Recently Alan D. Sokal gave a very short and completely elementary proof of
the uniform boundedness principle. The aim of this note is to point out that by
using a similiar technique one can give a considerably short and simple proof
of a stronger statement, namely a principle of condensation of singularities
for certain double-sequences of non-linear operators on quasi-Banach spaces,
which is a bit more general than a result of I.\,S. G\'al.Comment: 7 page

Hardtke, Jan-David

Zurnal matematiceskoj fiziki analiza geometrii

English

arXiv

Recently Alan D. Sokal in Amer. Math. Monthly 118 (2011), No. 5, 450-452, gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similar technique one can give a short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I. S. Gál from Duke Math. J. 20 (1953), No. 1, 27-35.Не так давно Алан Д. Сокал в Amer. Math. Monthly 118 (2011), No. 5, 450-452 представил очень короткое и простое доказательство принципа равномерной ограниченности. Цель настоящей заметки - показать, что, используя подобную технику, можно дать короткое и простое доказательство для более сильного утверждения, а именно принципа сгущения сингулярностей для определенных двойных последовательностей нелинейных операторов на квазибанаховых пространствах, которое является немного более общим, чем результат И.С. Гала из Duke Math. J. 20 (1953), No. 1, 27-35

Hardtke, J.-D.

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Journal of Mathematical Physics, Analysis, Geometry2013, vol. 9, No. 4, pp. 448–454A Remark on Condensation of SingularitiesJ.-D. HardtkeDepartment of Mathematics Freie Universita¨t BerlinArnimallee 6, Berlin 14195, GermanyE-mail: hardtke@math.fu-berlin.deReceived November 13, 2012Recently Alan D. Sokal in Amer. Math. Monthly 118 (2011), No. 5,450–452, gave a very short and completely elementary proof of the uniformboundedness principle. The aim of this note is to point out that by usinga similar technique one can give a short and simple proof of a strongerstatement, namely a principle of condensation of singularities for certaindouble-sequences of non-linear operators on quasi-Banach spaces, which isa bit more general than a result of I. S. Ga´l from Duke Math. J. 20 (1953),No. 1, 27–35.Key words: uniform boundedness principle, principle of condensation ofsingularities, quasi-Banach spaces, non-linear operators.Mathematics Subject Classification 2000: 46A16, 47H99.Let us begin by recalling that a quasi-norm on a linear space X over the fieldK = R or C is a map || . || : X → [0,∞) such that(i) ‖x‖ = 0 ⇒ x = 0,(ii) ‖λx‖ = |λ| ‖x‖ ∀x ∈ X,∀λ ∈ K,(iii) ∃K ≥ 1 such that ‖x+ y‖ ≤ K(‖x‖+ ‖y‖) ∀x, y ∈ X.The least constant K ≥ 1 which fulfils (iii) is sometimes called the modulus ofconcavity of the quasi-norm, and the pair (X, ‖ . ‖) is called a quasi-normed space.If p ∈ (0, 1], then a map ‖ . ‖ : X → [0,∞) is called a p-norm if it satisfies (i)and (ii) and the condition(iv) ‖x+ y‖p ≤ ‖x‖p + ‖y‖p ∀x, y ∈ X.The pair (X, ‖ . ‖) is then called a p-normed space. Every p-norm is a quasi-normwith K = 21/p−1. The standard examples of p-normed spaces are of course thespaces Lp(µ) with the p-norm‖f‖p =(∫S|f |p dµ)1/pc© J.-D. Hardtke, 2013A Remark on Condensation of Singularitiesfor any measure space (S,A, µ), which includes in particular the sequence spaces `p.The Aoki–Rolewicz-theorem (cf. [1, 11] or [9, Lemma 1.1]) states that if ‖ . ‖is a quasi-norm on X with modulus of concavity ≤ K and if p ∈ (0, 1] is definedby (2K)p = 2, then for any x1, . . . , xn ∈ X the inequality∣∣∣∣∣∣∣∣∣∣n∑i=1xi∣∣∣∣∣∣∣∣∣∣p≤ 4n∑i=1‖xi‖pholds.Then an equivalent p-norm on X can be defined by|||x||| = inf{( n∑i=1‖xi‖p)1/p: n ∈ N, x1, . . . , xn ∈ X, x =n∑i=1xi},cf. [1, 11] or [9, Theorems 1.2 and 1.3].The topology τ on X induced by the quasi-norm ‖ . ‖ is defined by declaringa subset O ⊆ X to be open if for every x ∈ O there exists ε > 0 such that{y ∈ X : ‖x− y‖ < ε} ⊆ O, which is the same as the topology induced by themetric d that is defined by d(x, y) = |||x− y|||p. The space (X, τ) is a topologicalvector space and (X, ‖ . ‖) is called a quasi-Banach space if (X, τ) is complete (orequivalently if X is complete with respect to the above metric d or any othertranslation-invariant metric on X that induces the topology τ).A linear operator T between two quasi-normed spaces X and Y is continuousif and only if there is a constantM ≥ 0 such that ‖Tx‖ ≤M‖x‖ for every x ∈ X.The space of continuous linear operators from X into Y is denoted by L(X,Y ).A quasi-norm on L(X,Y ) can be defined by ‖T‖ = sup{‖Tx‖ : x ∈ X, ‖x‖ ≤ 1}.The modulus of concavity of this quasi-norm is less than or equal to the oneof Y , and L(X,Y ) is complete if Y is complete, in particular the dual spaceX∗ = L(X,K) is always a Banach space.It should be mentioned that the Hahn–Banach-theorem fails in general forquasi-normed spaces, some of them do not even possess a separating dual, forexample, it is a well-known result that for p ∈ (0, 1) the dual space of Lp[0, 1]contains only 0. However, the classical results on Banach spaces like the openmapping and the closed graph theorem, as well as the uniform boundednessprinciple, still hold for quasi-Banach spaces with analogous proofs using the Bairecategory theorem.*The original proofs of the uniform boundedness principle by Banach and Hahn(cf. [2] and [7]) made no use of the Baire category theorem, but of a techniquewhich has come to be known as ”gliding hump” method. The Baire categoryargument was originally introduced into this theory by S. Saks.*For more information on quasi-Banach spaces the reader is referred to [8] and referencestherein.Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 449J.-D. HardtkeIn 1927, Banach and Steinhaus proved a by now well-known generalizationof the uniform boundedness principle, the so-called principle of condensation ofsingularities, which reads as follows.Theorem. (Principle of condensation of singularities, [3]). Let X be a Banachspace and (Yn)n∈N a sequence of normed linear spaces. If Gn ⊆ L(X,Yn) isunbounded for every n ∈ N, then there exists an element x ∈ X such thatsupT∈Gn‖Tx‖ =∞ ∀n ∈ N.The proof from [3] also uses the Baire theorem.Over the years many generalizations of the uniform boundedness principleand the principle of condensation of singularities to suitable classes of non-linearoperators on Banach and quasi-Banach spaces have been studied. In [5], I. S. Ga´lproved a principle of uniform boundedness for sequences of bounded homogenousoperators (see below for definitions) on Banach spaces that satisfy certain condi-tions of ”asymptotic subadditivity” and later in [6] he generalized his results to aprinciple of condensation of singularities for double-sequences of the same type.The authors of [4] considered quantitative versions of the principle of conden-sation of singularities for double-sequences of operators on quasi-Banach spacesthat fulfil similar conditions. The proofs of these results use refinements of theusual gliding hump argument. A proof of Ga´l’s results via the Baire categorytheorem can be found in [10] (it even works for some non-complete spaces, oneonly needs to assume that the space is of second category in itself).In [12], Alan D. Sokal gave a very short and elegant proof of the classicaluniform boundedness principle for linear operators on Banach spaces, which isalso completely elementary (in particular, it does not use the Baire categorytheorem).We wish to point out here that by using a suitable refinement of Sokal’s ideawe can give a proof of a theorem slightly more general than the aforementionedresult of Ga´l that is shorter than the original proof from [6]. The technique wewill use in the proof also resembles the one from [4] but since we are not interestedin quantitative versions it is less complicated.To formulate the result we first recall that a map T : X → Y , where X andY are of course quasi-normed spaces, is called homogenous if‖T (λx)‖ = |λ|‖Tx‖ ∀x ∈ X, ∀λ ∈ Kand it is called bounded if sup{‖Tx‖ : x ∈ X, ‖x‖ ≤ 1} is finite. For a boundedhomogenous map we denote this supremum again by ‖T‖. It follows that ‖Tx‖ ≤‖T‖ ‖x‖ for every x ∈ X and ‖T‖ is the best possible such constant.450 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4A Remark on Condensation of SingularitiesThe result then reads as follows.Theorem. Let X be a quasi-Banach space and (Ynm)(n,m)∈N2 a double-sequence of quasi-normed spaces, as well as (Tnm : X → Ynm)(n,m)∈N2 a double-sequence of bounded homogenous operators satisfyingsupn∈N‖Tnm‖ =∞ ∀m ∈ N. (+)Suppose further that there are two sequences (Cm)m∈N and (Lm)m∈N of positivereal numbers, a sequence (Nm)m∈N in N and two double-sequences (cnm)(n,m)∈N2and (fnm)(n,m)∈N2 of functions from X into [0,∞) such that(i) for every x ∈ X, each y ∈ X with ‖y‖ ≤ 1 and all n,m ∈ N with n ≥ Nmwe have‖Tnm(x+ y)‖ ≤ Cm(‖Tnmx‖+ ‖Tnm‖ ‖y‖+ fnm(x)),(ii) for every x ∈ X we have fnm(x) = O(1) as n→∞ uniformly in m ∈ N,(iii) for every x ∈ X, each y ∈ X with ‖y‖ ≤ 1 and all n,m ∈ N the inequality‖Tnmy‖ ≤ Lm(‖Tnm(x+ y)‖+ ‖Tnmx‖+ cnm(x)‖Tnm‖)holds,(iv) for every x ∈ X and every m ∈ N we have cnm(x)→ 0 as n→∞.Then there is an element x ∈ X such thatsupn∈N‖Tnmx‖ =∞ ∀m ∈ N.As we said, the conditions (i)–(iv) include in particular the case of Ga´l’sasymptotically subadditive double-sequences (essentially only the functions fnmare new here). For some examples, where such conditions of asymptotic subad-ditivity occur naturally (e. g., the so-called metric-mean interpolations), we referthe reader to [5] and [6].Before we can come to the main proof, we need an easy lemma.Lemma. For every sequence (αn)n∈N of positive real numbers and everyβ > 0 there is a sequence (βn)n∈N of positive real numbers such that∞∑i=n+1βi < αnβn and βn ≤ β ∀n ∈ N.Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 451J.-D. HardtkeP r o o f. Just choose inductively 0 < βn ≤ β such thatn∑i=m+1βi <αmβm2∀m ∈ {1, . . . , n− 1}.P r o o f of the Theorem. By the Aoki–Rolewicz-theorem we may assumewithout loss of generality that X is p-normed for some p ∈ (0, 1] (actually this isnot necessary for the proof but it is more convenient).Define ψ : N→ N by (ψ(1), ψ(2), . . . ) = (1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . . . ).It easily follows from (iii) that for any r ∈ (0, 1], each x ∈ X and all n ≥ Nmwe haver‖Tnm‖Lm≤ supy∈Br(x)‖Tnmy‖+ ‖Tnmx‖+ cnm(x)‖Tnm‖≤ 2 supy∈Br(x)‖Tnmy‖+ cnm(x)‖Tnm‖, (1)where Br(x) = {y ∈ X : ‖x− y‖ ≤ r} (this is an analogue of the lemma from[12]).According to the above lemma, we can find a sequence (β˜n)n∈N in (0, 1] suchthat ∞∑i=n+1β˜i <β˜n8pLpψ(n)Cpψ(n)∀n ∈ N. (2)For every n ∈ N we put βn = β˜1/pn andγn =βn8Lψ(n)Cψ(n)−( ∞∑i=n+1βpi)1/p, (3)which by (2) is strictly positive. Let us also put x0 = 0.Because of (iv) and our assumption (+), we can find n1 ≥ Nψ(1) such thatcn1ψ(1)(x0) ≤ β1/2Lψ(1) and ‖Tn1ψ(1)‖ ≥ 1/γ1.It follows from (1) thatβ14Lψ(1)‖Tn1ψ(1)‖ ≤ supy∈Bβ1 (x0)‖Tn1ψ(1)y‖and hence there is some x1 ∈ X such that ‖x1 − x0‖ ≤ β1 and ‖Tn1ψ(1)x1‖ ≥β1‖Tn1ψ(1)‖/8Lψ(1).Next we use (iv) and (+) to choose an index n2 > max{n1, Nψ(2)} suchthat cn2ψ(2)(x1) ≤ β2/2Lψ(2) and ‖Tn2ψ(2)‖ ≥ 2/γ2 and then find (using (1)) anelement x2 ∈ X with ‖x2 − x1‖ ≤ β2 and ‖Tn2ψ(2)x2‖ ≥ β2‖Tn2ψ(2)‖/8Lψ(2).452 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4A Remark on Condensation of SingularitiesContinuing in this way, we obtain a strictly increasing sequence (nk)k∈N in Nand a sequence (xk)k∈N in X such that for every k ∈ N we have nk ≥ Nψ(k) and‖xk − xk−1‖ ≤ βk, (4)‖Tnkψ(k)xk‖ ≥βk8Lψ(k)‖Tnkψ(k)‖, (5)‖Tnkψ(k)‖ ≥kγk. (6)From (4) it follows that d(xn, xm) = ‖xn − xm‖p ≤∑ni=m+1 βpi for every n >m and hence (xk)k∈N is a Cauchy sequence. By completeness, the limit x =limk→∞ xk exists and it follows that‖x− xm‖p ≤∞∑i=m+1βpi ∀m ∈ N. (7)Now for sufficiently large k ∈ N we have by (i), (5), (7), (3) and (6)fnkψ(k)(x) + ‖Tnkψ(k)x‖ ≥‖Tnkψ(k)xk‖Cψ(k)− ‖Tnkψ(k)‖ ‖xk − x‖≥ ‖Tnkψ(k)‖(βk8Lψ(k)Cψ(k)−( ∞∑i=k+1βpi)1/p)= ‖Tnkψ(k)‖γk ≥ k.Together with (ii) this implies ‖Tnkψ(k)x‖ → ∞ for k → ∞. Since for everym ∈ N the set ψ−1({m}) is infinite, it follows that supn∈N ‖Tnmx‖ =∞ for everym ∈ N.Let us remark that if we only wish to prove the usual principle of condensationof singularities for linear operators as it is stated on page 450, then all we haveto do is to replace ”Tn ∈ F” by ”Tn ∈ Gψ(n)” at the beginning of Sokal’s originalproof from [12] (where ψ is defined as in our preceding proof) and the rest of theargument remains exactly the same.Acknowledgements. The author would like to express his gratitude to theanonymous referee for pointing out to him the paper [10].References[1] T. Aoki, Locally Bounded Linear Topological Spaces. — Proc. Imp. Acad. Tokyo18 (1942), No. 10, 588–594.[2] S. Banach, Sur les operations dans les ensembles abstraits et leur application auxe´quations integrales. — Fund. Math. 3 (1922), 133–181.Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4 453J.-D. Hardtke[3] S. Banach and S. Steinhaus, Sur le principe de la condensation des singularite´s.—Fund. Math. 9 (1927), 50–61.[4] W. Dickmeis, R.J. Nessel, and E. van Wickeren, On Nonlinear Condensation Prin-ciples with Rates.— Manuskripta Math. 52 (1985), 1–20.[5] I.S. Ga´l, Sur la me´thode de re´sonance et sur un theore`me concernant les espaces detype (B).— Annales de l’Institut Fourier 3 (1951), 23–30.[6] I.S. Ga´l, The Principle of Condensation of Singularities.— Duke Math. J. 20 (1953),No. 1, 27–35.[7] H. Hahn, U¨ber Folgen Linearer Operationen.— Monatsh. Math. Phys. 32 (1922),3–88.[8] N. Kalton, Quasi-Banach Spaces. In: Handbook of the Geometry of Banach Spaces.W.B. Johnson and J. Lindenstrauss (eds.), Vol. 2, North-Holland, Amsterdam–NewYork, 2003, 1099–1130.[9] N.J. Kalton, N.T. Peck, and J.W Roberts, An F -Space Sampler. London Math.Soc. Lect. Notes 89, Cambridge Univ. Press, Cambridge, 1985.[10] J. Kolomy´, A Note on Uniform Boundedness Principle for Nonlinear Operators.—Comm. Math. Univ. Carol. 10 (1969), No. 2, 207–216.[11] S. Rolewicz, On a Certain Class of Linear Metric Spaces.— Bull. Polon. Acad. Sci.Se´r. Sci. Math. Astrono. Phys. 5 (1957), 471–473.[12] A.D. Sokal, A Really Simple Elementary Proof of the Uniform Boundedness Theo-rem. — Amer. Math. Monthly 118 (2011), No. 5, 450–452.454 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 4

A Remark on Condensation of Singularities

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A remark on condensation of singularities

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