Marcinkiewicz Interpolation Theorem by Daniel Baczkowski
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1 1 Introduction Marcinkiewicz Interpolation Theorem b Daniel Baczkowski Let (, µ be a measure space. Let M denote the collection of all extended real valued µ-measurable functions on. Also, let M denote the class of functions from M that are finite µ-almost everwhere. In this section, we assume that the functions are in M. For a function f M (, µ, we denote b m( = m f ( its distribution function, namel for > define m( = m f ( = µ { x : f(x > }. It is eas to see that m( is monotone decreasing and also true that it is continuous on the right for all. The decreasing rearrangement of f is the function f defined on [, b f (t = inf { : m f ( t } for t. (1 sing the convention that inf =, notice that if m f ( happens to be continuous and strictl decreasing then f is the inverse of m f. The basic propert of f is that (1 implies that f is equimeasurable with f, namel, m f ( m f ( for >. We sa that T : L p (, µ L q (V, ν is of strong tpe (p, q if T is bounded; i.e. if there exists a c > such that T f q c f p for all f L p. For 1 p <, we sa that T : L p (, µ M (V, ν is of weak tpe (p, p if there exists a c > such that m (T f ( = ν { x V : (T f(x > } c f p p for >. We sa a map T : A B is subadditive if for an f and g A, T (f + g T (f + T (g Theorem 1. (Marcinkiewicz Let 1 p < p 1 and assume that T : L p (, µ L p 1 (, µ M (V, ν is subadditive. Also, suppose that T is of weak tpe (p, p and (p 1, p 1. Then, for each p such that p < p < p 1, the map T is of strong tpe (p, p. Theorem. (Marcinkiewicz Let 1 p < and assume that T : L p (, µ L (, µ M (V, ν is subadditive. Also, suppose that T is of weak tpe (p, p and of strong tpe (,. Then, for each p such that p < p <, the map T is of strong tpe (p, p. 1
2 Lemma 1. If f L p, 1 p <, then and m f ( f p p p for > ( f p p = p t p 1 m f (tdt. (3 Proof. B using the convenient notation { f > } to denote {x : f(x > }, we have f p p = f(x p dµ f(x p dµ p µ{ f > } = p m f ( in which ( easil follows. Next, we introduce a set A [, which we define b A = {(x, s : f(x p > s}. sing Fubini s Theorem, we obtain f(x p f(x p dµ = 1 dsdµ = χ A (x, sdsdµ [, = χ A (x, sdµds = µ{x : f(x p > s}ds. se the substitution s = t p to obtain f(x p dµ = p t p 1 µ{x : f(x > t}dt = p t p 1 m f (tdt. For 1 p <, denote b L p, the weak L p space which is the set of functions in M (, µ such that m f ( M/ p, for all > where M > is some constant. Let f L p = sup > m f ( 1/p. Notice that b Lemma 1, L p L p with f L p f Lp, but L p L p. Example: Let f(x = 1/x 1/p where x (, 1] and 1 p <. Then, m f ( = {x : 1/x 1/p > } = { 1 if < 1 1/ p if > 1 and hence f L p(, 1]. But, f p = ( 1 1/x dx 1/p =. So, f / Lp. Lemma. If f L p where 1 p <, then f p dµ = m f ( p + p f p dµ = m f ( p + p { f } t p 1 m f (tdt (4 t p 1 m f (tdt (5
3 Proof. First notice that (3 minus (5 ields (4, hence it suffice to prove (4. Proceeding analogousl as in the proof of Lemma 1, f p dµ = = p f(x p 1 dµ + 1 dsdµ = f(x p p 1 dsdµ + 1 dsdµ = p m f ( + p f(x p p [ p, 1 dsdµ χ A (x, sdsdµ where A = {(x, s : p s f(x p }. Now, using Fubini s Theorem, then the substitution s = t p, we get χ A (x, sdsdµ = µ{x : f(x p > s}ds = p t p 1 µ{x : f(x p > t p }dt [ p, which ields (4. = p p t p 1 m f (tdt Proofs of the theorems Proof of Theorem 1: We will consider for simplicit the case when (, µ = (V, ν = ([a, b], µ where µ denotes the Lebesgue measure on the interval [a, b] (i.e. dµ = dx. For a given >, define { f(x if f(x g(x = g (x = if f(x > { if f(x h(x = h (x = f(x if f(x > Since f(x = g(x + h(x and T is subadditive, we have { x : T f(x > } { x : T g(x > / } { x : T h(x > / }. So, where m ( F, = m F (. m ( T f, m ( T g, / + m ( T h, / (6 Since T is of weak tpe (p 1, p 1 and (p, p, we have, respectivel, that m ( T g, / ( p1 c 1 g p 1 dx = c 1 p 1 p 1 f p 1 dx [a,b] { f } and m ( T h, / ( p c h p dx = c p p [a,b] f p dx. 3
4 For simplicit, use m(t = m f (t. Now, using ( through (6 it follows that T f p p = p p 1 m ( T f, d ( pc 1 p 1 p 1 p p 1 1 t p1 1 m(tdt d ( ( + c p p p 1 m(d + p t p p 1 t p 1 m(tdt d ( = c p f p p + c 1 pp 1 p 1 t p1 1 m(t p p1 1 d dt t ( t + c pp p t p 1 m(t p p 1 d dt = c p f p p + c 1 p 1 p 1 p = c f p p. 1 p p 1 t p 1 m(tdt + c p p p Thus, T is of strong tpe (p, p, and the theorem follows in this case. 1 p p t p 1 m(tdt Proof of Theorem : Now suppose that 1 p < and p 1 =. We will again consider when (, µ = (V, ν = ([a, b], µ where µ denotes the Lebesgue measure. For a given >, define f(x if f(x g(x = g (x = c 1 sign ( f(x if f(x > c 1 c 1 where c 1 > is from the definition of T being of weak tpe (p 1, p 1. Let h(x = h (x = f(x g(x. Also b the definition of T, T g c 1 g / since g /(c 1. So, { x : (T g(x > / } =. As (6 also holds in this case, we have from (1 that m ( T f, = µ { T f > } µ { T h > / } ( p c h p p ( ( c p c p f(x p dx + { f >/(c 1 } ( p { f >/(c 1 } c 1 p µ { f > /(c 1 } p f(x p dx + (c 1 p m f ( /(c1 One onl needs now to proceed exactl in the same fashion as in the proof for Theorem 1. 4
5 3 Application of Marcinkiewicz Interpolation Theorem It is quite interesting that the Marcinkiewicz Interpolation Theorem is used to prove man pertinent theorems in Approximation Theor and Analsis. In this section, we simpl state just one application of this powerful tool. We present a generalization of the Hausdorff- Young inequalit due to Pale. The main difference between the theorems being that Pale introduced a weight function into his inequalit and resorted to the theorem of Marcinkiewicz. In what follows, we consider the measure space (R n, µ where µ denotes the Lebesgue measure (dµ = dx. Let F denote the Fourier transform which is defined b (Ff(ξ = ˆf(ξ = f(x exp{ i x, ξ }dx where x, ξ = x 1 ξ x n ξ n (note here x = (x 1,..., x n and ξ = (ξ 1,..., ξ n. Also, we assume that w(x is a weight function, i.e. a positive measurable function on R n. Letting L p (w denote the L p space with respect to wdx, the norm on L p (w is Theorem 3. Assume that 1 p. Then, ( 1/p f Lp(w = f(x p w(xdx. R n Ff L ( ξ n( p C p f Lp. Proof. Consider the map defined b (T f(ξ = ξ n ˆf(ξ. B Parseval s formula, T f L ( ξ n = ˆf L C f L. Simpl b the definition of weak L p spaces we have that T f L ( ξ n T f L ( ξ n. Altogether, we have that T is of weak tpe (,. We now work towards showing that T is of weak tpe (1, 1. Thus, the Marcinkiewicz interpolation theorem implies the theorem. Now, consider the set E = {ξ : ξ n ˆf(ξ > }. For simplicit, we let ν denote the measure ξ n dξ and assume that f L1 = 1. Then, ˆf(ξ 1. Therefore, for ξ E we have ξ n. This forces m ( T f, = ν ( E = ξ E n dξ ξ n dξ. {ξ: ξ n } Substituting ξ = 1/n u gives ξ n = u n and dξ = du; thus, ξ n dξ = 1 u n du = C 1. {ξ: ξ n } {u: u 1} So, we have that m ( T f, C 1. Now, we have shown that m ( T f, C f L1 which implies T is of weak tpe (1, 1. 5
6 References: Bergh, J. and Löfström, J., An Introduction to Interpolation Spaces, Springer-Verlag, Petrushev, P., handout on the Marcinkiewicz Interpolation Theorem. 6
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