Sums and averages Stephen Semmes Rice University Abstract These informal notes are concerned with sums and averages in various situations in analysis. Contents 1 Real and complex numbers 3 2 Cesaro means 4 3 Admissible series 5 4 Abel summability 6 5 Cauchy products 8 6 Norms on vector spaces 9 7 Inner product spaces 10 8 Convexity 11 9 A few estimates 12 10 Operator norms 13 arXiv:1008.2467v1 [math.CA] 14 Aug 2010 11 Linear mappings 14 12 Convergence 15 13 Completeness 16 14 The supremum norm 17 15 Algebras 18 16 Banach algebras 19 1 17 Invertibility 20 18 Submultiplicative sequences 21 19 Invertibility, 2 21 20 Spectrum 22 21 Averages in normed algebras 23 22 Invertibility, 3 24 23 The open mapping theorem 25 24Theuniformboundednessprinciple 26 25 Strong operator convergence 27 26 Convergence of averages 27 27 Hilbert spaces 28 28 Unitary transformations 30 29 Rotations 31 30 Fourier series 31 31Measure-preservingtransformations 32 32 Sequence spaces 33 33 Maximal functions 34 34 Lp estimates 35 35 Maximal functions, 2 36 36 Discrete maximal functions 37 37 Another variant 37 38 Transference 38 39 Almost-everywhere convergence 40 40 Multiplication operators 42 41 Doubling spaces 43 2 42 Ultrametrics 43 43 Sequence spaces, 2 44 44 Snowflake transforms 44 45 Invariant measures 45 References 47 1 Real and complex numbers Let R be the field of real numbers. The absolute value of a real number x is denoted |x| and defined to be equal to x when x ≥ 0 and to −x when x ≤ 0. Thus |x|≥ 0, (1.1) |x + y| ≤ |x| + |y|, and (1.2) |xy| = |x| |y| for every x, y ∈ R. If A is a nonempty set of real numbers with an upper bound, then there is a unique real number which is the least upper bound or supremum of A, denoted sup A. Similarly, a nonempty set A of real numbers with a lower bound has a greatest lower bound or infimum, denoted inf A. It is sometimes convenient to put sup A = +∞ when A has no upper bound, or inf A = −∞ when A has no lower bound. Normally we shall only be concerned with infima of sets of nonnegative real numbers here, which are also nonnegative. Let C be the field of complex numbers. A complex number z can be ex- pressed as x + yi, where x, y ∈ R and i2 = −1. In this case, x and y are known as the real and imaginary parts of z, and are denoted Re z, Im z, respectively. The complex conjugate z of z is defined by (1.3) z = x − y i. In particular, z = z and z + z z − z (1.4) Re z = , Im z = . 2 2 i Observe that (1.5) z + w = z + w and (1.6) z w = z w for every z, w ∈ C. The modulus of z = x + yi is the nonnegative real number given by (1.7) |z| = (x2 + y2)1/2. 3 Equivalently, (1.8) |z|2 = z z, and hence (1.9) |z w| = |z| |w| for every z, w ∈ C. Of course, | Re z|, | Im z| ≤ |z| for every z ∈ C. If z, w ∈ C, then (1.10) |z + w|2 = (z + w)(z + w) = |z|2 + z w + z w + |w|2 = |z|2 +2 Re z w + |w|2, (1.11) since z w = z w. This implies that (1.12) |z + w|2 ≤ |z|2 +2 |z| |w| + |w|2 = (|z| + |w|)2, and therefore (1.13) |z + w| ≤ |z| + |w|. 2 Cesaro means ∞ As usual, a sequence {zj}j=0 of complex numbers converges to z ∈ C if for every ǫ> 0 there is a nonnegative integer L such that (2.1) |zj − z| <ǫ for every j ≥ L. In this case, one can show that the sequence of averages z + ··· + z (2.2) ζ = 1 n n n +1 ∞ also converges to z as n → ∞. However, there are also sequences {zj}j=0 of complex numbers that do not converge, but for which the corresponding ∞ j sequence {ζn}n=0 of averages does converge. For example, if zj = (−1) , then ζn = 0 when n is odd and ζn =1/(n + 1) when n is even, and limn→∞ ζn = 0. If z is any complex number, then n (2.3) (z − 1) zj = zn+1 − 1, Xj=0 where zj = 1 for every z ∈ C when j = 0. Hence n zn+1 − 1 (2.4) zj = z − 1 Xj=0 4 when z 6= 1. It follows that 1 n (2.5) lim zj =0 n→∞ n +1 Xj=0 when |z| = 1 and z 6= 1. This extends the case of z = −1 described in the previous paragraph. ∞ Let j=0 aj be an infinite series of complex numbers, and consider the sequenceP of partial sums l (2.6) bl = aj . Xj=0 ∞ ∞ By definition, j=0 aj converges if {bl}l=0 converges, in which event P ∞ (2.7) aj = lim bl. l→∞ Xj=0 The average βn of b0,...,bn is also given by n n +1 − j (2.8) β = a . n n +1 j Xj=0 ∞ ∞ The series j=0 aj is said to be Cesaro summable if {βn}n=0 converges. ConsiderP the case of a geometric series ∞ (2.9) aj , Xj=0 j j where a ∈ C and a = 1 when j = 0 again. If |a| < 1, then limj→∞ a = 0, and the corresponding geometric series converges with ∞ 1 (2.10) aj = 1 − a Xj=0 by the previous computations. If |a| ≥ 1, then |aj | = |a|j ≥ 1 for every j, and the geometric series does not converge in the conventional sense. It is Cesaro summable with sum 1/(1 − a) when |a| = 1 and a 6= 1, by the earlier computations for the partial sums applied twice to estimate their averages too. 3 Admissible series ∞ Let j=0 aj be an infinite series of complex numbers. If P ∞ j (3.1) aj t Xj=0 5 converges for some real number t ≤ 1, then j (3.2) lim aj t =0, j→∞ j ∞ j ∞ which implies that {aj t }j=0 is a bounded sequence. Conversely, if {aj t }j=0 is bounded, then ∞ j (3.3) |aj | r Xj=0 converges for 0 ≤ r<t, by comparison with a convergent geometric series. ∞ Let us say that an infinite series j=0 aj of complex numbers is admissible if any of the previous conditions holdsP for every positive real number r < 1 or t< 1, as appropriate, so that each of the other conditions also holds for every 0 ≤ r< 1 or0 ≤ t< 1. This is the same as saying that ∞ j (3.4) f(z)= aj z Xj=0 l has radius of convergence greater than or equal to 1. If bl = j=0 aj are the ∞ ∞ j partial sums of j=0 aj , then l=0 bl is admissible too. Indeed,P aj = O(R ) l implies that bl =PO(R ) for eachPR> 1. Put b−1 = 0, so that n n n n j j j j (3.5) aj z = (bj − bj−1) z = bj z − bj−1 z . Xj=0 Xj=0 Xj=0 Xj=0 Since n n n−1 j j j+1 (3.6) bj−1 z = bj−1 z = bj z , Xj=0 Xj=1 Xj=0 we get that n n−1 n−1 j j j+1 n j n (3.7) aj z = bj (z − z )+ bn z = (1 − z) bj z + bn z . Xj=0 Xj=0 Xj=0 ∞ n If |z| < 1, then admissibility of n=0 bn implies that limn→∞ bn z = 0, and P ∞ j (3.8) f(z)=(1 − z) bj z . Xj=0 4 Abel summability ∞ If j=0 aj converges, then it is well known that P ∞ j (4.1) lim aj r r→1− Xj=0 6 ∞ ∞ exists and is equal to j=0 aj. An admissible series j=0 aj of complex num- bers is said to be AbelP summable when this limit exists.P ∞ j For example, j=0 a is admissible for any complex number a with |a| = 1, and P ∞ 1 (4.2) aj zj = 1 − az Xj=0 ∞ j for every z ∈ C with |z| < 1, which implies that j=0 a is Abel summable to ∞ j 1/(1 − a) when a 6= 1. Similarly, j=0(j + 1) a isP admissible when |a| = 1, and P ∞ 1 (4.3) (j + 1) aj zj = (1 − az)2 Xj=0 ∞ j 2 for |z| < 1, so that j=0 ja is Abel summable to 1/(1 − a) when a 6= 1. ∞ Let {bj}j=0 be aP sequence of complex numbers, and consider n 1 (4.4) β = b . n n +1 j Xj=0 ∞ If {βn}n=0 converges, then n 1 n−1 (4.5) β = b n +1 n−1 n +1 j Xj=0 converges to the same value. This implies that b n (4.6) n = β − β → 0 n +1 n n +1 n−1 ∞ as n → ∞. If j=0 aj is a Cesaro summable series of complex numbers, then n one cam applyP this to bn = j=0 aj to get that P a b − b (4.7) n = n n−1 → 0 n +1 n +1 ∞ j as n → ∞. In particular, j=0(j + 1) a is not Cesaro summable for any a ∈ C with |a| = 1. P ∞ ∞ If j=0 aj is Cesaro summable, then j=0 aj is an admissible series, since ∞ an = OP(n).