summary:Given a rotation invariant measure in $\Bbb R^n$, we define the maximal operator over circular sectors. We prove that it is of strong type $(p,p)$ for $p>1$ and we give necessary and sufficient conditions on the measure for the weak type $(1,1)$ inequality. Actually we work in a more general setting containing the above and other situations

Aimar, H.

Forzani, L.

Naibo, V.

Commentationes Mathematicae Universitatis Carolinae

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeHugo A. Aimar; Liliana Forzani; Virginia NaiboOn maximal functions over circular sectors with rotation invariant measuresCommentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 2, 311--318Persistent URL: http://dml.cz/dmlcz/119245Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 42,2 (2001)311–318 311On maximal functions over circularsectors with rotation invariant measuresH. Aimar, L. Forzani, V. NaiboAbstract. Given a rotation invariant measure in Rn, we define the maximal operatorover circular sectors. We prove that it is of strong type (p, p) for p > 1 and we givenecessary and sufficient conditions on the measure for the weak type (1, 1) inequality.Actually we work in a more general setting containing the above and other situations.Keywords: maximal functions, spaces of homogeneous typeClassification: 42B25Let X be a topological space and µ be a Borel measure on X . By Lpµ(X)we will denote the space of all real valued measurable functions on X such that‖f‖p,µ = (∫X |f(x)|p dµ(x))1/p < ∞ if 1 ≤ p < ∞, and ‖f‖∞,µ = inf{a : µ({x ∈X : |f(x)| > a}) = 0} < ∞, if p = +∞. By Lµ,loc(X) we mean the set of all realvalued measurable functions on X which are integrable on compact sets.An operator T defined on Lpµ(X) is said to be of strong type (p, p), 1 ≤p ≤ +∞, if there exists a positive constant Ap such that for every f ∈ Lpµ(X),‖Tf‖p,µ ≤ Ap‖f‖p,µ. The operator T is of weak type (p, p), 1 ≤ p < +∞, ifthere exists a constant Bp such that for every f ∈ Lpµ(X) and λ > 0, µ({x ∈ X :|Tf(x)| > λ}) ≤ Bp‖f‖p,µλ−p.Let y ∈ Sn−1, the unit sphere in Rn, α ∈ (0, π) and 0 ≤ r ≤ R ≤ +∞. Theset S = S(y, α, r, R) = {x ∈ Rn : arg(x, y) < α and r < |x| < R}, where arg(x, y)is the angle between x and y, is called a circular sector in Rn. If G is a Borelsubset of Sn−1, G(r, R) will denote the set {ρx : x ∈ G, r < ρ < R} and σr willmean the surface measure on the sphere of radius r.Let µ be a non-negative rotation invariant Radon measure. If f ∈ Lµ,loc(Rn)and x 6= 0 we define(1) Msµf(x) = supx∈S1µ(S)∫S|f(y)|dµ(y)Supported by CONICET and Universidad Nacional del Litoral.312 H.Aimar, L. Forzani, V.Naibowhere the supremum is taken over all bounded sectors S in Rn with non-zeromeasure containing x. Set Msµf(0) = 0.It is not hard to prove that if G is a Borel set in Sn−1, 0 ≤ r ≤ R ≤ +∞ andf ∈ L1µ(Rn) then∫G(r,R)f(y)dµ(y) =1σ1(Sn−1)∫G∫ Rrf(ρy′)dν(ρ)dσ1(y′)=1σ1(Sn−1)∫ Rr∫Gf(ρy′)dσ1(y′)dν(ρ),where ν is the measure in R+ defined by ν((a, b)) = µ(Sn−1(a, b)), with the abovenotation and G = Sn−1, for 0 ≤ a ≤ b ≤ +∞. So, the average of f over circularsectors is obtained by considering averages over the intervals in R, with respectto ν, and over the “balls” in Sn−1, with respect to the surface measure σ1. Thisfact allows us dominate Msµf by an iteration of two maximal functions: one in Rwith respect to an abstract measure ν, and the other in Sn−1 with respect to its“doubling” usual measure σ1.This situation is analogous to the classical one of the maximal operator overrectangles with sides parallel to the coordinate axes, with respect to the Lebesguemeasure. In this case the strong type inequality holds for p > 1 but the weaktype (1, 1) inequality is no longer true (see [4]).We are interested in the boundedness of Msµ. The strong type (p, p) for p > 1will be obtained as a consequence of the strong type inequality for the maximalfunctions in (Sn−1, σ1) and in (R+, ν). Our main result is a characterization ofthe measures ν for which the weak type (1,1) inequality holds. We shall work in amore general setting that includes the above situations. The basic result of thesenotes concerning Msµ will be the following corollary of Theorems 1 and 2 below.Given a non-negative rotation invariant Radon measure µ, we have(a) Msµ is of strong type (p, p) for 1 < p ≤ ∞;(b) Msµ is of weak type (1, 1) if and only if µ is a finite linear combination ofsurface measures; i.e., µ =∑Ni=1 aiσri , with ai, ri > 0, N ∈ N.Let X be a set; a nonnegative symmetric function d on X × X is called aquasi-distance if there exits a constant k such thatd(x, y) ≤ k [d(x, z) + d(z, y)]for every x, y, x ∈ X , and d(x, y) = 0 if and only if x = y.The balls B(x, r) = {y : d(x, y) < r} form a basis of neighborhoods of x forthe topology induced by d.On maximal functions 313Consider (X, d, λ), where d is a quasi-distance on X and λ a non-negativemeasure defined on a σ-algebra of subsets of X which contains the balls. Forf ∈ Lλ,loc(X) we define the Hardy-Littlewood maximal function operatorMλf(x) = supx∈B1λ(B)∫B|f(x)|dλ(x),where B is a ball containing x with positive measure.We shall say that (X, d, λ) is a space of homogeneous type if there exists aconstant A such that(2) 0 < λ(B(x, 2r)) ≤ Aλ(B(x, r)) < ∞holds for every x ∈ X and r > 0.For these spaces, the weak type (1, 1) and the strong type (p, p) for p > 1 ofMλ hold (see [2]).Let (X, d, λ) be a space of homogeneous type and ν a Lebesgue-Stieltjes mea-sure on R. Consider the product space X ×R endowed with the product measureµ = λ × ν. For f ∈ Lµ,loc(X × R) we consider the maximal function(3) Mµf(x, t) = sup(x,t)∈S1µ(S)∫S|f(y, s)|dµ(y, s)where the supremum is taken over all the cylinders S = B × (r, R) in X ×R withpositive measure containing (x, t), B being a ball in X .Theorem 1. Mµ is of strong type (p, p) for p > 1.Proof: Since the case p = +∞ is trivial we consider 1 < p < ∞. By Tonellitheorem we have thatMµf(x, t) = sup(x,t)∈S1µ(S)∫S|f(y, s)|dµ(y, s)= sup(x,t)∈S1λ(B)∫B(1ν((r, R))∫ Rr|f(y, s)|dν(s))dλ(y)≤ sup(x,t)∈S1λ(B)∫BMν(f(y, .))(t)dλ(y)≤ Mλ[(Mν(f(., .)))(t)](x).The maximal function Mλ is Lpλ(X)-bounded because (X, d, λ) is a space ofhomogeneous type and, even without any doubling condition for ν, the maxi-mal function Mν is Lpν(R)-bounded because the basic space is one-dimensional(see [1]). Then314 H.Aimar, L. Forzani, V.Naibo∫X×R|Mµf(x, t)|pdµ(x, t) ≤∫X×R|Mλ[(Mν(f(., .)))(t)](x)|pdµ(x, t)=∫R∫X|Mλ[(Mν(f(., .)))(t)](x)|pdλ(x)dν(t)≤ C∫R∫X|Mν(f(x, .))(t)|pdλ(x)dν(t)= C∫X∫R|Mν(f(x, .))(t)|pdν(t)dλ(x)≤ C∫X∫R|f(x, t)|pdν(t)dλ(x)= C∫X×R|f(x, t)|pdµ(x, t).Remarks. (1) Observe that we have just used the strong type inequalities forMλ and Mν . In fact, if (X1, d1, λ1) and (X2, d2, λ2) are spaces for which theirmaximal operators are bounded, then Theorem 1 holds for the product spaceX1 × X2.(2) Following Zygmund, Marcinkiewicz and Jensen one can prove a strongerresult: If (X1, d1, λ1) and (X2, d2, λ2) are spaces for which their maximal oper-ators are of weak type (1, 1), for each f ∈ Lµ,loc(X1 × X2) and each λ > 0 wehaveµ({(x, t) ∈ X1 × X2 :Mµf(x, t) > λ}) ≤c∫X1×X2|f(x,t)|λ (1 + log+ |f(x,t)|λ ) dµ(x, t).See, for example, [4, p. 161].The weak type (1, 1) inequality ofMµ is a bit more delicate. We shall now givea characterization of the measures ν for which Mµ is of weak type (1, 1) whendealing with a special class of spaces of homogeneous type.Let us first observe that if ν is a finite linear combination of Dirac measures theweak type inequality holds. This easily follows from the weak type for the Hardy-Littlewood maximal operator in spaces of homogeneous type. What is importantis that the converse is true in most of the spaces of homogeneous type of interest,such as the unit sphere with the surface area measure or the euclidean space withnon-isotropic parabolic distances and Lebesgue measure.The fact that every non-atomic space of homogeneous type can be normal-ized in such a way that each ball of radius r has measure of order r, for r smallOn maximal functions 315enough, was proved in [5]. In [6] it is shown that the more restrictive normaliza-tion λ(B(x, r)) = r, can be obtained in several interesting cases containing thegeneralized parabolic distances or some regular and smooth Vitali families.The proof of Theorem 2 is an extension of the standard construction of acounterexample given in [4, p. 165].Theorem 2. Let (X, d, λ) be a space of homogeneous type normalized in sucha way that λ(B(x, r)) = r for x ∈ X and r small enough. If Mµ is of weaktype (1, 1) then ν is a finite linear combination of Dirac measures; in other words,ν =∑Ni=1 aiδri , ai > 0, ri ∈ R, N ∈ N.Proof: Let ν be a measure which is not a finite linear combination of Diracmeasures. We will show that Mµ is not of weak type (1, 1). Fix x0 ∈ X . Weconsider two cases.Case 1. Suppose first that(4) there exists z ∈ R such that ν((z, z + 1/i)) 6= 0 for every i ∈ Nand define Ψ(i) = 1/ν((z, z + 1/i)). Consider, for each N ∈ N, the setsJNi = {(x, t) ∈ X × R : z < t < z + 1/i, x ∈ B(x0,Ψ(i)Ψ(N))} i = 1, . . . , N ;EN =N⋂i=1JNi = {(x, t) ∈ X × R : z < t < z + 1/N, x ∈ B(x0,Ψ(1)Ψ(N))}.Then, by the definitions of µ and Ψ, we haveµ(JNi ) = λ(B(x0,Ψ(i)Ψ(N))) ν((z, z + 1/i)) =1Ψ(N);µ(EN ) = λ(B(x0,Ψ(1)Ψ(N))) ν((z, z + 1/N)) =Ψ(1)(Ψ(N))2.Observe thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) >CΨ(N)}for some constant C. In fact, if (x, t) ∈ JNi thenMµ(χEN )(x, t) ≥1µ(JNi )∫JNiχEN (y, s)dµ(y, s)=µ(EN )µ(JNi )=Ψ(1)Ψ(N).316 H.Aimar, L. Forzani, V.NaiboThenµ({(x, t) ∈X × R :Mµ(χEN )(x, t) >CΨ(N)}) ≥ µ(N⋃i=1JNi )=1Ψ(N)+N∑i=2Ψ(i)−Ψ(i − 1)Ψ(N)1Ψ(i),sinceN⋃i=1JNi ={(x, t) ∈ X × R : z < t < z + 1, x ∈ B(x0,Ψ(1)Ψ(N))}⋃N⋃i=2{(x, t) ∈ X × R : z < t < z + 1i , x ∈ B(x0,Ψ(i)Ψ(N))− B(x0,Ψ(i−1)Ψ(N))}.So,Ψ(N)µ({(x, t) ∈ X × R :Mµ(χEN )(x, t) >CΨ(N)}) ≥ 1 +N∑i=2(1−Ψ(i − 1)Ψ(i)).Observe that limN→+∞∏Ni=2Ψ(i−1)Ψ(i)= 0, so the above series diverges to +∞.We conclude that Mµ is not of weak type (1, 1).Observe that the same construction with the obvious changes works if insteadof condition (4) we have(5) there exists z ∈ R such that ν((z − 1/i, z)) 6= 0 for every i ∈ N.Case 2. Suppose now that neither (4) nor (5) hold. Then ν is a discrete measuresupported on an infinite set whose atoms do not accumulate; that is, there existsan infinite sequence {ri}i∈N of real numbers with no accumulation points such thatν(A) =∑ri∈Aai for every measurable subset A ⊂ R, where ν({ri}) = ai > 0. Infact, for each z ∈ R there exists iz ∈ N such that ν(z−1/iz, z+1/iz) = ν(z). Nowif K ⊂ R is compact, the set {z ∈ K : ν(z) > 0} is finite. So, we obtain a sequence{ri} with no accumulation points such that ν(ri) = ai > 0 and ν(A) =∑ri∈Aaifor every measurable subset A ⊂ R. Observe that {ri} is not finite because ν isnot a linear combination of Dirac measures. We assume, without loss of generality,that 0 < ri < ri+1.We consider two cases. (i) Assume∑i∈N ai = +∞. Define for each N ∈ NJNi = {(x, t) ∈ X × R : 0 < t < ri+1, x ∈ B(x0,1Pij=1 aj)} i = 1, . . . , N ;EN =N⋂i=1JNi = {(x, t) ∈ X × R : 0 < t < r2, x ∈ B(x0,1PNj=1 aj)}.On maximal functions 317Then, for (x, t) ∈ JNi , we have(6) Mµ(χEN )(x, t) ≥1µ(JNi )∫JNiχEN (y, s)dµ(y, s) =a1∑Nj=1 aj.So, there is a constant C > 0 such thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) >CPNj=1 aj}.Thenµ({(x, t) ∈ X × R :Mµ(χEN )(x, t) >CPNj=1 aj}) ≥ µ(N⋃i=1JNi ) =N∑i=1ai∑ij=1 aj,sinceN⋃i=1JNi =N⋃i=1{(x, t) ∈ X × R : ri−1 < t ≤ ri, x ∈ B(x0,1Pij=1 aj)}.This last series diverges since∑i∈N ai = +∞, see for example page 18 of [3].Then Mµ is not of weak type (1, 1).(ii) Assume now that∑i∈N ai < +∞ and for each N ∈ N defineJNi = {(x, t) ∈ X × R : ri−1 < t, x ∈ B(x0,Pj≥N ajPj≥i aj)} i = 1, . . . , N (r0 = 0);EN =N⋂i=1Ji ={(x, t) ∈ X × R : rN−1 < t, x ∈ B(x0,Pj≥N ajPj≥1 aj)}.As before, there exists a constant C such thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) > C∑j≥Naj}.Then we have1∑j≥N ajµ{(x, t) ∈ X ×R :Mµ(χEN )(x, t) > C∑j≥Naj}≥ c(1+N−1∑i=1ai∑j≥i aj)for some constant c. The series above diverges to +∞ so we conclude that Mµ isnot of weak type (1, 1). 318 H.Aimar, L. Forzani, V.NaiboReferences[1] Sjögren P., A remark on the maximal functions for measures in Rn, Amer. J. Math. 105(1983), 1231–1233.[2] Coifman R., Weiss G., Analyse harmonique non-commutative sur certains espaces ho-mogènes, étude de certaines intégrales singulières, Lectures Notes in Math., Vol 242, Sprin-ger-Verlag, 1971.[3] Pólya G., Szegö G., Problems and Theorems in Analysis, Volume I, Springer-Verlag, Berlin-Heidelberg-New York, 1972.[4] de Guzmán M., Real Variable Methods in Fourier Analysis, North Holland, Amsterdam,1981.[5] Maćıas R., Segovia C., Lipschitz functions on spaces of homogeneous type, Advances inMathematics 33 (1979), 257–270.[6] Aimar H., Harboure E., Iaffei B., Extensions of a theorem of Stein and Zygmund, preprint.Departamento de Matemática, Facultad de Ingenieŕıa Qúımica, UniversidadNacional del Litoral, IMAL, CONICET, Güemes 3450, 3000 Santa Fe, ArgentinaE-mail : haimar@alpha.arcride.edu.arlforzani@intec.unl.edu.arvnaibo@intec.unl.edu.ar(Received March 7, 2000)

English

On maximal functions over circular sectors with rotation invariant measures

Given a rotation invariant measure in Rn, we define the maximal operator over circular sectors. We prove that it is of strong type (p,p) for p>1 and we give necessary and sufficient conditions on the measure for the weak type (1,1) inequality. Actually we work in a more general setting containing the above and other situations.Fil: Aimar, Hugo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Forzani, Liliana Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Naibo, Virginia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentin

Aimar, Hugo Alejandro

Forzani, Liliana Maria

Naibo, Virginia

CONICET Digital

Comment.Math.Univ.Carolin. 42,2 (2001)311–318 311On maximal functions over circularsectors with rotation invariant measuresH. Aimar, L. Forzani, V. NaiboAbstract. Given a rotation invariant measure in Rn, we define the maximal operatorover circular sectors. We prove that it is of strong type (p, p) for p > 1 and we givenecessary and sufficient conditions on the measure for the weak type (1, 1) inequality.Actually we work in a more general setting containing the above and other situations.Keywords: maximal functions, spaces of homogeneous typeClassification: 42B25Let X be a topological space and µ be a Borel measure on X . By Lpµ(X)we will denote the space of all real valued measurable functions on X such that‖f‖p,µ = (∫X |f(x)|p dµ(x))1/p < ∞ if 1 ≤ p < ∞, and ‖f‖∞,µ = inf{a : µ({x ∈X : |f(x)| > a}) = 0} <∞, if p = +∞. By Lµ,loc(X) we mean the set of all realvalued measurable functions on X which are integrable on compact sets.An operator T defined on Lpµ(X) is said to be of strong type (p, p), 1 ≤p ≤ +∞, if there exists a positive constant Ap such that for every f ∈ Lpµ(X),‖Tf‖p,µ ≤ Ap‖f‖p,µ. The operator T is of weak type (p, p), 1 ≤ p < +∞, ifthere exists a constant Bp such that for every f ∈ Lpµ(X) and λ > 0, µ({x ∈ X :|Tf(x)| > λ}) ≤ Bp‖f‖p,µλ−p.Let y ∈ Sn−1, the unit sphere in Rn, α ∈ (0, pi) and 0 ≤ r ≤ R ≤ +∞. Theset S = S(y, α, r, R) = {x ∈ Rn : arg(x, y) < α and r < |x| < R}, where arg(x, y)is the angle between x and y, is called a circular sector in Rn. If G is a Borelsubset of Sn−1, G(r,R) will denote the set {ρx : x ∈ G, r < ρ < R} and σr willmean the surface measure on the sphere of radius r.Let µ be a non-negative rotation invariant Radon measure. If f ∈ Lµ,loc(Rn)and x 6= 0 we define(1) Msµf(x) = supx∈S1µ(S)∫S|f(y)|dµ(y)Supported by CONICET and Universidad Nacional del Litoral.312 H.Aimar, L. Forzani, V.Naibowhere the supremum is taken over all bounded sectors S in Rn with non-zeromeasure containing x. Set Msµf(0) = 0.It is not hard to prove that if G is a Borel set in Sn−1, 0 ≤ r ≤ R ≤ +∞ andf ∈ L1µ(Rn) then∫G(r,R)f(y)dµ(y) =1σ1(Sn−1)∫G∫ Rrf(ρy′)dν(ρ)dσ1(y′)=1σ1(Sn−1)∫ Rr∫Gf(ρy′)dσ1(y′)dν(ρ),where ν is the measure in R+ defined by ν((a, b)) = µ(Sn−1(a, b)), with the abovenotation and G = Sn−1, for 0 ≤ a ≤ b ≤ +∞. So, the average of f over circularsectors is obtained by considering averages over the intervals in R, with respectto ν, and over the “balls” in Sn−1, with respect to the surface measure σ1. Thisfact allows us dominate Msµf by an iteration of two maximal functions: one in Rwith respect to an abstract measure ν, and the other in Sn−1 with respect to its“doubling” usual measure σ1.This situation is analogous to the classical one of the maximal operator overrectangles with sides parallel to the coordinate axes, with respect to the Lebesguemeasure. In this case the strong type inequality holds for p > 1 but the weaktype (1, 1) inequality is no longer true (see [4]).We are interested in the boundedness of Msµ. The strong type (p, p) for p > 1will be obtained as a consequence of the strong type inequality for the maximalfunctions in (Sn−1, σ1) and in (R+, ν). Our main result is a characterization ofthe measures ν for which the weak type (1,1) inequality holds. We shall work in amore general setting that includes the above situations. The basic result of thesenotes concerning Msµ will be the following corollary of Theorems 1 and 2 below.Given a non-negative rotation invariant Radon measure µ, we have(a) Msµ is of strong type (p, p) for 1 < p ≤ ∞;(b) Msµ is of weak type (1, 1) if and only if µ is a finite linear combination ofsurface measures; i.e., µ =∑Ni=1 aiσri , with ai, ri > 0, N ∈ N.Let X be a set; a nonnegative symmetric function d on X × X is called aquasi-distance if there exits a constant k such thatd(x, y) ≤ k [d(x, z) + d(z, y)]for every x, y, x ∈ X , and d(x, y) = 0 if and only if x = y.The balls B(x, r) = {y : d(x, y) < r} form a basis of neighborhoods of x forthe topology induced by d.On maximal functions 313Consider (X, d, λ), where d is a quasi-distance on X and λ a non-negativemeasure defined on a σ-algebra of subsets of X which contains the balls. Forf ∈ Lλ,loc(X) we define the Hardy-Littlewood maximal function operatorMλf(x) = supx∈B1λ(B)∫B|f(x)|dλ(x),where B is a ball containing x with positive measure.We shall say that (X, d, λ) is a space of homogeneous type if there exists aconstant A such that(2) 0 < λ(B(x, 2r)) ≤ Aλ(B(x, r)) <∞holds for every x ∈ X and r > 0.For these spaces, the weak type (1, 1) and the strong type (p, p) for p > 1 ofMλ hold (see [2]).Let (X, d, λ) be a space of homogeneous type and ν a Lebesgue-Stieltjes mea-sure on R. Consider the product space X×R endowed with the product measureµ = λ× ν. For f ∈ Lµ,loc(X × R) we consider the maximal function(3) Mµf(x, t) = sup(x,t)∈S1µ(S)∫S|f(y, s)|dµ(y, s)where the supremum is taken over all the cylinders S = B× (r,R) in X ×R withpositive measure containing (x, t), B being a ball in X .Theorem 1. Mµ is of strong type (p, p) for p > 1.Proof: Since the case p = +∞ is trivial we consider 1 < p < ∞. By Tonellitheorem we have thatMµf(x, t) = sup(x,t)∈S1µ(S)∫S|f(y, s)|dµ(y, s)= sup(x,t)∈S1λ(B)∫B(1ν((r,R))∫ Rr|f(y, s)|dν(s))dλ(y)≤ sup(x,t)∈S1λ(B)∫BMν(f(y, .))(t)dλ(y)≤Mλ[(Mν(f(., .)))(t)](x).The maximal function Mλ is Lpλ(X)-bounded because (X, d, λ) is a space ofhomogeneous type and, even without any doubling condition for ν, the maxi-mal function Mν is Lpν(R)-bounded because the basic space is one-dimensional(see [1]). Then314 H.Aimar, L. Forzani, V.Naibo∫X×R|Mµf(x, t)|pdµ(x, t) ≤∫X×R|Mλ[(Mν(f(., .)))(t)](x)|pdµ(x, t)=∫R∫X|Mλ[(Mν(f(., .)))(t)](x)|pdλ(x)dν(t)≤ C∫R∫X|Mν(f(x, .))(t)|pdλ(x)dν(t)= C∫X∫R|Mν(f(x, .))(t)|pdν(t)dλ(x)≤ C∫X∫R|f(x, t)|pdν(t)dλ(x)= C∫X×R|f(x, t)|pdµ(x, t).Remarks. (1) Observe that we have just used the strong type inequalities forMλ and Mν . In fact, if (X1, d1, λ1) and (X2, d2, λ2) are spaces for which theirmaximal operators are bounded, then Theorem 1 holds for the product spaceX1 ×X2.(2) Following Zygmund, Marcinkiewicz and Jensen one can prove a strongerresult: If (X1, d1, λ1) and (X2, d2, λ2) are spaces for which their maximal oper-ators are of weak type (1, 1), for each f ∈ Lµ,loc(X1 × X2) and each λ > 0 wehaveµ({(x, t) ∈ X1 ×X2 : Mµf(x, t) > λ}) ≤c∫X1×X2|f(x,t)|λ (1 + log+ |f(x,t)|λ ) dµ(x, t).See, for example, [4, p. 161].The weak type (1, 1) inequality ofMµ is a bit more delicate. We shall now givea characterization of the measures ν for which Mµ is of weak type (1, 1) whendealing with a special class of spaces of homogeneous type.Let us first observe that if ν is a finite linear combination of Dirac measures theweak type inequality holds. This easily follows from the weak type for the Hardy-Littlewood maximal operator in spaces of homogeneous type. What is importantis that the converse is true in most of the spaces of homogeneous type of interest,such as the unit sphere with the surface area measure or the euclidean space withnon-isotropic parabolic distances and Lebesgue measure.The fact that every non-atomic space of homogeneous type can be normal-ized in such a way that each ball of radius r has measure of order r, for r smallOn maximal functions 315enough, was proved in [5]. In [6] it is shown that the more restrictive normaliza-tion λ(B(x, r)) = r, can be obtained in several interesting cases containing thegeneralized parabolic distances or some regular and smooth Vitali families.The proof of Theorem 2 is an extension of the standard construction of acounterexample given in [4, p. 165].Theorem 2. Let (X, d, λ) be a space of homogeneous type normalized in sucha way that λ(B(x, r)) = r for x ∈ X and r small enough. If Mµ is of weaktype (1, 1) then ν is a finite linear combination of Dirac measures; in other words,ν =∑Ni=1 aiδri , ai > 0, ri ∈ R, N ∈ N.Proof: Let ν be a measure which is not a finite linear combination of Diracmeasures. We will show that Mµ is not of weak type (1, 1). Fix x0 ∈ X . Weconsider two cases.Case 1. Suppose first that(4) there exists z ∈ R such that ν((z, z + 1/i)) 6= 0 for every i ∈ Nand define Ψ(i) = 1/ν((z, z + 1/i)). Consider, for each N ∈ N, the setsJNi = {(x, t) ∈ X × R : z < t < z + 1/i, x ∈ B(x0,Ψ(i)Ψ(N))} i = 1, . . . , N ;EN =N⋂i=1JNi = {(x, t) ∈ X × R : z < t < z + 1/N, x ∈ B(x0,Ψ(1)Ψ(N))}.Then, by the definitions of µ and Ψ, we haveµ(JNi ) = λ(B(x0,Ψ(i)Ψ(N))) ν((z, z + 1/i)) =1Ψ(N);µ(EN ) = λ(B(x0,Ψ(1)Ψ(N))) ν((z, z + 1/N)) =Ψ(1)(Ψ(N))2.Observe thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) >CΨ(N)}for some constant C. In fact, if (x, t) ∈ JNi thenMµ(χEN )(x, t) ≥1µ(JNi )∫JNiχEN (y, s)dµ(y, s)=µ(EN )µ(JNi )=Ψ(1)Ψ(N).316 H.Aimar, L. Forzani, V.NaiboThenµ({(x, t) ∈X × R :Mµ(χEN )(x, t) >CΨ(N)}) ≥ µ(N⋃i=1JNi )=1Ψ(N)+N∑i=2Ψ(i)−Ψ(i− 1)Ψ(N)1Ψ(i),sinceN⋃i=1JNi ={(x, t) ∈ X × R : z < t < z + 1, x ∈ B(x0,Ψ(1)Ψ(N))}⋃ N⋃i=2{(x, t) ∈ X × R : z < t < z + 1i , x ∈ B(x0,Ψ(i)Ψ(N))−B(x0,Ψ(i−1)Ψ(N))}.So,Ψ(N)µ({(x, t) ∈ X × R :Mµ(χEN )(x, t) >CΨ(N)}) ≥ 1 +N∑i=2(1−Ψ(i− 1)Ψ(i)).Observe that limN→+∞∏Ni=2Ψ(i−1)Ψ(i)= 0, so the above series diverges to +∞.We conclude that Mµ is not of weak type (1, 1).Observe that the same construction with the obvious changes works if insteadof condition (4) we have(5) there exists z ∈ R such that ν((z − 1/i, z)) 6= 0 for every i ∈ N.Case 2. Suppose now that neither (4) nor (5) hold. Then ν is a discrete measuresupported on an infinite set whose atoms do not accumulate; that is, there existsan infinite sequence {ri}i∈N of real numbers with no accumulation points such thatν(A) =∑ri∈Aai for every measurable subset A ⊂ R, where ν({ri}) = ai > 0. Infact, for each z ∈ R there exists iz ∈ N such that ν(z−1/iz, z+1/iz) = ν(z). Nowif K ⊂ R is compact, the set {z ∈ K : ν(z) > 0} is finite. So, we obtain a sequence{ri} with no accumulation points such that ν(ri) = ai > 0 and ν(A) =∑ri∈Aaifor every measurable subset A ⊂ R. Observe that {ri} is not finite because ν isnot a linear combination of Dirac measures. We assume, without loss of generality,that 0 < ri < ri+1.We consider two cases. (i) Assume∑i∈N ai = +∞. Define for each N ∈ NJNi = {(x, t) ∈ X × R : 0 < t < ri+1, x ∈ B(x0,1Pij=1 aj)} i = 1, . . . , N ;EN =N⋂i=1JNi = {(x, t) ∈ X × R : 0 < t < r2, x ∈ B(x0,1PNj=1 aj)}.On maximal functions 317Then, for (x, t) ∈ JNi , we have(6) Mµ(χEN )(x, t) ≥1µ(JNi )∫JNiχEN (y, s)dµ(y, s) =a1∑Nj=1 aj.So, there is a constant C > 0 such thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) >CPNj=1 aj}.Thenµ({(x, t) ∈ X × R : Mµ(χEN )(x, t) >CPNj=1 aj}) ≥ µ(N⋃i=1JNi ) =N∑i=1ai∑ij=1 aj,sinceN⋃i=1JNi =N⋃i=1{(x, t) ∈ X × R : ri−1 < t ≤ ri, x ∈ B(x0,1Pij=1 aj)}.This last series diverges since∑i∈N ai = +∞, see for example page 18 of [3].Then Mµ is not of weak type (1, 1).(ii) Assume now that∑i∈N ai < +∞ and for each N ∈ N defineJNi = {(x, t) ∈ X × R : ri−1 < t, x ∈ B(x0,Pj≥N ajPj≥i aj)} i = 1, . . . , N (r0 = 0);EN =N⋂i=1Ji ={(x, t) ∈ X × R : rN−1 < t, x ∈ B(x0,Pj≥N ajPj≥1 aj)}.As before, there exists a constant C such thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) > C∑j≥Naj}.Then we have1∑j≥N ajµ{(x, t) ∈ X ×R :Mµ(χEN )(x, t) > C∑j≥Naj}≥ c(1+N−1∑i=1ai∑j≥i aj)for some constant c. The series above diverges to +∞ so we conclude that Mµ isnot of weak type (1, 1). 318 H.Aimar, L. Forzani, V.NaiboReferences[1] Sjo¨gren P., A remark on the maximal functions for measures in Rn, Amer. J. Math. 105(1983), 1231–1233.[2] Coifman R., Weiss G., Analyse harmonique non-commutative sur certains espaces ho-moge`nes, e´tude de certaines inte´grales singulie`res, Lectures Notes in Math., Vol 242, Sprin-ger-Verlag, 1971.[3] Po´lya G., Szego¨ G., Problems and Theorems in Analysis, Volume I, Springer-Verlag, Berlin-Heidelberg-New York, 1972.[4] de Guzma´n M., Real Variable Methods in Fourier Analysis, North Holland, Amsterdam,1981.[5] Mac´ıas R., Segovia C., Lipschitz functions on spaces of homogeneous type, Advances inMathematics 33 (1979), 257–270.[6] Aimar H., Harboure E., Iaffei B., Extensions of a theorem of Stein and Zygmund, preprint.Departamento de Matema´tica, Facultad de Ingenier´ıa Qu´ımica, UniversidadNacional del Litoral, IMAL, CONICET, Gu¨emes 3450, 3000 Santa Fe, ArgentinaE-mail : haimar@alpha.arcride.edu.arlforzani@intec.unl.edu.arvnaibo@intec.unl.edu.ar(Received March 7, 2000)

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Comment.Math.Univ.Carolin. 42,2 (2001)311–318 311On maximal functions over circularsectors with rotation invariant measuresH. Aimar, L. Forzani, V. NaiboAbstract. Given a rotation invariant measure in Rn, we define the maximal operatorover circular sectors. We prove that it is of strong type (p, p) for p > 1 and we givenecessary and sufficient conditions on the measure for the weak type (1, 1) inequality.Actually we work in a more general setting containing the above and other situations.Keywords: maximal functions, spaces of homogeneous typeClassification: 42B25Let X be a topological space and µ be a Borel measure on X . By Lpµ(X)we will denote the space of all real valued measurable functions on X such that‖f‖p,µ = (∫X |f(x)|p dµ(x))1/p < ∞ if 1 ≤ p < ∞, and ‖f‖∞,µ = inf{a : µ({x ∈X : |f(x)| > a}) = 0} < ∞, if p = +∞. By Lµ,loc(X) we mean the set of all realvalued measurable functions on X which are integrable on compact sets.An operator T defined on Lpµ(X) is said to be of strong type (p, p), 1 ≤p ≤ +∞, if there exists a positive constant Ap such that for every f ∈ Lpµ(X),‖Tf‖p,µ ≤ Ap‖f‖p,µ. The operator T is of weak type (p, p), 1 ≤ p < +∞, ifthere exists a constant Bp such that for every f ∈ Lpµ(X) and λ > 0, µ({x ∈ X :|Tf(x)| > λ}) ≤ Bp‖f‖p,µλ−p.Let y ∈ Sn−1, the unit sphere in Rn, α ∈ (0, π) and 0 ≤ r ≤ R ≤ +∞. Theset S = S(y, α, r, R) = {x ∈ Rn : arg(x, y) < α and r < |x| < R}, where arg(x, y)is the angle between x and y, is called a circular sector in Rn. If G is a Borelsubset of Sn−1, G(r, R) will denote the set {ρx : x ∈ G, r < ρ < R} and σr willmean the surface measure on the sphere of radius r.Let µ be a non-negative rotation invariant Radon measure. If f ∈ Lµ,loc(Rn)and x 6= 0 we define(1) Msµf(x) = supx∈S1µ(S)∫S|f(y)|dµ(y)Supported by CONICET and Universidad Nacional del Litoral.312 H.Aimar, L. Forzani, V.Naibowhere the supremum is taken over all bounded sectors S in Rn with non-zeromeasure containing x. Set Msµf(0) = 0.It is not hard to prove that if G is a Borel set in Sn−1, 0 ≤ r ≤ R ≤ +∞ andf ∈ L1µ(Rn) then∫G(r,R)f(y)dµ(y) =1σ1(Sn−1)∫G∫ Rrf(ρy′)dν(ρ)dσ1(y′)=1σ1(Sn−1)∫ Rr∫Gf(ρy′)dσ1(y′)dν(ρ),where ν is the measure in R+ defined by ν((a, b)) = µ(Sn−1(a, b)), with the abovenotation and G = Sn−1, for 0 ≤ a ≤ b ≤ +∞. So, the average of f over circularsectors is obtained by considering averages over the intervals in R, with respectto ν, and over the “balls” in Sn−1, with respect to the surface measure σ1. Thisfact allows us dominate Msµf by an iteration of two maximal functions: one in Rwith respect to an abstract measure ν, and the other in Sn−1 with respect to its“doubling” usual measure σ1.This situation is analogous to the classical one of the maximal operator overrectangles with sides parallel to the coordinate axes, with respect to the Lebesguemeasure. In this case the strong type inequality holds for p > 1 but the weaktype (1, 1) inequality is no longer true (see [4]).We are interested in the boundedness of Msµ. The strong type (p, p) for p > 1will be obtained as a consequence of the strong type inequality for the maximalfunctions in (Sn−1, σ1) and in (R+, ν). Our main result is a characterization ofthe measures ν for which the weak type (1,1) inequality holds. We shall work in amore general setting that includes the above situations. The basic result of thesenotes concerning Msµ will be the following corollary of Theorems 1 and 2 below.Given a non-negative rotation invariant Radon measure µ, we have(a) Msµ is of strong type (p, p) for 1 < p ≤ ∞;(b) Msµ is of weak type (1, 1) if and only if µ is a finite linear combination ofsurface measures; i.e., µ =∑Ni=1 aiσri , with ai, ri > 0, N ∈ N.Let X be a set; a nonnegative symmetric function d on X × X is called aquasi-distance if there exits a constant k such thatd(x, y) ≤ k [d(x, z) + d(z, y)]for every x, y, x ∈ X , and d(x, y) = 0 if and only if x = y.The balls B(x, r) = {y : d(x, y) < r} form a basis of neighborhoods of x forthe topology induced by d.On maximal functions 313Consider (X, d, λ), where d is a quasi-distance on X and λ a non-negativemeasure defined on a σ-algebra of subsets of X which contains the balls. Forf ∈ Lλ,loc(X) we define the Hardy-Littlewood maximal function operatorMλf(x) = supx∈B1λ(B)∫B|f(x)|dλ(x),where B is a ball containing x with positive measure.We shall say that (X, d, λ) is a space of homogeneous type if there exists aconstant A such that(2) 0 < λ(B(x, 2r)) ≤ Aλ(B(x, r)) < ∞holds for every x ∈ X and r > 0.For these spaces, the weak type (1, 1) and the strong type (p, p) for p > 1 ofMλ hold (see [2]).Let (X, d, λ) be a space of homogeneous type and ν a Lebesgue-Stieltjes mea-sure on R. Consider the product space X ×R endowed with the product measureµ = λ × ν. For f ∈ Lµ,loc(X × R) we consider the maximal function(3) Mµf(x, t) = sup(x,t)∈S1µ(S)∫S|f(y, s)|dµ(y, s)where the supremum is taken over all the cylinders S = B × (r, R) in X ×R withpositive measure containing (x, t), B being a ball in X .Theorem 1. Mµ is of strong type (p, p) for p > 1.Proof: Since the case p = +∞ is trivial we consider 1 < p < ∞. By Tonellitheorem we have thatMµf(x, t) = sup(x,t)∈S1µ(S)∫S|f(y, s)|dµ(y, s)= sup(x,t)∈S1λ(B)∫B(1ν((r, R))∫ Rr|f(y, s)|dν(s))dλ(y)≤ sup(x,t)∈S1λ(B)∫BMν(f(y, .))(t)dλ(y)≤ Mλ[(Mν(f(., .)))(t)](x).The maximal function Mλ is Lpλ(X)-bounded because (X, d, λ) is a space ofhomogeneous type and, even without any doubling condition for ν, the maxi-mal function Mν is Lpν(R)-bounded because the basic space is one-dimensional(see [1]). Then314 H.Aimar, L. Forzani, V.Naibo∫X×R|Mµf(x, t)|pdµ(x, t) ≤∫X×R|Mλ[(Mν(f(., .)))(t)](x)|pdµ(x, t)=∫R∫X|Mλ[(Mν(f(., .)))(t)](x)|pdλ(x)dν(t)≤ C∫R∫X|Mν(f(x, .))(t)|pdλ(x)dν(t)= C∫X∫R|Mν(f(x, .))(t)|pdν(t)dλ(x)≤ C∫X∫R|f(x, t)|pdν(t)dλ(x)= C∫X×R|f(x, t)|pdµ(x, t).Remarks. (1) Observe that we have just used the strong type inequalities forMλ and Mν . In fact, if (X1, d1, λ1) and (X2, d2, λ2) are spaces for which theirmaximal operators are bounded, then Theorem 1 holds for the product spaceX1 × X2.(2) Following Zygmund, Marcinkiewicz and Jensen one can prove a strongerresult: If (X1, d1, λ1) and (X2, d2, λ2) are spaces for which their maximal oper-ators are of weak type (1, 1), for each f ∈ Lµ,loc(X1 × X2) and each λ > 0 wehaveµ({(x, t) ∈ X1 × X2 :Mµf(x, t) > λ}) ≤c∫X1×X2|f(x,t)|λ (1 + log+ |f(x,t)|λ ) dµ(x, t).See, for example, [4, p. 161].The weak type (1, 1) inequality ofMµ is a bit more delicate. We shall now givea characterization of the measures ν for which Mµ is of weak type (1, 1) whendealing with a special class of spaces of homogeneous type.Let us first observe that if ν is a finite linear combination of Dirac measures theweak type inequality holds. This easily follows from the weak type for the Hardy-Littlewood maximal operator in spaces of homogeneous type. What is importantis that the converse is true in most of the spaces of homogeneous type of interest,such as the unit sphere with the surface area measure or the euclidean space withnon-isotropic parabolic distances and Lebesgue measure.The fact that every non-atomic space of homogeneous type can be normal-ized in such a way that each ball of radius r has measure of order r, for r smallOn maximal functions 315enough, was proved in [5]. In [6] it is shown that the more restrictive normaliza-tion λ(B(x, r)) = r, can be obtained in several interesting cases containing thegeneralized parabolic distances or some regular and smooth Vitali families.The proof of Theorem 2 is an extension of the standard construction of acounterexample given in [4, p. 165].Theorem 2. Let (X, d, λ) be a space of homogeneous type normalized in sucha way that λ(B(x, r)) = r for x ∈ X and r small enough. If Mµ is of weaktype (1, 1) then ν is a finite linear combination of Dirac measures; in other words,ν =∑Ni=1 aiδri , ai > 0, ri ∈ R, N ∈ N.Proof: Let ν be a measure which is not a finite linear combination of Diracmeasures. We will show that Mµ is not of weak type (1, 1). Fix x0 ∈ X . Weconsider two cases.Case 1. Suppose first that(4) there exists z ∈ R such that ν((z, z + 1/i)) 6= 0 for every i ∈ Nand define Ψ(i) = 1/ν((z, z + 1/i)). Consider, for each N ∈ N, the setsJNi = {(x, t) ∈ X × R : z < t < z + 1/i, x ∈ B(x0,Ψ(i)Ψ(N))} i = 1, . . . , N ;EN =N⋂i=1JNi = {(x, t) ∈ X × R : z < t < z + 1/N, x ∈ B(x0,Ψ(1)Ψ(N))}.Then, by the definitions of µ and Ψ, we haveµ(JNi ) = λ(B(x0,Ψ(i)Ψ(N))) ν((z, z + 1/i)) =1Ψ(N);µ(EN ) = λ(B(x0,Ψ(1)Ψ(N))) ν((z, z + 1/N)) =Ψ(1)(Ψ(N))2.Observe thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) >CΨ(N)}for some constant C. In fact, if (x, t) ∈ JNi thenMµ(χEN )(x, t) ≥1µ(JNi )∫JNiχEN (y, s)dµ(y, s)=µ(EN )µ(JNi )=Ψ(1)Ψ(N).316 H.Aimar, L. Forzani, V.NaiboThenµ({(x, t) ∈X × R :Mµ(χEN )(x, t) >CΨ(N)}) ≥ µ(N⋃i=1JNi )=1Ψ(N)+N∑i=2Ψ(i)−Ψ(i − 1)Ψ(N)1Ψ(i),sinceN⋃i=1JNi ={(x, t) ∈ X × R : z < t < z + 1, x ∈ B(x0,Ψ(1)Ψ(N))}⋃N⋃i=2{(x, t) ∈ X × R : z < t < z + 1i , x ∈ B(x0,Ψ(i)Ψ(N))− B(x0,Ψ(i−1)Ψ(N))}.So,Ψ(N)µ({(x, t) ∈ X × R :Mµ(χEN )(x, t) >CΨ(N)}) ≥ 1 +N∑i=2(1−Ψ(i − 1)Ψ(i)).Observe that limN→+∞∏Ni=2Ψ(i−1)Ψ(i)= 0, so the above series diverges to +∞.We conclude that Mµ is not of weak type (1, 1).Observe that the same construction with the obvious changes works if insteadof condition (4) we have(5) there exists z ∈ R such that ν((z − 1/i, z)) 6= 0 for every i ∈ N.Case 2. Suppose now that neither (4) nor (5) hold. Then ν is a discrete measuresupported on an infinite set whose atoms do not accumulate; that is, there existsan infinite sequence {ri}i∈N of real numbers with no accumulation points such thatν(A) =∑ri∈Aai for every measurable subset A ⊂ R, where ν({ri}) = ai > 0. Infact, for each z ∈ R there exists iz ∈ N such that ν(z−1/iz, z+1/iz) = ν(z). Nowif K ⊂ R is compact, the set {z ∈ K : ν(z) > 0} is finite. So, we obtain a sequence{ri} with no accumulation points such that ν(ri) = ai > 0 and ν(A) =∑ri∈Aaifor every measurable subset A ⊂ R. Observe that {ri} is not finite because ν isnot a linear combination of Dirac measures. We assume, without loss of generality,that 0 < ri < ri+1.We consider two cases. (i) Assume∑i∈N ai = +∞. Define for each N ∈ NJNi = {(x, t) ∈ X × R : 0 < t < ri+1, x ∈ B(x0,1Pij=1 aj)} i = 1, . . . , N ;EN =N⋂i=1JNi = {(x, t) ∈ X × R : 0 < t < r2, x ∈ B(x0,1PNj=1 aj)}.On maximal functions 317Then, for (x, t) ∈ JNi , we have(6) Mµ(χEN )(x, t) ≥1µ(JNi )∫JNiχEN (y, s)dµ(y, s) =a1∑Nj=1 aj.So, there is a constant C > 0 such thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) >CPNj=1 aj}.Thenµ({(x, t) ∈ X × R :Mµ(χEN )(x, t) >CPNj=1 aj}) ≥ µ(N⋃i=1JNi ) =N∑i=1ai∑ij=1 aj,sinceN⋃i=1JNi =N⋃i=1{(x, t) ∈ X × R : ri−1 < t ≤ ri, x ∈ B(x0,1Pij=1 aj)}.This last series diverges since∑i∈N ai = +∞, see for example page 18 of [3].Then Mµ is not of weak type (1, 1).(ii) Assume now that∑i∈N ai < +∞ and for each N ∈ N defineJNi = {(x, t) ∈ X × R : ri−1 < t, x ∈ B(x0,Pj≥N ajPj≥i aj)} i = 1, . . . , N (r0 = 0);EN =N⋂i=1Ji ={(x, t) ∈ X × R : rN−1 < t, x ∈ B(x0,Pj≥N ajPj≥1 aj)}.As before, there exists a constant C such thatN⋃i=1JNi ⊂{(x, t) ∈ X × R :Mµ(χEN )(x, t) > C∑j≥Naj}.Then we have1∑j≥N ajµ{(x, t) ∈ X ×R :Mµ(χEN )(x, t) > C∑j≥Naj}≥ c(1+N−1∑i=1ai∑j≥i aj)for some constant c. The series above diverges to +∞ so we conclude that Mµ isnot of weak type (1, 1). 318 H.Aimar, L. Forzani, V.NaiboReferences[1] Sjögren P., A remark on the maximal functions for measures in Rn, Amer. J. Math. 105(1983), 1231–1233.[2] Coifman R., Weiss G., Analyse harmonique non-commutative sur certains espaces ho-mogènes, étude de certaines intégrales singulières, Lectures Notes in Math., Vol 242, Sprin-ger-Verlag, 1971.[3] Pólya G., Szegö G., Problems and Theorems in Analysis, Volume I, Springer-Verlag, Berlin-Heidelberg-New York, 1972.[4] de Guzmán M., Real Variable Methods in Fourier Analysis, North Holland, Amsterdam,1981.[5] Maćıas R., Segovia C., Lipschitz functions on spaces of homogeneous type, Advances inMathematics 33 (1979), 257–270.[6] Aimar H., Harboure E., Iaffei B., Extensions of a theorem of Stein and Zygmund, preprint.Departamento de Matemática, Facultad de Ingenieŕıa Qúımica, UniversidadNacional del Litoral, IMAL, CONICET, Güemes 3450, 3000 Santa Fe, ArgentinaE-mail : haimar@alpha.arcride.edu.arlforzani@intec.unl.edu.arvnaibo@intec.unl.edu.ar(Received March 7, 2000)

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On maximal functions over circular sectors with rotation invariant measures

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