INTERPOLATION, MAXIMAL OPERATORS, AND THE HILBERT TRANSFORM MICHAEL WONG Abstract. Real-variable methods are used to prove the Marcinkiewicz Inter- polation Theorem, boundedness of the dyadic and Hardy-Littlewood maximal operators, and the Calder´on-Zygmund Covering Lemma. The Hilbert trans- form is defined, and its boundedness is investigated. All results lead to a final theorem on the pointwise convergence of the truncated Hilbert transform Contents 1. Introduction 1 2. Preliminaries 2 3. Marcinkiewicz Interpolation Theorem 4 4. The Dyadic Maximal Operator and Calder´on-Zygmund Decomposition 5 5. The Hardy-Littlewood Maximal Operator 7 6. Schwartz Functions and Tempered Distributions 9 7. The Hilbert Transform 12 8. The Truncated Hilbert Transform 16 9. Conclusion 19 Acknowledgments 20 References 20 1. Introduction The Hilbert transform H on R is formally defined by (1.1) Hf(x) = lim H(x) !0+ where H is the truncated Hilbert transform at > 0, Z f(x − y) Hf(x) = dy jyj> y and dx is the Lebesgue measure. While H is well-defined for a large class of functions, the principal value integral implicit in Equation (1.1) is finite for only well-behaving functions. However, H has boundedness properties by which it can be extended to larger function spaces. In the following exposition, we build the real-variable tools needed to extend the Hilbert transform to Lp(R; µ) for all p 2 [1; 1), where µ is the Lebesgue measure. Date: August 31, 2010. 1 2 MICHAEL WONG These instruments include the Marcinkiewicz Interpolation Theorem, Calder´on- Zygmund decomposition, Schwartz functions, and tempered distributions. In the end, we prove that Equation (1.1) is valid for f 2 Lp, p 2 [1; 1), up to a µ-null set. 2. Preliminaries We begin by defining different types of boundedness. The basic domain consid- ered in this paper is Lp(X; µ), where p 2 (0; 1], X is an arbitrary set, and µ is a nonnegative, extended real-valued measure. Intuitively, the image of Lp under a given operator T determines the strength of its boundedness: does T map Lp to q- integrable functions, 0 < q ≤ 1, or to functions satisfying only a weaker condition? One such weaker condition is that which defines the weak-Lp space. Definition 2.1. Let (X; µ) be a measure space, and let f : X ! C be a measurable function. For measurable A ⊆ X, we denote µ(A) by jAj. The distributional function of f is the function df : R+ ! R given by df (λ) = jfx 2 X : jf(x)j > λgj If X = Rn, we set µ to be the Lebesgue measure. Note that df is a measurable function. Definition 2.2. For p 2 (0; 1), weak-Lp(X; µ), denoted by Lp;1(X; µ), is the space of all measurable functions f : X ! C such that p C 1 p jjfjjp;1 := inf fC > 0 : df (λ) ≤ p g = supfλdf (λ) g < 1 λ>0 λ λ>0 1 1 We set weak-L (X; µ) to be L (X; µ) and jj · jj1;1 to be jj · jj1. p;1 The map jj · jjp;1 is a quasinorm on the linear space L (X; µ)=C. Chebyshev's inequality shows that Lp ⊆ Lp;1. If X = Rn, a counterexample demonstrating that this containment is strict is f(x) = jxj−n=p. So Lp and Lp;1 determine two types of boundedness. Definition 2.3. An operator T from Lp(X; µ) to the space of complex-valued, measurable functions on a measure space (Y; ν) is sublinear if (1) 8f; g 2 Lp(X; µ); jT (f + g)(y)j ≤ jT f(y)j + jT g(y)j (2) 8α 2 C; jT (αf)(y)j = jαjjT f(y)j Definition 2.4. A sublinear operator T is weakly bounded from p to q, 0 < p; q ≤ 1, if there exists C > 0 such that p jjT fjjq;1 ≤ Cjjfjjp 8f 2 L (X; µ) We say that such an operator T is weak (p; q) for short. T is strongly bounded from p to q if there exists a C > 0 such that p jjT fjjq ≤ Cjjfjjp 8f 2 L (X; µ) We say that T is strong (p; q). Remark 2.5. Note that by our definitions, weak (p; 1) boundedness is identical to strong (p; 1) boundedness. We will say that an operator satisfying Definition 2.4 for q = 1 is bounded (p; 1). INTERPOLATION, MAXIMAL OPERATORS, AND THE HILBERT TRANSFORM 3 In other words, T is strong (p; q) if T maps Lp into Lq; T is weak (p; q) if T maps Lp only into Lq;1. By the fact that Lq ⊆ Lq;1, T is strong (p; q) implies T is weak (p; q). Note that T is weak (p; q) if and only if for all f 2 Lp and λ > 0, C d (λ) ≤ ( jjfjj )q T f λ p The following theorem about the pointwise convergence of linear operators, based only on the above definitions, will be of use later. p Theorem 2.6. Let fTtg be a family of linear operators mapping L (X; µ) into the space of complex-valued, measurable functions over (X; µ). Define the maximal operator T ∗ by ∗ T f(x) = supfjTtf(x)jg t If T ∗ is weak (p; q), then the set p A = ff 2 L : lim Ttf(x) exists a.e.g t!t0 is closed in Lp. Proof. Assume that Ttf is real-valued. If Ttf is complex-valued, apply the following argument to the real and imaginary parts of Ttf separately. First, observe that for all f 2 Lp, ∗ (2.7) lim sup Ttf(x) − lim inf Ttf(x) ≤ 2T f(x) t!t t!t0 0 p Now suppose ffng ⊂ A converges in L norm to f. Each fn satisfies jfx 2 X : lim sup Ttfn(x) − lim inf Ttfn(x) > 0gj = 0 t!t t!t0 0 and it suffices to show f satisfies the same equation. For all λ > 0, jfx 2 X : lim sup Ttf(x) − lim inf Ttf(x) > λgj t!t t!t0 0 ≤ jfx 2 X : lim sup Tt(f − fn)(x) − lim inf Tt(f − fn)(x) > λgj t!t t!t0 0 ∗ ≤ jfx 2 X : 2T (f − fn)(x) > λgj (Equation (2.7)) λ 2C q = d ∗ ( ) ≤ ( jjf − f jj ) (T is weak (p; q)) T (f−fn) 2 λ n p The limit of the last term as n ! 1 is 0. Hence, jfx 2 X : lim sup Ttf(x) − lim inf Ttf(x) > 0gj t!t t!t0 0 1 X 1 ≤ jfx 2 X : lim sup Ttf(x) − lim inf Ttf(x) > gj = 0 t!t t!t0 k k=1 0 Remark 2.8. By a similar proof, one could show that the set 0 p A = ff 2 L : lim Ttf(x) = f(x) a.e.g t!t0 p is closed in L . One would disregard Equation (2.7) and thereafter replace lim inf Ttfn(x) with fn(x) and lim inf Ttf(x) with f(x). 4 MICHAEL WONG 3. Marcinkiewicz Interpolation Theorem The next step is to determine the p 2 (0; 1] for which a given sublinear op- erator is bounded (p; p). The Marcinkiewicz interpolation theorem asserts strong boundedness for all values of p between two values for which weak boundedness is established. In our proof of the theorem, the following two lemmas will be used. We set dµ(x) = dx. p0 p1 Lemma 3.1. Let L (X; µ) + L (X; µ), 1 ≤ p0 < p1 ≤ 1, be the direct sum of p0 p1 p L (X; µ) and L (X; µ). If p0 < p < p1 and f 2 L (X; µ), then f = f0 + f1 for p0 p1 some f0 2 L and f1 2 L . p Proof. Fix λ > 0. Given f 2 L , define f0 and f1 by f0 = fχfx:jf(x)|≥cλg f1 = fχfx:jf(x)j<cλg where the value of the constant c will be chosen in Theorem 3.3. To see that po f0 2 L , observe that Z Z cλ p0 p0 p−p0 jf0(x)j dx = jf(x)j ( ) dx X fjf(x)|≥cλg cλ Z jf(x)j ≤ jf(x)jp0 ( )p−p0 dx fjf(x)|≥cλg cλ 1 ≤ ( )p−p0 jjfjjp cλ p p1 The fact that f1 2 L is shown similarly. p p R 1 p−1 Lemma 3.2. If f 2 L (X; µ), 1 ≤ p < 1, then jjfjjp = 0 pλ df (λ) dλ. Proof. Z p p jjfjjp = jf(x)j dx X Z Z jf(x)j = pλp−1 dλdx X 0 Z 1 Z = pλp−1 dxdλ (Fubini's theorem) 0 fx:jf(x)j>λg Z 1 p−1 = pλ df (λ) dλ 0 Theorem 3.3 (Marcinkiewicz Interpolation Theorem). Suppose T is a sublinear p0 p1 operator from L (X; µ) + L (X; µ), 1 ≤ p0 < p1 ≤ 1, to the space of complex- valued, measurable functions on (Y; ν). If T is weak (p0; p0) and weak (p1; p1), then T is strong (p; p) for all p 2 (p0; p1). p Proof. Fix λ > 0. Given f 2 L , define f0 and f1 as in Lemma 3.1. Because T is sublinear, λ λ (3.4) d (λ) ≤ d ( ) + d ( ) T f T f0 2 T f1 2 We choose c by cases for the value of p1. INTERPOLATION, MAXIMAL OPERATORS, AND THE HILBERT TRANSFORM 5 First, suppose p1 < 1, and let c = 1.