r -e: = q 1 e 1-e -= - + q s (14.1) Define ~ and E by 1 'ii" = 1 p First suppose that q < s. 1 1 (it1- . 2) s Then, (13.4) shows that (14.3) Applying the functional to both sides of (14.3), we obtain, by (14.1 ), (14.2) and a change of variable in both sides, (14.l+ ) 'l'he definition (14. 1) shows that p < u < r and q < v < s. 1.' hus, Theorem 6. 1 shows that in the expression ( l l1-. 4 ), the operators 82 Ap , Br , Cq and Ds are all equivalent to the identity. Hence, (14.4) is equivalent to the inequality !!Tf ·· l!·va·ex -< c\\f!!· ua·ex ' ' This is precisely the desired conclusion The case q > s is proved similarly utilizing (13.5). This completes the proof of Theorem A. The following theorem gives an interpolation result for the auxiliary Lorentz-Zygmund spaces of Section 11. THEOREM 14.1: Suppose 0 < p < r ~ oo and 0 < q,s ~ oo with q f s. Let T be a quasilinear operator of weak type (p,q;r,s). (i) If ex + 1/a > O and 6 + 1/b > o, then (ii) If ex+ 1/a < O and ~ + 1/b < o, then T a ex n ....sb 6 -:r.na (log ;/,)ex n -:J'.,..rb (log -/,) R -+ ...JJ '.'f/6 (log '71) ·:11 (log ~) . Proof: Let ~ and £ be defined by (14.2). q < s. The inequality (13.4) then holds: (14.6 ) Let us first assume that (14. 7 ) First, suppose a+ 1/a > 0 and ~ + 1/b > O. Applying the functional to both sides of ( 1LL 7), we obtain, after a change of variable in each side, I qa;a ([C q * - pa;a ([Ap +B]f). * +D](Tf))<cI s r (14. 8) 1 Multiplying both sides of (14.7) by t- , the identities (13. 10) show (14, 9) Applying the functional to both sides of (14.9), we have, after a change of variable in each side, (14.10 ) The sum of the inequalities (14.8) and (14.10), in conjunction with Theorem 11.5, shows that \\Tf\\ a a b R ~ cl\f\I pa a b I' '?7fl (log ?7?) + '!if (log ?7?) ·· J!. (log .£) + J! (log .£) This is precisely the assertion (14.5). When a+ 1/a and R + 1/b are negative, the statement (14.6) is proved similarly by first applying the functional 84 t o both sides of (14 . 9); then, we apply the functional to both sides of (14.7). The sum of the resulting inequalities, together with Theorem 11.5, produces the desired result (14.6). The case q > s is proved similarly using the fundamental inequality (13, 5). The proof is now complete. Proof of THEOREM B: Recall that O < p < r q f s. :S oo and 0 < q,s :S oo with T is a quasilinear operator of weak type (p,q;r,s). For 1 :S a :S b :S oo and 1 :S c :S d :S oo, the desired results are: (a) If a + 1/a = R + 1/b > 0 and v + 1/c = 5 + 1/d > o, then T : Lpa(log L)a+l + Lrc(log L)v+ 1 ~ Lqb(log L) 8 + Lsd(log L) 0 . (b) If a + 1/a = R + 1/b < 0 and y + 1/c = 5 + 1/d < o, then T : Lpa(log L)a+l n Lrc(log L)y+ 1 ~ Lqb(log L) 8 n L8 d(log L) 0 • If a+ 1/a > O and y + 1/c > o, Theorem 14.1.i shows that (14.11) But, the embedding theorem of Section 12 (cf. (12.1)) shows that ,,pa .re )y (14.12 ) Lpa (logL)a+ 1 +Lre (logL) y+ 1 C-;r: (log-;/, )a +-:!'(log./!,. The hypotheses (11.16.iv) now hold. Hence, we may use (11.24) and the embedding theorem of Section 12 to show (14.13) qb :: L (log L) ~ + L sd (log L) 5 • The inclusions (14.12) and (14.13), together with the result (14.11), prove the assertion of part (a). Part (b) of Theorem B is proved similarly using (14.6), (12.2 ) and ( 11 • 2 5 ) • REMARKS 14.2 (i): Using the full force of the inclusion relations given by Theorem 10.5, Theorem B remains valid for various other choices of the parameters a, b, c, d and a, A, y, 5. suppose a > b ~ 1 and a + 1/a > 8 + 1/b. For example, Then (10.6) gives If 9 + 1/b > O and 1 ~ c ~ d ~ oo with y + 1/c = 5 + 1/d > o, then Theorem B shows that This result and the inclusion (14.14) prove that part (a) of Theorem B holds when a > b ~ 1, a + 1/a > $ + 1/b > o, 1 < c < d < and y + 1/ c = A + 1 / d > 0. oo 86 (ii). The proofs of 'lll.eorems Bl and B2 are now simple consequences of 'llleorem B and 'llleorem 9.5. Proof of THEOREM C: Recall that 0 < p < r ~ oo, 0 < q,s ~ oo with q f s and T is a quasilinear operator of weak type (p,q;r,s). also have 1 ~ a ~ b ~ oo, O < c ~ oo, -oo We < a,~, v < oo and O < e < 1 • The parameters u and v are given by 1 e 1-e - = -+ u p r 1 e 1-~ - - = - + -v q s Theorem A implies (14.15) Applying 'llleorem B (with c = d = 1, V = 5 = O), we obtain T : 1pa (log 1 )a+l + 1 r1 (log 1) ~ 1qb (log 1) R + 1 s1 • (14.16) 'llle results (14.15) and (14.16) show that T : ( 1pa (log 1)a+ 1 + 1 rl (log 1) ) + 1uc (log 1) V (14.17) An examination of the norms of the spaces involved (cf. (6.1), (9.3)) shows that the spaces in (14.17) reduce to Lpa(log L)a+l + Luc(log 1)V and Lqb(log L)R + 1vc(log 1)V, respectively. (14.17) reduces to Hence, the statement 87 T pa a+1 UC v qb R vc v L (log L) + L (log L) ~ L (log L) + L (log L) , which is precisely the desired conclusion of Theorem C.a.i. remaining statements in Theorem C are proved similarly. The 88 REFERENCES [1] BENNETT, c., Intermediate spaces and the class L log+ L. Ark. Mat., 11(1973), 215-228. [2 J - - - - -, Estimates for weak-type operators. Bull. Amer. M.ath. Soc. 79(1973), 933-935. [3) , Banach function spaces and interpolation methods. I. The abstract theory. J. Functional Anal.,17(1974), 409-440 [4) , Banach function spaces and interpolation methods. II. Interpolation of weak-type operators. "Linear Operators and Approximation. II," Proc. Confer. Oberwolfach, P.L. Butzer and B. Sz. Nagy, Eds., ISNM 25, Birkhauser Verlag, Baselstuttgart, 1974, 129-139. , Banach function spaces and interpolation methods. 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