Maximal estimates for a generalized spherical mean Radon transform acting on radial functions
MMAXIMAL ESTIMATES FORA GENERALIZED SPHERICAL MEAN RADON TRANSFORMACTING ON RADIAL FUNCTIONS ´OSCAR CIAURRI, ADAM NOWAK, AND LUZ RONCAL
Abstract.
We study a generalized spherical means operator, viz. generalized spherical meanRadon transform, acting on radial functions. As the main results, we find conditions for theassociated maximal operator and its local variant to be bounded on power weighted Lebesguespaces. This translates, in particular, into almost everywhere convergence to radial initial dataresults for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and waveequations. Moreover, our results shed some new light on the interesting and important questionof optimality of the yet known L p boundedness results for the maximal operator in the generalnon-radial case. It appears that these could still be notably improved, as indicated by ourconjecture of the ultimate sharp result. Preliminaries and statement of results
This paper is a natural continuation of our recent research from [2]. We study a generalizedspherical means operator acting on radial functions. In [2] we viewed this operator as a familyof integral transforms { M α,βt : t > } acting on profile functions on R + and found fairly preciseestimates of the associated integral kernels K α,βt ( x, z ). This enabled us to prove two-weight L p − L q ( L rt ) estimates for f (cid:55)→ M α,βt f , with 1 ≤ p, q ≤ ∞ and 1 ≤ r < ∞ . In the present workwe focus on the more subtle limiting case r = ∞ and restrict to p = q , which in the abovelanguage of mixed norm estimates corresponds to weighted L p -boundedness of the maximal operator f (cid:55)→ sup t> | M α,βt f | . Obtaining such results requires a different, in fact more tricky, approach fromthat used in [2]. It is worth emphasizing that both our works, [2] and this one, were to large extentmotivated by connections of the generalized spherical means with solutions to a number of classicalinitial-value PDE problems being of physical and practical importance; see e.g. [2, Section 7] andreferences given there.Let n ≥ M β f ( x, t ) = F − (cid:0) m β ( t | · | ) F f (cid:1) ( x ) , where F is the Fourier transform in R n and the radial multiplier is given via m β ( s ) = 2 β + n/ − Γ( β + n/ J β + n/ − ( s ) s β + n/ − , s > , Mathematics Subject Classification.
Primary: 44A12; Secondary: 42B37, 35L15, 35B07, 35L05, 35Q05.
Key words and phrases.
Spherical Radon transform, spherical mean, maximal operator, radial function, weightedestimate, wave equation, Euler-Poisson-Darboux equation, axially symmetric solution, convergence to initial data.The first-named author was supported by the grant PGC2018-096504-B-C32 from Spanish Government. Thesecond-named author was supported by the National Science Centre of Poland within the research project OPUS2017/27/B/ST1/01623. The third-named author was supported by the Basque Government through BERC 2018–2021 program, by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excel-lence accreditation SEV-2017-2018 and through the project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE)and acronym “HAQMEC”, and by 2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundation.The Foundation accepts no responsibility for the opinions, statements and contents included in the project and/orthe results thereof, which are entirely the responsibility of the authors. a r X i v : . [ m a t h . C A ] N ov ´O. CIAURRI, A. NOWAK, AND L. RONCAL with J ν denoting the Bessel function of the first kind and order ν . The parameter β can, in general,be a complex number excluding β = − n/ , − n/ − , − n/ − , . . . . For β = 0 one recovers theclassical spherical means M f ( x, t ) = (cid:90) S n − f ( x − ty ) dσ ( y ) , ( x, t ) ∈ R n × R + , where dσ is the normalized uniform measure on the unit sphere S n − ⊂ R n . Clearly, M f ( x, t )returns the mean value of f on the sphere centered at x and of radius t .For the maximal operator M β ∗ f = sup t> | M β f ( · , t ) | Stein [8] proved the following.
Theorem 1.1 ([8]) . Let n ≥ . Then M β ∗ is bounded on L p ( R n ) provided that (1.1) 1 < p ≤ and β > − n + np , or p > and β > − np . This result was enhanced in the sense of admitted parameters and dimensions by subsequentauthors: Bourgain [1], Mockenhoupt, Seeger and Sogge [7] and recently by Miao, Yang and Zheng[6], see the historical comments in [6, p. 4272]. All these refinements can be stated altogether asfollows, cf. [6, Theorem 1.1].
Theorem 1.2 ([6]) . Let n ≥ . Then M β ∗ is bounded on L p ( R n ) provided that (1.2) 2 < p ≤ n + 2 n − and β > − n − n p , or p > n + 2 n − and β > − np . The range of β in Theorem 1.2, for p >
2, is strictly wider than in Theorem 1.1; see Figure 1below. However, according to our best knowledge, it is not known whether it is already optimal.We strongly believe it is not, see our Conjecture 1.9 below. We remark that both Theorems 1.1and 1.2 were originally proved for complex β , but for our purposes it is enough to state them forreal values of the parameter.A restriction of M β to radially symmetric functions is still of interest and, moreover, admits amore explicit finer analysis that potentially leads to more general or stronger theorems. In thisconnection we invoke two quite recent results of Duoandikoetxea, Moyua and Oruetxebarria [4, 5].The first of them is a characterization of weighted L p rad -boundedness of M ∗ with radial powerweights involved; here and elsewhere the subscript “rad” indicates the subspace of radial functions. Theorem 1.3 ([4]) . Let n ≥ . The operator M ∗ is bounded on L p rad ( R n , | x | γ dx ) for < p < ∞ (for < p < ∞ in case n = 2 ) if and only if − n ≤ γ < p ( n − − n. Another result in this spirit provides sufficient conditions for weighted L p rad -boundedness of M (3 − n ) / ∗ , again with radial power weights involved. Theorem 1.4 ([5]) . Let n ≥ and < p < ∞ . Then the operator M (3 − n ) / ∗ is bounded on L p rad ( R n , | x | γ dx ) provided that n − p + 1 − n < γ < n + 12 p − n, where in case n = 2 the lower bound for γ should be replaced by − / , and with the first inequalityweakened for odd dimensions n . Here and elsewhere by weakening a strict inequality we mean replacing “ < ” by “ ≤ ”. We take thisopportunity to note that the statement of [5, Theorem 1.1] (an unweighted specification of Theorem1.4) contains a small error, the upper bound for p in even dimensions higher than 3 should be givenby strict inequality, cf. [5, Lemma 4.3]. This problem affects the abstract of [5] as well. The radialimprovement for the unweighted estimates occurs only in odd dimensions higher than 3.We omit here discussion of more sophisticated mapping properties of M β ∗ , like boundedness from L to weak L , or from the Lorentz space L , to weak L (which correspond to weak and restricted AXIMAL ESTIMATES FOR SPHERICAL MEANS 3 weak type estimates for the maximal operator, respectively). For this kind of results, especially inthe radial case, we refer to [4, 5] and references given there.The maximal operator M α,β ∗ we shall study generalizes the restriction of M β ∗ to radial functions,since, in a sense, it covers a continuous range of dimensions n = 2 α + 2, α > −
1. Our main result,Theorem 1.5 below, contains Theorems 1.3 and 1.4 as special cases; in particular, we deliver a partlyalternative and seemingly simpler proof of Theorem 1.4. Moreover, our more general perspectivesheds new light on the discrepancy between n = 2 and higher dimensions in Theorems 1.3 and 1.4,as well as on the discrepancy between odd and even dimensions in Theorem 1.4. Finally, we gainsome intuition that enables us to conjecture an optimal result in the spirit of Theorems 1.1 and 1.2,see Conjecture 1.9 below.Let α > − α + β > − /
2. For each t > M α,βt f ( x ) = (cid:90) ∞ K α,βt ( x, z ) f ( z ) dµ α ( z ) , x ∈ R + , with the measure dµ α ( x ) = x α +1 dx and the kernel given by K α,βt ( x, z ) = 2 α + β Γ( α + β + 1) t α + β ( xz ) α (cid:90) ∞ J α + β ( ty ) J α ( xy ) J α ( zy ) y − α − β dy. This kernel is well defined for ( t, x, z ) ∈ R such that, in general, t (cid:54) = | x − z | and t (cid:54) = x + z . Notethat the integral here converges absolutely when α + β > /
2, but otherwise the convergence at ∞ is only conditional, in the Riemann sense.For a radial function f = f ( |·| ) in L ( R n ), n ≥ M β f ( x, t ) is for each t > x ∈ R n whose profile is given by M n/ − ,βt f ; see [2, Corollary 4.2]. Clearly, the maximal operators M β ∗ and M α,β ∗ f ( x ) = sup t> (cid:12)(cid:12) M α,βt f ( x ) (cid:12)(cid:12) are connected in the same way. Thus M β ∗ is bounded on L p rad ( R n , | x | γ dx ) if and only if M n/ − ,β ∗ is bounded on L p ( R + , x γ dµ n/ − ).We now formulate the main result of this paper. Theorem 1.5.
Assume that α > − and α + β > − / . Let < p < ∞ and δ ∈ R . Then themaximal operator M α,β ∗ is bounded on L p ( R + , x δ dx ) if (1.3) 1 p < α + β + 12 and (a) in case β ≥ , (1.4) − < δ < (2 α + 2) p − , (b) in case < β < , (1.5) − β [ β ∨ ( α + β + 1 / ∧ < δ < (2 α + β + 1) p − , with the first inequality weakened when α + β > / , (c) in case β ≤ , (1.6) − βp < δ < (2 α + β + 1) p − , with the first inequality weakened when − β ∈ N or α + β > / . We presume that Theorem 1.5 is sharp, but at the moment we are not able to give a completejustification of this statement. Nevertheless, we can show that Theorem 1.5 is optimal, or very closeto optimal, for most choices of the parameters α, β . ´O. CIAURRI, A. NOWAK, AND L. RONCAL Proposition 1.6.
Assume that α > − and α + β > − / . Let < p < ∞ and δ ∈ R . If M α,β ∗ isbounded on L p ( R + , x δ dx ) , then (1.3) holds and, moreover, − < δ, − βp ≤ δ, δ < (2 α + 2) p − , δ < (2 α + β + 1) p − . It follows that Theorem 1.5 gives sharp conditions when β ≥ β ≤ − β ∈ N or α + β > /
2] or when β ∈ (0 ,
1) and α ≤ − /
2. For β ≤ − β / ∈ N and α + β ≤ / δ in (1.6); thisremains to be settled. In case β ∈ (0 ,
1) and α > − /
2, the upper bound for δ in (1.5) is sharp,while optimality of the lower one remains an open question. See also Remark 4.3 at the end.Note that in Theorem 1.5 condition (1.3) is meaningful only when α + β < /
2. Further,considering the lower bound in (1.5), notice that − β [ β ∨ ( α + β + 1 / ∧ − β, α + β ≥ / , − βα + β +1 / , α + β < / α > − / , − , α + β < / α ≤ − / . Observe also that (1.4) is in fact the A p condition for the power weight x δ − (2 α +1) in the context ofthe space of homogeneous type ( R + , dµ α , | · | ).In the circumstances of Theorem 1.5, assuming in addition that 2 α + β > −
1, for each p satisfying(1.3) there is always a non-trivial interval of δ for which the L p ( x δ dx ) boundedness holds. On theother hand, when 2 α + β ≤ − α, β ) is inside or on the rightmost side of the small trianglewith vertices ( − / , − , / − , L p ( x δ dx ) boundedness (the rangeof δ is empty for each p ).Taking δ = γ + 2 α + 1 and α = n/ − n ≥ β = 0 or β = (3 − n ) / δ = 2 α + 1 and get the following. Corollary 1.7.
Assume that α > − and α + β > − / . Let < p < ∞ . Then M α,β ∗ is boundedon L p ( R + , dµ α ) provided that condition (1.3) is satisfied and (1.7) − β α + 1 < p < − − β α + 2 , with the first inequality weakened if − β ∈ N or α + β > / . These conditions are optimal up toa possible weakening of the strict inequality in the lower bound in (1.7) when < − β / ∈ N and α + β ≤ / . Observe that, in the context of Corollary 1.7, condition (1.7) is always satisfied in case β ≥ ≤ β < p for which theboundedness holds if either β ∈ (0 ,
1) and 2 α + β + 1 ≤ β < α + β ≤ − / − / (16 α + 12).Otherwise, there is always a non-trivial interval of p for which M α,β ∗ is bounded on L p ( dµ α ).Taking α = n/ − n ≥
2, in Corollary 1.7 we obtain
Corollary 1.8.
Let n ≥ and < p < ∞ . Then M β ∗ is bounded on L p rad ( R n ) provided that (1.8) p ≤ n − n − and β > − n + n/p, or p > n − n − and β > − np , with the last inequality weakened when − β ∈ N or β > − n . This condition is optimal up to apossible weakening of the last inequality in (1.8) in case − β / ∈ N and β ≤ − n . We believe that weakening the inequality in Corollary 1.8 reflects radial improvement of mappingproperties of M β ∗ . On the other hand, the fact that for 2 < p < n + 1) / ( n −
1) condition (1.8)is strictly less restrictive than condition (1.2), see Figure 1 below, most probably indicates non-optimality of the yet known results on L p -boundedness of M β ∗ stated in Theorem 1.2; cf. [6, Section3, Problem (1)]. AXIMAL ESTIMATES FOR SPHERICAL MEANS 5 Oβ p − ( n − n − n − n +2 12 − ( n − n +22 − n n − n − − n A C B
Figure 1.
Regions
OAB and
ABC visualize differences between Stein’s condition(1.1), the less restrictive condition (1.2) due to Miao et al., and the least restric-tive condition (1.8) from our radial case and Conjecture 1.9 for the general case,respectively. Picture for n = 4, with different axes scaling.Taking into account Corollary 1.8 and known sharp results on boundedness of M ∗ on L p ( R n ) westate the following. Conjecture 1.9.
Let n ≥ and < p < ∞ . The operator M β ∗ is bounded on L p ( R n ) if and onlyif condition (1.8) is satisfied. Finally, we comment on pointwise almost everywhere convergence M α,βt f → f as t → + . This isan important issue, due to connections of M α,βt with solutions to certain initial-value PDE problems.It is well known that the key ingredient leading to such results are suitable mapping properties of M α,β ∗ , or actually of a smaller truncated maximal operator M α,β ∗ , tru f ( x ) = sup Assume that α > − and α + β > − / . Then (1.9) M α,β ∗ , tru f ( x ) (cid:46) (cid:40) Lf ( x ) , if α + β ≥ / ,x β [ L ( z − β f ) θ ( x )] / (1+ θ ) , if α + β < / , uniformly in locally integrable functions f on R + and x ∈ R + , with any fixed θ > such that θ < α + β + 1 / . ´O. CIAURRI, A. NOWAK, AND L. RONCAL Consequently, if < p < ∞ satisfies condition (1.3) then M α,β ∗ , tru is bounded on L p ( R + , x δ dx ) forany δ ∈ R . Moreover, in case α + β ≥ / , M α,β ∗ , tru is of weak type (1 , with respect to the measurespace ( R + , x δ dx ) with any δ ∈ R . Corollary 1.11. Let α, β and p be as in Proposition 1.10. Then M α,βt f ( x ) → f ( x ) for a.a. x ∈ R + as t → + for any locally integrable function f on R + in case α + β ≥ / , or for any f locally in L p ( R + , dx ) with p satisfying condition (1.3) in case α + β < / . Consequently, we obtain almost everywhere convergence to initial data results in contexts ofdifferential problems whose solutions express via M α,βt . In particular, this pertains to the followingCauchy initial-value problems, see e.g. [2, Section 7].(i) Euler-Poisson-Darboux equation in R n with a locally integrable/locally in L p radial initialposition and null initial speed.(ii) Wave equation in R n with null initial position and a locally integrable radial initial speed(this case was already treated before, see [5, Section 5]).(iii) Euler-Poisson-Darboux equation based on the one-dimensional Bessel operator with a lo-cally integrable/locally in L p initial position and null initial speed.(iv) Wave equation based on the one-dimensional Bessel operator with null initial position anda locally integrable/locally in L p initial speed.In (i), (iii) and (iv) by “locally integrable/locally in L p ” we mean local integrability in case α + β ≥ / 2, and being locally in L p for some p satisfying condition (1.3) in case − / < α + β < / Structure of the paper. The rest of the paper is organized as follows. Section 2 is a technicalpreparation for the proof of Theorem 1.5. It provides estimates of the integral kernel K α,βt ( x, z ),decomposition of the ( t, x, z ) space into suitable regions and definitions of special auxiliary operatorstogether with description of their L p -behavior. Section 3 contains the proof of Theorem 1.5. Themain part is preceded by establishing suitable control in terms of the special operators. At theend of Section 3 a brief justification of Proposition 1.10 is given. Finally, in Section 4 we constructsuitable counterexamples in order to prove Proposition 1.6. Notation. Throughout the paper we use a fairly standard notation. Thus N = { , , , . . . } and R + = (0 , ∞ ). The symbols “ ∨ ” and “ ∧ ” mean the operations of taking maximum and minimum,respectively. We write X (cid:46) Y to indicate that X ≤ CY with a positive constant C independent ofsignificant quantities. We shall write X (cid:39) Y when simultaneously X (cid:46) Y and Y (cid:46) X .For the sake of brevity, we often omit R + when denoting L p spaces related to the measure spaces( R + , x δ dx ) and ( R + , dµ α ). We write L p rad ( . . . ) for the subspace of L p ( . . . ) consisting of radialfunctions. Acknowledgment. We thank the referee for a constructive criticism that improved the presenta-tion and led us to enrich the paper with the sharpness results.2. Technical preparation Define the main regions E = (cid:8) ( t, x, z ) ∈ R : | x − z | < t < x + z (cid:9) ,F = (cid:8) ( t, x, z ) ∈ R : x + z < t (cid:9) . Following the strategy used in [5], see also [4], we shall estimate the kernel K α,βt ( x, z ) and thensplit the bounds according to suitably defined sub-regions of E and F . Then we will estimate theresulting maximal operators independently and get control in terms of a number of special operatorswhose mapping properties are essentially known. Altogether, this will give an overall control of themaximal operator M α,β ∗ . AXIMAL ESTIMATES FOR SPHERICAL MEANS 7 In preparatory subsections 2.1–2.3 below we gather the kernel estimates, suitable decompositionsof E and F and definitions of the special operators, respectively.2.1. Kernel estimates. We first rephrase [2, Theorem 3.3] in a way more convenient for ourpresent purposes; see also [2, Section 2.3]. In particular, we neglect parts concerning sharpness ofthe bounds for certain α and β . Theorem 2.1 ([2]) . Assume that α > − and α + β > − / . Let t, x, z > . The kernel K α,βt ( x, z ) vanishes when t < | x − z | , whereas in E ∪ F the following uniform estimates hold. (1) If − β ∈ N , then the kernel vanishes in F and | K α,βt ( x, z ) | (cid:46) ( xz ) − α − β t α +2 β (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:2) ( x + z ) − t (cid:3)(cid:17) α + β − / in E. (2) If α + β > / and neither − β ∈ N nor α + β = 0 , then | K α,βt ( x, z ) | (cid:46) (cid:40) ( xz ) − α − / t α +2 β (cid:2) t − ( x − z ) (cid:3) α + β − / in E, t α +2 β (cid:2) t − ( x − z ) (cid:3) β − in F. (3) If α + β > / and α + β = 0 , then | K α,βt ( x, z ) | (cid:46) t − α (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:12)(cid:12) t − ( x + z ) (cid:12)(cid:12)(cid:17) − α − / in E ∪ F. (4) If α + β = 1 / and neither − β ∈ N nor β = 1 , then | K α,βt ( x, z ) | (cid:46) ( xz ) − α − / t log xz ( x + z ) − t in E, t (cid:2) t − ( x − z ) (cid:3) − α − / log (cid:16) t − ( x − z ) t − ( x + z ) (cid:17) in F. (5) If α + β = 1 / and β = 1 , then K α,βt ( x, z ) (cid:39) t in E ∪ F. (6) If α + β < / and neither − β ∈ N nor β = 1 nor α + 1 / ∈ N , then | K α,βt ( x, z ) | (cid:46) ( xz ) − α − β t α +2 β (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:2) ( x + z ) − t (cid:3)(cid:17) α + β − / in E, t α +2 β (cid:2) t − ( x − z ) (cid:3) − α − / (cid:2) t − ( x + z ) (cid:3) α + β − / in F. (7) If α + β < / and β = 1 , then | K α,βt ( x, z ) | (cid:46) ( xz ) − α − t α +2 (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:2) ( x + z ) − t (cid:3)(cid:17) α +1 / in E, t α +2 in F. (8) If α + β < / and α + 1 / ∈ N , then | K α,βt ( x, z ) | (cid:46) (cid:40) ( xz ) − α − / t α +2 β (cid:2) t − ( x − z ) (cid:3) α + β − / in E, t α +2 β (cid:2) t − ( x − z ) (cid:3) − α − / (cid:2) t − ( x + z ) (cid:3) α + β − / in F. The above theorem distinguishes cases, or rather segments and lines in the ( α, β ) plane, wherethe estimates are better comparing to other neighboring ( α, β ). However, this does not seem tobe reflected in power weighted L p , 1 < p < ∞ , boundedness of M α,β ∗ . This claim is based onour detailed analysis of the maximal operator in each of the cases related to items (1)–(8) ofTheorem 2.1. Even though we could often obtain seemingly better control of K α,β ∗ in terms ofthe auxiliary special operators, it led us to the same mapping properties as stated in Theorem 1.5.Therefore, we simplify things already at this stage and derive less accurate kernel bounds, but withsimpler structure (in particular, containing no logarithmic expressions), that will be sufficient forour purpose. However, this strategy might not be appropriate for studying more subtle mappingproperties of M α,β ∗ , like weak or restricted weak type estimates. ´O. CIAURRI, A. NOWAK, AND L. RONCAL For notational convenience, define auxiliary kernelsΦ α,βt ( x, z ) = χ E ( t, x, z ) ( xz ) − α − / t α +2 β (cid:2) t − ( x − z ) (cid:3) α + β − / + χ F ( t, x, z ) 1 t α +2 β (cid:2) t − ( x − z ) (cid:3) β − , Ψ α,βt ( x, z ) = χ E ( t, x, z ) ( xz ) − α − β t α +2 β (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:2) ( x + z ) − t (cid:3)(cid:17) α + β − / + χ F ( t, x, z ) 1 t α +2 β (cid:2) t − ( x − z ) (cid:3) − α − / (cid:2) t − ( x + z ) (cid:3) α + β − / . Theorem 2.2. Assume that α > − and α + β > − / . Fix an arbitrary ε > . The followingestimates hold uniformly in ( t, x, z ) ∈ E ∪ F : (cid:12)(cid:12) K α,βt ( x, z ) (cid:12)(cid:12) (cid:46) Φ α,βt ( x, z ) , if α + β > / , Ψ α,β − εt ( x, z ) , if α + β = 1 / , Ψ α,βt ( x, z ) , if α + β < / , Ψ α,βt ( x, z ) χ E ( t, x, z ) , if − β ∈ N . Moreover, χ E ( t, x, z )Ψ α,βt ( x, z ) (cid:46) χ E ( t, x, z )Φ α,βt ( x, z ) when α + β ≥ / .Proof. All the asserted bounds are straightforward consequences of Theorem 2.1. Clearly, | K α,βt ( x, z ) | (cid:46) χ E ( t, x, z )Ψ α,βt ( x, z ) if − β ∈ N , for ( t, x, z ) ∈ E ∪ F , is just Theorem 2.1(1). We can write χ E ( t, x, z )Ψ α,βt ( x, z ) = χ E ( t, x, z )Φ α,βt ( x, z ) (cid:20) ( x + z ) − t xz (cid:21) α + β − / and, since the expression in square brackets is bounded from above by a constant, we infer that(2.1) χ E ( t, x, z )Ψ α,βt ( x, z ) (cid:46) χ E ( t, x, z )Φ α,βt ( x, z ) when α + β ≥ / . Next, assume that α + β > / α + β = 0 (see Theorem 2.1(3)), which forces α < − / t, x, z ) ∈ E ∪ F we have1 t − α (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:12)(cid:12) t − ( x + z ) (cid:12)(cid:12)(cid:17) − α − / = χ E ( t, x, z )Φ α,βt ( x, z ) (cid:20) xz ( x + z ) − t (cid:21) α +1 / + χ F ( t, x, z )Φ α,βt ( x, z ) (cid:20) t − ( x − z ) t − ( x + z ) (cid:21) α +1 / . Here the expressions in square brackets are bounded from below by a positive constant. Hence, inview of Theorem 2.1(1)–(3) and (2.1), we see that | K α,βt ( x, z ) | (cid:46) Φ α,βt ( x, z ) if α + β > / . Let now α + β = 1 / ε > 0. For ( t, x, z ) ∈ E we have (see Theorem 2.1(4))( xz ) − α − / t log 8 xz ( x + z ) − t (cid:46) ( xz ) − α − / t (cid:20) ( x + z ) − t xz (cid:21) α + β − ε − / = Ψ α,β − εt ( x, z ) (cid:20) t t − ( x − z ) (cid:21) α + β − ε − / ≤ Ψ α,β − εt ( x, z ) , AXIMAL ESTIMATES FOR SPHERICAL MEANS 9 where we used the fact that log y grows slower than the positive power y − ( α + β − ε − / (here y positive and separated from zero). Further, for ( t, x, z ) ∈ F ,1 t (cid:2) t − ( x − z ) (cid:3) − α − / log (cid:18) t − ( x − z ) t − ( x + z ) (cid:19) (cid:46) t (cid:2) t − ( x − z ) (cid:3) − α − / (cid:20) t − ( x + z ) t − ( x − z ) (cid:21) α + β − ε − / = Ψ α,β − εt ( x, z ) (cid:20) t t − ( x − z ) (cid:21) α + β − ε − / ≤ Ψ α,β − εt ( x, z ) . This shows that (see Theorem 2.1(4),(5)) | K α,βt ( x, z ) | (cid:46) Ψ α,β − εt ( x, z ) if α + β = 1 / t, x, z ) ∈ E ∪ F (to be precise, the case coming from Theorem 2.1(5), i.e. ( α, β ) = ( − / , α + β < / 2, see Theorem 2.1(6)–(8). If β = 1 then automatically α < − / t, x, z ) ∈ F we have (see Theorem 2.1(7))1 t α +2 = Ψ α,βt ( x, z ) (cid:20) t − ( x − z ) t − ( x + z ) (cid:21) α +1 / ≤ Ψ α,βt ( x, z ) . If α + 1 / ∈ N then for ( t, x, z ) ∈ E (see Theorem 2.1(8))( xz ) − α − / t α +2 β (cid:2) t − ( x − z ) (cid:3) α + β − / = Ψ α,βt ( x, z ) (cid:20) ( x + z ) − t xz (cid:21) − ( α + β − / . Altogether, this implies for ( t, x, z ) ∈ E ∪ F | K α,βt ( x, z ) | (cid:46) Ψ α,βt ( x, z ) if α + β < / . The proof of Theorem 2.2 is complete. (cid:3) Decompositions of E and F . Inspired by the analysis presented in [5], we define the fol-lowing sub-regions of E and F , see Figure 2 below, E = E ∩ (cid:8) ( t, x, z ) ∈ R : t < x/ (cid:9) ,E = E ∩ (cid:8) ( t, x, z ) ∈ R : x/ ≤ t < x (cid:9) ,E = E ∩ (cid:8) ( t, x, z ) ∈ R : 3 x ≤ t (cid:9) and F = F ∩ (cid:8) ( t, x, z ) ∈ R : x < t < x (cid:9) ,F = F ∩ (cid:8) ( t, x, z ) ∈ R : 3 x ≤ t (cid:9) ,F (cid:48) = F ∩ (cid:8) ( t, x, z ) ∈ R : z < ( t − x ) / (cid:9) ,F (cid:48)(cid:48) = F ∩ (cid:8) ( t, x, z ) ∈ R : ( t − x ) / ≤ z < t − x (cid:9) . Clearly, E = E ∪ E ∪ E , F = F ∪ F , F = F (cid:48) ∪ F (cid:48)(cid:48) , all the sums being disjoint.In what follows we will use uniform estimates of expressions in t, x, z that hold specifically in one(or more) of the above regions, cf. [5]. We list them below for an easy further reference. Verificationof these relations is straightforward and left to the reader.in E : x (cid:39) z, (2.2) in E : t − ( x − z ) (cid:46) tz, (2.3) in E : t − ( x − z ) (cid:46) tx, z (cid:39) t, (2.4) in F : t − ( x − z ) (cid:39) x ( t − x ) (cid:39) x ( t − x + z ) , t − ( x + z ) (cid:39) x ( t − x − z ) , (2.5) t zx/ x x/ x x x x x x x x x x F (cid:48) F (cid:48)(cid:48) E F E E z = x − t z = t + x z = t − x z = t − x Figure 2. Sections of regions E i and F i given x > F : t − ( x − z ) (cid:39) t ( t + x − z ) , t − ( x + z ) (cid:39) t ( t − x − z ) , (2.6) in F (cid:48) : t − ( x ± z ) (cid:39) t , (2.7) in F (cid:48)(cid:48) : z (cid:39) t (cid:39) t − x (cid:39) t − ( x − z ) , t + x − z (cid:39) t − z. (2.8)2.3. Special operators. We will use the following auxiliary operators whose mapping propertiesare essentially known, see [4, 5] and also references given there. A variant of the Hardy-Littlewood maximal operator (denoted by E in [4]) Df ( x ) = sup ≤ a The maximal operator N . For positive η , let N η f ( x ) = sup t>x t η (cid:90) t z η − | f ( z ) | dz. The operator N η is bounded on L p ( x γ dx ), 1 < p < ∞ , if − < γ < ηp − 1. At this point it isperhaps in order to note that the range given in [5, p. 1545] for the boundedness of N η is wrong;consequently some statements in the proof of [5, Theorem 4.3] are not correct, nevertheless theresult claimed there is true. The maximal operator T . For any η ∈ R , consider the operator(2.9) T η f ( x ) = sup t> x (cid:90) tt/ z η − | f ( z ) | ( t − z + x ) η dz. Boundedness of T η on power weighted L p spaces is shown in the following lemma, which is a simpleextension of [5, Lemma 2.1]. Actually, this is even more than we need, we state item (b) only forthe sake of completeness. Lemma 2.3. Let < p < ∞ .(a) For η > , T η is bounded on L p ( x γ dx ) for γ ≥ ( η − p .(b) For η = 1 , T η is bounded on L p ( x γ dx ) for γ > .(c) For < η < , T η is bounded on L p ( x γ dx ) for γ ≥ η − .(d) For η ≤ , T η is bounded on L p ( x γ dx ) for γ > − .Proof. Items (a) and (c) are proved in [5], while (b) is commented there.To show (d), assume that η ≤ T η f ( x ) = sup t> x (cid:90) tt/ (cid:18) t − z + xz (cid:19) − η | f ( z ) | z dz (cid:46) sup t> x (cid:90) tt/ | f ( z ) | z dz ≤ (cid:90) ∞ x | f ( z ) | z dz. Thus T η is controlled in terms of the dual Hardy operator f ( x ) (cid:55)→ (cid:82) ∞ x f ( z ) z dz , which is well knownto be bounded on L p ( x γ dx ) if and only if γ > − 1; see e.g. [2, Lemma 6.3]. (cid:3) Proof of the main theorem From now on we will consider only non-negative functions f . This will be enough, since ourapproach is based on absolute estimates of the kernel.Let G ⊂ R . We denoteΦ α,β ∗ ,G f ( x ) = sup t> (cid:90) ∞ χ G ( t, x, z )Φ α,βt ( x, z ) f ( z ) dµ α ( z ) , Φ α,β ∗ f ( x ) = Φ α,β ∗ , R f ( x )and analogously for the Ψ counterparts. Clearly, Φ α,β ∗ f ≤ Φ α,β ∗ ,E f + Φ α,β ∗ ,F f and similarly in case ofthe Ψ operators.3.1. Control of Φ α,β ∗ . In this subsection we will establish a control of Φ α,β ∗ in terms of the specialoperators defined in Section 2.3. This will be done under the assumption α + β ≥ / Lemma 3.1. Let α > − and α + β ≥ / . Then Φ α,β ∗ ,E f ( x ) (cid:46) Lf ( x ) + H α + β +1 f (4 x ) + x β R (cid:0) z − β f (cid:1) ( x ) , Φ α,β ∗ ,F f ( x ) (cid:46) χ { β ≥ } H α +2 f (2 x ) + χ { β< } H α + β +1 f (2 x ) + N α +2 f ( x ) + T − β f ( x ) , uniformly in f ≥ and x > . Proof. We will bound the maximal operators related to restrictions of Φ α,βt ( x, z ) to the sub-regionsof E and F specified in Section 2.2. Region E . In view of (2.2), taking into account that α + β − / ≥ 0, we haveΦ α,β ∗ ,E f ( x ) (cid:46) sup t 0, we getΦ α,β ∗ ,E f ( x ) (cid:46) sup x/ ≤ t< x x α + β +1 (cid:90) x + t | x − t | z α + β f ( z ) dz ≤ H α + β +1 f (4 x ) . Region E . Using (2.4) and α + β − / ≥ α,β ∗ ,E f ( x ) (cid:46) sup t ≥ x x β x (cid:90) t + xt − x z − α − β − / f ( z ) z α +1 / dz (cid:46) x β R (cid:0) z − β f (cid:1) ( x ) . Region F . Now t (cid:39) x . Using the fact that t + x − z (cid:39) x in F , see (2.5), we can writeΦ α,β ∗ ,F f ( x ) (cid:46) sup x 1, and by z β − otherwise.This leads directly to the boundΦ α,β ∗ ,F f ( x ) (cid:46) χ { β ≥ } H α +2 f (2 x ) + χ { β< } H α + β +1 f (2 x ) . Region F (cid:48) . In view of (2.7),Φ α,β ∗ ,F (cid:48) f ( x ) (cid:46) sup t ≥ x t α +2 (cid:90) ( t − x ) / f ( z ) z α +1 dz ≤ N α +2 f (3 x ) ≤ N α +2 f ( x ) . Region F (cid:48)(cid:48) . Here we use (2.8) and (2.6) obtainingΦ α,β ∗ ,F (cid:48)(cid:48) f ( x ) (cid:46) sup t ≥ x (cid:90) t − x ( t − x ) / z − β f ( z )( t − z ) − β dz = sup t ≥ x (cid:90) tt/ z − β f ( z )( t + x − z ) − β dz = T − β f ( x ) . The bounds of Lemma 3.1 follow. (cid:3) Control of Ψ α,β ∗ ,E . We will prove the following. Lemma 3.2. Let α > − and − / < α + β < / . Then, given any θ > such that θ <α + β + 1 / , Ψ α,β ∗ ,E f ( x ) (cid:46) x β (cid:2) D ( z − β f ) θ ( x ) (cid:3) θ + x β (cid:2) R ( z − β f ) θ ( x ) (cid:3) θ , uniformly in f ≥ and x > .Proof. We first reduce the task to the special case β = 0. Observe thatΨ α,β ∗ ,E f ( x ) = sup t> x β ( tx ) α +2 β (cid:90) x + t | x − t | (cid:16)(cid:2) t − ( x − z ) (cid:3)(cid:2) ( x + z ) − t (cid:3)(cid:17) α + β − / z − β f ( z ) z dz. Thus Ψ α,β ∗ ,E f ( x ) = x β Ψ α + β, ∗ ,E (cid:0) z − β f (cid:1) ( x )and hence it is enough to show that, given − / < α < / α, ∗ ,E f ( x ) (cid:46) (cid:2) Df θ ( x ) (cid:3) θ + (cid:2) Rf θ ( x ) (cid:3) θ with any fixed θ > θ < α + 1 / AXIMAL ESTIMATES FOR SPHERICAL MEANS 13 The proof of (3.1) is a straightforward generalization of the reasoning from [4], see the proofof [4, Theorem 3.1(b)]. We present briefly some details for the reader’s convenience. Using theidentity(3.2) (cid:2) t − ( x − z ) (cid:3)(cid:2) ( x + z ) − t (cid:3) = (cid:2) ( t + x ) − z (cid:3)(cid:2) z − ( x − t ) (cid:3) and then H¨older’s inequality one gets the boundΨ α, ∗ ,E f ( x ) ≤ sup t> (cid:34) xt ) α (cid:18) (cid:90) t + x | t − x | zf θ ( z ) dz (cid:19) θ × (cid:18) (cid:90) t + x | t − x | (cid:16)(cid:2) ( t + x ) − z (cid:3)(cid:2) z − ( x − t ) (cid:3)(cid:17) (2 α − θ θ z dz (cid:19) θ θ (cid:35) . The second integral here converges when (2 α − θ θ > − 1, i.e. when θ < α + 1 / 2. In suchthe case it is comparable to ( xt ) (2 α − θθ +1 , see Lemma 4.2 in Section 4 below. Consequently, wearrive at the bound Ψ α, ∗ ,E f ( x ) (cid:46) sup t> (cid:18) xt (cid:90) t + x | t − x | zf θ ( z ) dz (cid:19) θ . From here one proceeds by splitting the supremum into t ≤ x and t > x (which correspondsessentially to estimating separately Ψ α, ∗ ,E ∪ E and Ψ α, ∗ ,E ). In the first case | t − x | ≤ x and we getthe control by [ Df θ ( x )] / (1+ θ ) . On the other hand, if t > x , then z (cid:39) t and this part of themaximal operator is controlled by [ Rf θ ( x )] / (1+ θ ) . (cid:3) Control of Ψ α,β ∗ ,F . We finally establish a control of Ψ α,β ∗ ,F in terms of the special operators. Lemma 3.3. Let α > − and − / < α + β < / . Then, given any θ > such that θ <α + β + 1 / , Ψ α,β ∗ ,F f ( x ) (cid:46) χ { β ≥ } (cid:2) H ( α +1 / − β )(1+ θ )+1 (cid:0) f θ (cid:1) (2 x ) (cid:3) θ + χ { β< } (cid:2) H ( α +1 / θ )+1 (cid:0) f θ (cid:1) (2 x ) (cid:3) θ + N α +2 f ( x ) + (cid:2) T ( α +1 / θ ) (cid:0) f θ (cid:1) ( x ) (cid:3) θ , uniformly in f ≥ and x > .Proof. We will bound suitably the restricted maximal operators Ψ α,β ∗ ,F , Ψ α,β ∗ ,F (cid:48) and Ψ α,β ∗ ,F (cid:48)(cid:48) . Region F . Here t (cid:39) x . Then, by (2.5), we haveΨ α,β ∗ ,F f ( x ) (cid:46) sup x 1, the second integral here is finite if and only if θ < α + β + 1 / 2. If this isthe case, it is comparable with ( t − x ) (2 α + β ) θθ +1 , as verified by changing the variable z (cid:55)→ ( t − x ) z .This gives Ψ α,β ∗ ,F f ( x ) (cid:46) sup x 2) we use H¨older’s inequality with the splitting of z α +1 so that z α +1 / − β is attached to f . This leads to the boundΨ α,β ∗ ,F f ( x ) (cid:46) (cid:2) H ( α +1 / − β )(1+ θ )+1 (cid:0) f θ (cid:1) (2 x ) (cid:3) θ under the condition θ < α + β + 1 / 2. The assumption β ≥ t − x occurring there is non-negative. Region F (cid:48) . In view of (2.7),Ψ α,β ∗ ,F (cid:48) f ( x ) (cid:46) sup t ≥ x t α +2 (cid:90) ( t − x ) / f ( z ) z α +1 dz ≤ N α +2 f ( x ) . Region F (cid:48)(cid:48) . Now we use (2.6) and getΨ α,β ∗ ,F (cid:48)(cid:48) f ( x ) (cid:46) sup t ≥ x t α + β +1 (cid:90) t − x ( t − x ) / ( t + x − z ) − α − / ( t − x − z ) α + β − / f ( z ) z α +1 dz. By means of H¨older’s inequality we can control the above integral by (cid:18) (cid:90) t − x ( t − x ) / (cid:20)(cid:18) zt + x − z (cid:19) α +1 / f ( z ) (cid:21) θ dz (cid:19) θ (cid:18) (cid:90) t − x ( t − x ) / (cid:2) ( t − x − z ) α + β − / z α +1 / (cid:3) θθ dz (cid:19) θ θ . The second integral here is finite if and only if θ < α + β + 1 / 2. In this case it is comparablewith ( t − x ) (2 α + β ) θθ +1 . This implies, see (2.8),Ψ α,β ∗ ,F (cid:48)(cid:48) f ( x ) (cid:46) sup t ≥ x ( t − x ) α + β + θ θ t α + β +1 (cid:18) (cid:90) t − x ( t − x ) / (cid:20)(cid:18) zt + x − z (cid:19) α +1 / f ( z ) (cid:21) θ dz (cid:19) θ (cid:46) sup u ≥ x (cid:18) (cid:90) uu/ z ( α +1 / θ ) − ( u + x − z ) ( α +1 / θ ) f θ ( z ) dz (cid:19) θ = (cid:2) T ( α +1 / θ ) (cid:0) f θ (cid:1) ( x ) (cid:3) θ . Now Lemma 3.3 follows. (cid:3) Proof of Theorem 1.5. We distinguish several cases described by conditions on α and β .Altogether, they imply Theorem 1.5. In what follows we always assume that α > − α + β > − / 2, 1 < p < ∞ , and x ∈ R + . Case 1. α + β > / − β / ∈ N . By Theorem 2.2, M α,β ∗ f ( x ) (cid:46) Φ α,β ∗ f ( x ). Thus from Lemma3.1 we have the control M α,β ∗ f ( x ) (cid:46) Lf ( x ) + H α + β +1 f (4 x ) + H α +2 f (2 x ) + N α +2 f ( x )+ x β R (cid:0) z − β f (cid:1) ( x ) + T − β f ( x ) . The operator L is bounded on L p ( x δ dx ) for any δ ∈ R . The conditions for L p ( x δ dx ) boundednessof the remaining operators controlling M α,β ∗ are as follows, see Section 2.3: H α + β +1 : δ < (2 α + β + 1) p − ,H α +2 : δ < (2 α + 2) p − ,N α +2 : − < δ < (2 α + 2) p − ,f (cid:55)→ x β R ( z − β f )( x ) : − βp ≤ δ,T − β : − < δ, if β ≥ , − β ≤ δ, if β ∈ (0 , , − βp ≤ δ, if β < . AXIMAL ESTIMATES FOR SPHERICAL MEANS 15 We conclude that M α,β ∗ is bounded on L p ( x δ dx ) if δ satisfies − min(1 , β, βp ) < δ < min (cid:16) (2 α + β + 1) p − , (2 α + 2) p − (cid:17) , with the first inequality weakened when β < 1. Heremin(1 , β, βp ) = , if β ≥ ,β, if β ∈ (0 , ,βp, if β < , while min (cid:16) (2 α + β + 1) p − , (2 α + 2) p − (cid:17) = (cid:40) (2 α + β + 1) p − , if β < , (2 α + 2) p − , if β ≥ . Case 2. α + β < / − β / ∈ N . In view of Theorem 2.2, M α,β ∗ f ( x ) (cid:46) Ψ α,β ∗ f ( x ). Consequently,from Lemmas 3.2 and 3.3 we get the control M α,β ∗ f ( x ) (cid:46) x β (cid:2) D ( z − β f ) θ ( x ) (cid:3) θ + x β (cid:2) R ( z − β f ) θ ( x ) (cid:3) θ + χ { β< } (cid:2) H ( α +1 / θ )+1 (cid:0) f θ (cid:1) (2 x ) (cid:3) θ + χ { β ≥ } (cid:2) H ( α +1 / − β )(1+ θ )+1 (cid:0) f θ (cid:1) (2 x ) (cid:3) θ + N α +2 f ( x ) + (cid:2) T ( α +1 / θ ) (cid:0) f θ (cid:1) ( x ) (cid:3) θ , with any fixed θ > θ < α + β + 1 / /p < α + β + 1 / 2, conditions for L p ( x δ dx ) boundedness of the operators con-trolling M α,β ∗ are as follows, see Section 2.3: f (cid:55)→ x β (cid:2) D ( z − β f ) θ ( x ) (cid:3) θ : − βp − < δ < (2 α + β + 1) p − ,f (cid:55)→ x β (cid:2) R ( z − β f ) θ ( x ) (cid:3) θ : − βp ≤ δ,f (cid:55)→ χ { β< } (cid:2) H ( α +1 / θ )+1 ( f θ )( x ) (cid:3) θ : δ < (2 α + β + 1) p − ,f (cid:55)→ χ { β ≥ } (cid:2) H ( α +3 / − β )(1+ θ )+1 ( f θ )( x ) (cid:3) θ : δ < (2 α + 2) p − ,N α +2 : − < δ < (2 α + 2) p − ,f (cid:55)→ (cid:2) T ( α +1 / θ ) ( f θ )( x ) (cid:3) θ : − βp < δ, if β < , − < δ, if β > α ≤ − / , − βα + β +1 / < δ, if β > α > − / . Treatment of the last operator will be explained in a moment; the cases of the other θ -operators aresimpler and left to the reader. In all these conditions we understand that given δ from the indicatedrange, there exists θ > θ < α + β + 1 / L p ( x δ dx ) boundedness holds.Intersecting the above conditions we see that M α,β ∗ is bounded on L p ( x δ dx ) if 1 /p < α + β + 1 / β ≥ − < δ < (2 α + 2) p − < β < − β max(0 ,α +1 / β < δ < (2 α + β + 1) p − β < − βp < δ < (2 α + β + 1) p − . We now explain the analysis related to T in greater detail. Observe that, for p > θ , theoperator f (cid:55)→ (cid:2) T ( α +1 / θ ) ( f θ )( x ) (cid:3) θ is bounded on L p ( x δ dx ) if and only if T ( α +1 / θ ) isbounded on L p θ ( x δ dx ). If β < α > − / θ > /p < θ < α + β + 1 / /p < α + β + 1 / θ < α + 1 / T ( α +1 / θ ) falls under Lemma 2.3(a): it is bounded on L p θ ( x δ dx ) when δ ≥ (cid:2) ( α + 1 / θ ) − (cid:3) p θ = ( α + 1 / p − p θ . Choosing θ so that θ is sufficiently close to α + β + 1 / δ arbitrarily close to ( α + 1 / p − ( α + β + 1 / p = − βp .Let now β > 0. If α ≤ − / 2, then we choose any θ > /p < θ < α + β + 1 / α > − / α ∈ ( − / , / θ > < α + 1 / < θ < α + β + 1 / 2. By Lemma 2.3(c) the condition for theboundedness is δ ≥ ( α + 1 / θ ) − 1. One can choose θ so that θ is arbitrarily close to α + β + 1 / 2, which covers all δ > α +1 / α + β +1 / − − βα + β +1 / . Case 3. α + β = 1 / − β / ∈ N . By Theorem 2.2, M α,β ∗ f ( x ) (cid:46) Ψ α,β − ε ∗ f ( x ) with any fixed ε > β replaced by β − ε , we infer that M α,β ∗ is bounded on L p ( x δ dx )if 1 /p < α + β − ε + 1 / β − ε ≥ − < δ < (2 α + 2) p − < β − ε < − ( β − ε )max(0 ,α +1 / β − ε < δ < (2 α + β − ε + 1) p − β − ε < − ( β − ε ) p < δ < (2 α + β − ε + 1) p − . Since ε can be chosen arbitrarily small, the above conditions with ε = 0 also imply L p ( x δ dx )boundedness of M α,β ∗ , as can be easily verified. Case 4. − β ∈ N . In view of Theorem 2.2, we have the bound M α,β ∗ f ( x ) (cid:46) Φ α,β ∗ ,E f ( x ) in case α + β ≥ / 2, and M α,β ∗ f ( x ) (cid:46) Ψ α,β ∗ ,E f ( x ) when α + β < / 2. Therefore, Lemmas 3.1 and 3.2 providethe control M α,β ∗ f ( x ) (cid:46) (cid:40) Lf ( x ) + H α + β +1 f (4 x ) + x β R ( z − β f )( x ) , if α + β ≥ / ,x β (cid:2) D ( z − β f ) θ ( x ) (cid:3) θ + x β (cid:2) R ( z − β f ) θ ( x ) (cid:3) θ , if α + β < / , with any fixed θ > θ < α + β + 1 / L p ( x δ dx ) boundedness of the component operators in-volved, see Cases 1 and 2 above, we conclude that M α,β ∗ is bounded on L p ( x δ dx ) if 1 /p < α + β +1 / α + β ≥ / 2) and − βp ≤ δ < (2 α + β + 1) p − . The proof of Theorem 1.5 is complete. (cid:3) Proof of Proposition 1.10. To begin with, observe that M α,β ∗ , tru arises from M α,β ∗ by restrict-ing the kernel to the region E .When α + β ≥ / 2, we have Φ α,β ∗ ,E f ( x ) (cid:46) Lf ( x ), see the proof of Lemma 3.1. Further, incase α + β < / 2, Ψ α,β ∗ ,E f ( x ) (cid:46) x β [ L ( z − β f ) θ ( x )] / (1+ θ ) with any fixed θ > θ <α + β + 1 / 2. This is implicitly contained in the proof of Lemma 3.2. The more precise bounds ofTheorem 2.1(4),(5) lead easily to the control M α,β ∗ , tru f ( x ) (cid:46) Lf ( x ) in case α + β = 1 / − β / ∈ N (when t < x/ δ ∈ R , L is bounded on L p ( x δ dx ), 1 < p < ∞ , and from L ( x δ dx ) to weak L ( x δ dx ). (cid:3) Counterexamples Our aim in this section is to prove Proposition 1.6. This will be done by constructing suitablecounterexamples, in the first step for certain auxiliary maximal operators. AXIMAL ESTIMATES FOR SPHERICAL MEANS 17 Counterexamples for auxiliary operators. For α > − α + β > − / t > R + : U α,βt, f ( x ) = x − α − β t α +2 β (cid:90) t + x | t − x | (cid:16)(cid:2) ( x + t ) − z (cid:3)(cid:2) z − ( x − t ) (cid:3)(cid:17) α + β − / z − β f ( z ) dz,U α,βt, f ( x ) = x − α − / t α +2 β (cid:90) t + x | t − x | (cid:2) t − ( x − z ) (cid:3) α + β − / z α +1 / f ( z ) dz, (cid:101) U α,βt, f ( x ) = x − α − / t α +2 β (cid:90) t + x | t − x | (cid:2) t − ( x − z ) (cid:3) α + β − / log (cid:18) xz ( x + z ) − t (cid:19) z α +1 / f ( z ) dz,V α,βt, f ( x ) = χ { x Let α > − , α + β > − / , δ ∈ R and < p < ∞ . (a) The following are necessary conditions for each of U α,β ∗ , , U α,β ∗ , , (cid:101) U α,β ∗ , to be well defined andbounded on L p ( R + , x δ dx ) : (a1) δ < (2 α + β + 1) p − , (a2) p < α + β + 1 / , (a3) − βp ≤ δ . (b) The following are necessary conditions for each of V α,β ∗ , , V α,β ∗ , , (cid:101) V α,β ∗ , to be well defined andbounded on L p ( R + , x δ dx ) : (b1) δ < (2 α + 2) p − , (b2) − < δ . In the proof of Lemma 4.1 we will need a simple technical result. Lemma 4.2. Let γ > − and ≤ A < B . Then (cid:90) BA (cid:16)(cid:2) B − z (cid:3)(cid:2) z − A (cid:3)(cid:17) γ z dz = Γ( γ + 1) γ + 2) (cid:0) B − A (cid:1) γ +1 . Proof. Changing the variable ( z − A ) / ( B − A ) = s we immediately see that the integral inquestion is equal to 12 (cid:0) B − A (cid:1) γ +1 (cid:90) s γ (1 − s ) γ ds. The last integral is the well known Euler beta integral. The conclusion follows. (cid:3) Proof of Lemma 4.1. We will construct suitable counterexamples. Part (a). Take f ( z ) = χ (1 , ( z ). Then for large x we have the bounds U α,βx − , f ( x ) (cid:38) x − α − β x α +2 β (cid:90) x α +2 β − ( z − α + β − / dz (cid:39) x α + β +1 ,U α,βx − , f ( x ) (cid:38) x − α − / x α +2 β (cid:90) x α + β − / ( z − α + β − / dz (cid:39) x α + β +1 . Thus a necessary condition for each of U α,β ∗ , , U α,β ∗ , and (cid:101) U α,β ∗ , to map L p ( x δ dx ) into itself is that thefunction x (cid:55)→ x − α − β − is in L p ( x δ dx ) for large x , which is equivalent to (a1).Next, choose f ( z ) = χ (1 , ( z )( z − − α − β − / / log z − . This function belongs to L p ( x δ dx ) when p ≥ α + β + 1 / 2. However, estimating similarly as above, for large x we get U α,βx − , f ( x ) (cid:38) x α + β +1 (cid:90) z − 1) log( z − ) dz = ∞ , and in the same way U α,βx − , f ( x ) = ∞ . This shows that (a2) is necessary for each of U α,β ∗ , , U α,β ∗ , and (cid:101) U α,β ∗ , to be well defined on L p ( x δ dx ).Now let f N ( z ) = χ ( N − ,N +1) ( z ) z β with N large. Then for x ∈ (1 , 2) we have the estimates U α,βN, f N ( x ) (cid:39) N α +2 β (cid:90) N +1 N − (cid:16)(cid:2) ( x + N ) − z (cid:3)(cid:2) z − ( x − N ) (cid:3)(cid:17) α + β − / z dz (cid:39) , where the last relation follows from Lemma 4.2. Furthermore, still for x ∈ (1 , U α,βN, f N ( x ) (cid:39) N α +2 β (cid:90) N +1 N − (cid:2) ( N − x + z )( N + x − z ) (cid:3) α + β − / N α +1 / N β dz (cid:39) (cid:90) N +1 N − ( N + x − z ) α + β − / dz (cid:39) . All these bounds hold uniformly in x and N . On the other hand, when δ < − βp it is straightforwardto check that the norms of f N in L p ( x δ dx ) tend to zero as N → ∞ . This gives the necessity of(a3). Part (b). Observe that for x > f ≥ V α,β x, f ( x ) (cid:39) x α +2 β (cid:90) x (cid:2) x − ( x − z ) (cid:3) − α − / (cid:2) x − ( x + z ) (cid:3) α + β − / z α +1 f ( z ) dz (cid:38) x α +2 β x − α − (cid:90) x/ (cid:2) x − ( x + z ) (cid:3) α + β − / z α +1 f ( z ) dz (cid:39) x α +2 (cid:90) x/ z α +1 f ( z ) dz,V α,β x, f ( x ) (cid:38) x α +2 β (cid:90) x/ (cid:2) x − ( x − z ) (cid:3) β − z α +1 f ( z ) dz (cid:39) x α +2 (cid:90) x/ z α +1 f ( z ) dz. Take now f ( z ) = χ (0 , ( z ) z − ( δ +1) /p / log z . As easily verified, this function belongs to L p ( x δ dx ).Hence, in view of the above estimates, a necessary condition for each of V α,β ∗ , , V α,β ∗ , and (cid:101) V α,β ∗ , to bewell defined on L p ( x δ dx ) is that the function z (cid:55)→ z α +1 z − ( δ +1) /p / log z is integrable at 0 + , whichis equivalent to (b1).Finally, consider f ( z ) = χ (0 , ( z ) z − δ . Assume that δ ≤ − 1. Then f ∈ L p ( x δ dx ). Further,choosing t = 2, for x ∈ (0 , 1) we have V α,β , f ( x ) (cid:39) (cid:90) − x (cid:2) − ( x − z ) (cid:3) − α − / (cid:2) − ( x + z ) (cid:3) α + β − / z α +1 − δ χ (0 , ( z ) dz AXIMAL ESTIMATES FOR SPHERICAL MEANS 19 (cid:38) (cid:90) / (cid:2) − ( x − z ) (cid:3) − α − / (cid:2) − ( x + z ) (cid:3) α + β − / z α +1 − δ dz (cid:39) (cid:90) / z α +1 − δ dz (cid:39) ,V α,β , f ( x ) (cid:39) (cid:90) − x (cid:2) − ( x − z ) (cid:3) β − z α +1 − δ χ (0 , ( z ) dz (cid:39) (cid:90) z α +1 − δ dz (cid:39) . This implies that none of V α,β ∗ , , V α,β ∗ , and (cid:101) V α,β ∗ , maps L p ( x δ dx ) into itself. The necessity of (b2)follows.The proof of Lemma 4.1 is complete. (cid:3) Proof of Proposition 1.6. Let α > − α + β > − / 2. For ε > E ε = (cid:8) ( t, x, z ) ∈ E : t − | x − z | < ε √ xz or x + z − t < ε √ xz (cid:9) ,F ε = (cid:8) ( t, x, z ) ∈ F : t − ( x + z ) < ε √ xz or t > ε − √ xz (cid:9) . From the results of [2], see [2, Theorem 3.3] and the accompanying comments, it follows that foreach pair α, β there exists ε = ε ( α, β ) > K α,βt ( x, z ) does not change sign in E ε and in F ε and, moreover, | K α,βt ( x, z ) | is comparable in E ε and in F ε with the correspondingexpressions on the right-hand side of the relevant bound from (1)–(8) in Theorem 2.1.Therefore the unboundedness results obtained for U α,β ∗ , , U α,β ∗ , , (cid:101) U α,β ∗ , , V α,β ∗ , , V α,β ∗ , , (cid:101) V α,β ∗ , in Lemma4.1 will also be valid for M α,β ∗ (with α, β suitably restricted according to the splitting in (1)–(8)in Theorem 2.1) provided that we assure that ( t, x, z ) involved in the counterexamples are locatedeither in E ε or in F ε .In the counterexamples from part (a) of the proof of Lemma 4.1 one can consider either x largeenough (in the first and the second counterexamples) or N large enough (in the third counterex-ample) so that all ( t, x, z ) involved are contained in E ε . In part (b), in the first counterexampleone considers x sufficiently large, while in the second one x in a sufficiently small neighborhood of0, so that all ( t, x, z ) involved belong to F ε .We conclude that necessary conditions for M α,β ∗ to be well defined and bounded on L p ( x δ dx )are (a1), (a2), (a3) of Lemma 4.1, and in case − β / ∈ N also (b1) and (b2). However, since when − β ∈ N conditions (b1) and (b2) are less restrictive than (a1) and (a3), respectively, actually all(a1), (a2), (a3), (b1), (b2) are always the necessary conditions. (cid:3) Remark 4.3. The discrepancy between the sufficient conditions in Theorem 1.5 and the necessaryconditions from Proposition 1.6 is related to region F (cid:48)(cid:48) and the special maximal operator T η . We donot know if the restriction of M α,β ∗ to F (cid:48)(cid:48) is always controlled by means of T η in an optimal way.Neither we know whether the description of mapping properties of T η from Lemma 2.3 is sharp.These issues remain to be investigated. References [1] J. Bourgain, Averages in the plane over convex curves and maximal operators , J. Anal. Math. (1986), 69–85.[2] ´O. Ciaurri, A. Nowak, L. Roncal Two-weight mixed norm estimates for a generalized spherical mean Radontransform acting on radial functions , SIAM J. Math. Anal. 49 (2017), 4402–4439.[3] L. Colzani, A. Cominardi, K. Stempak, Radial solutions to the wave equation , Ann. Mat. Pura Appl. 181 (2002),25–54.[4] J. Duoandikoetxea, A. Moyua, O. Oruetxebarria, The spherical maximal operator on radial functions , J. Math.Anal. Appl. 387 (2012), 655–666.[5] J. Duoandikoetxea, A. Moyua, O. Oruetxebarria, Estimates for radial solutions to the wave equation , Proc.Amer. Math. Soc. 144 (2016), 1543–1552.[6] C. Miao, J. Yang, J. Zheng, On local smoothing problems and Stein’s maximal spherical means , Proc. Amer.Math. Soc. 145 (2017), 4269–4282.[7] G. Mockenhaupt, A. Seeger, C. Sogge, Wave front sets, local smoothing and Bourgain’s circular maximaltheorem , Ann. of Math. 136 (1992), 207–218. [8] E. M. Stein, Maximal functions. I. Spherical means , Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174–2175.( ´O. Ciaurri) Departamento de Matem´aticas y Computaci´on, Universidad de La Rioja, 26004 Logro˜no,Spain E-mail address : [email protected] (A. Nowak) Institute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, 00-656 Warszawa,Poland E-mail address : [email protected] (L. Roncal) BCAM - Basque Center for Applied Mathematics, 48009 Bilbao, Spain and Ikerbasque,Basque Foundation for Science, 48011 Bilbao, Spain E-mail address ::