Weak Type (1, 1) Estimates for Maximal Operators Associated with Various Multi-Dimensional Systems of Laguerre Functions

Transcription

1 PREPRINT 2005:37 Weak Type 1, 1 Estimates for Maximal Operators Associated with Various Multi-Dimensional Systems of Laguerre Functions ADAM NOWAK PETER SJÖGREN Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY GÖTEBORG UNIVERSITY Göteborg Sweden 2005

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3 Preprint 2005:37 Weak Type 1, 1 Estimates for Maximal Operators Associated with Various Multi-Dimensional Systems of Laguerre Functions Adam Nowak and Peter Sjögren Department of Mathematical Sciences Division of Mathematics Chalmers University of Technology and Göteborg University SE Göteborg, Sweden Göteborg, October 2005

4 Preprint 2005:37 ISSN Matematiskt centrum Göteborg 2005

5 WEAK TYPE 1, 1 ESTIMATES FOR MAXIMAL OPERATORS ASSOCIATED WITH VARIOUS MULTI-DIMENSIONAL SYSTEMS OF LAGUERRE FUNCTIONS ADAM NOWAK AND PETER SJÖGREN Abstract. We prove that maximal operators associated with heat-diffusion semigroups corresponding to expansions with respect to different systems of multi-dimensional Laguerre functions are of weak type 1, Introduction and Preliminaries Consider the classical, d-dimensional heat semigroup T t = expt. As is well known, T t f = W t f, where W x = 4π d/2 exp x 2 /4 is the Gauss-Weierstrass kernel in R d and W t = t d W /t. Given f L p, it is a natural and important problem to determine whether T t f f a.e. as t 0 +. Such convergence holds indeed for all f L p, 1 p < and is a consequence of the mapping properties of the associated maximal operator T f = sup t>0 T t f ; precisely, T is bounded on L p, 1 < p < and of weak type 1, 1. This, in turn, follows from the fact that, since W is positive, radial and radially decreasing and W L 1 = 1, we have T f Mf, where M denotes the centered Hardy-Littlewood maximal operator, see [Du, Chapter 2]. An analogous situation occurs when one considers heat semigroups corresponding to various orthogonal expansions: almost everywhere convergence on the boundary can be obtained from suitable estimates for the associated maximal operators. Of particular interest are the so-called classical orthogonal expansions. Let T be a maximal operator corresponding to either the Jacobi, Hermite or Laguerre polynomial multi-dimensional semigroup. Since these are symmetric diffusion semigroups, it follows by Stein s general maximal theorem cf. [St] that T is bounded on L p Ω, dµ, 1 < p <, where Ω, dµ is an appropriate measure space. However, the case p = 1 is different: T fails to be L 1 -bounded and the natural candidate for a substitute result is weak type 1, 1. Unfortunately, no general tool analogous to Stein s maximal theorem is known to give weak type 1, 1, hence each of the three settings has to be handled separately. Furthermore, the corresponding heat kernels are not of convolution type, and proving weak type 1, 1 for 2000 Mathematics Subject Classification: primary 42C10; secondary 42B25 Key words and phrases: Laguerre semigroups, Laguerre functions, maximal operators, weak type 1, 1; Research of both authors supported by the European Commission via the Research Training Network Harmonic Analysis and Related Problems. 1

6 2 A.NOWAK AND P.SJÖGREN T is much more involved than in the case of the classical heat semigroup. For the onedimensional Hermite and Laguerre semigroups, the weak type 1, 1 of T was proved by Muckenhoupt [Mu]; however, the methods he used are inapplicable in higher dimensions. The first multi-dimensional result was obtained by one of the authors, for the Hermite semigroup [Sj1]. The case of the multi-dimensional Laguerre maximal operator is due to Dinger [Di]; her proof is long and technical. Considering the Jacobi setting, the weak type 1, 1 of T is not known even in dimension one the main reason is that, so far, no explicit formula for the Jacobi heat kernel is known. The same question about the mapping properties of T arises when one deals with classical orthogonal function systems, which arise from classical polynomial systems by multiplication by suitable factors and/or a change of variable. In the Hermite case there is essentially only one possibility, and the Hermite functions are defined in one dimension by, see [Th1], h k x = π2 k k! 1/2 H k xe x2 /2, x R, k = 0, 1, 2,..., H k being the kth Hermite polynomial, cf. [Le]. The multi-dimensional Hermite functions are simply tensor products of these h k. However, in the Laguerre case there are several possibilities. Let L α k denote the onedimensional Laguerre polynomials with parameter α > 1, defined on 0, for k = 0, 1,..., cf. [Le, Chapter 4]. It is reasonable to consider the following four function systems, all defined for x > 0 : l α k x = k!/γk + α + 1 1/2 L α k xe x/2, α > 1; ψ α k x = 2 α/2 l α k x 2 /2, α > 1; L α k x = l α k xx α/2, α 0; ϕ α k x = 2l α k x 2 x α+1/2, α 1/2. The corresponding multi-dimensional systems are formed by taking tensor products. For each of these function systems, there exists a natural measure for which the system constitutes an orthonormal basis in the corresponding L 2 space. More precisely, {h k } is connected with R d, dx, and {L α k } and {ϕα k } both with Rd +, dx, where we denote R + = 0,. Further, {l α k } is connected with Rd +, dµ α, where dµ α x = x α 1 1 x α d d dx, and ψk α with Rd +, dη α ; here dη α x = x 2α x 2α d+1 d dx. The function systems under consideration were investigated by many authors, including Askey, Markett, Muckenhoupt, Stempak, Thangavelu and Torrea, see [Th1], [Stem], [StTo] and references there. A motivation for a study of these function systems comes from the fact that the corresponding expansions behave, in some sense, better than polynomial expansions, cf. [St]. Moreover, most of the systems originate naturally elsewhere. For example, it is well known that the Hermite functions h k are eigenfunctions of the Fourier transform as well as of the harmonic oscillator L = + x 2, whereas the Laguerre

7 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 3 functions ϕ α k are eigenfunctions of the Hankel transform, and Lα k and ψα k appear in analysis on the Heisenberg group, cf. [Th1, Th2]. Furthermore, it is worth noting that if α has one of the two special forms α = ±1/2,..., 1/2, then expansions with respect to {ϕ α k } are directly related to Hermite function expansions, see for instance [NoSt]. A similar connection with the Hermite setting holds for {ψk α } and α = 1/2,..., 1/2. Now, let T be the maximal operator of the heat semigroup associated with one of these function expansions. In the Hermite case, it is easy to prove that T is bounded on L p, 1 < p and of weak type 1, 1 since, by the Feynman-Kac formula, the corresponding heat kernel is dominated by the Gauss-Weierstrass kernel, cf. [StTo, p. 453]. The case of the system {ϕ α k } is similar: the heat kernel can be estimated from above by a constant C times a dilation of the Gauss-Weierstrass kernel, see [No, p. 262]. Moreover, one can take C = 1 if all the coordinates of α are in [1/2,, and C = 2 d if α i { 1/2} [1/2,, i = 1,..., d, cf. [NoSt, Proposition 2.1]. However, the treatment of the three remaining systems is not so simple. For {l α k } and {L α k }, the quantity T f can be estimated by a constant times M s f, where M s is the strong maximal operator with respect to the measure µ α or Lebesgue measure, respectively, see [No]. This implies the L p boundedness, 1 < p <, of T, which, by a change of variable argument, also carries over to T for {ψk α}. Unfortunately, M s is not of weak type 1, 1 unless the setting is one-dimensional, in which case M s coincides with the Hardy- Littlewood maximal operator. Thus, it is known so far that in one dimension T is of weak type 1, 1 for {l α k } and {Lα k }, results originally obtained by Stempak [Stem]. In higher dimensions only partial results are known for {l α k } and {ψα k } : in [No] it was observed that for these systems the maximal operators are of weak type 1, 1 for a discrete set of half-integer type multi-indices α. This follows by transferring analogous result from the Hermite function setting. The purpose of the present paper is to prove weak type 1, 1 of T in arbitrary finite dimension for the three Laguerre systems in question. Let us state the main result. Theorem 1.1. Let T t be the d-dimensional, d 1, heat semigroup associated with one of the following orthonormal systems of Laguerre functions: {l α k }, {ψ α k }, {L α k }. Assume that α 1, d for the first and the second system, and that α [0, d in the case of the third system. Then the maximal operator T f = sup t>0 T t f is of weak type 1, 1 with respect to the appropriate measure µ α, η α or Lebesgue, respectively. Corollary 1.2. Let T t be as in the theorem above. Then lim T tf = f t 0 + a.e. for each f L 1 dω, ω being the appropriate measure.

8 4 A.NOWAK AND P.SJÖGREN The idea of the proof of Theorem 1.1 is in part similar to that of Dinger [Di], in the Laguerre polynomial case. However, the present settings are less sophisticated due to the fact that the corresponding measures possess the doubling property. This allows us, in comparison with [Di] to avoid many splittings, decompositions and computations, especially in the longest and most technical part of the proof. Instead, roughly speaking, after a suitable partition of the underlying space, we extract in a series of estimates anisotropically dilated convolution kernels and take advantage of the theory of hierarchical partitions from [Sj2]. The remaining part of the paper is devoted to the proof of Theorem 1.1. We remark that the weak type 1, 1 constants emerging from our proof depend strongly on the dimension. As a matter of fact, for any of the abovementioned settings, it is an important and open problem to obtain weak type 1, 1 constants for the heat semigroup maximal operator independent of the dimension. We shall use the following conventions. By c > 0 and C < we will always denote constants whose values may change from one occurrence to another; these constants will usually depend on the dimension d and the type multi-index α. Any other dependence will be indicated by means of suitable subscripts. If c f/g C for some c and C, we will write shortly f g. Similarly, we abbreviate f Cg to f g. Given a multi-index m = m 1,..., m d R d, its length m m d will be denoted by m notice that this quantity may be negative. 2. Proof of Theorem 1.1; the system {l α k } In this section we assume that α = α 1,..., α d 1, d. Recall that the system of Laguerre functions {l α k : k Nd } forms an orthonormal basis in L 2 R d +, dµ α, where R d + = 0, d, dµ α x = x α 1 1 x α d d dx. Each lα k is an eigenfunction of the differential operator d L = x 2 i + α i + 1 x i, x i 4 i=1 x 2 i which is symmetric in L 2 dµ α and positive. The corresponding eigenvalue is k + α +d/ 2. Moreover, L has a self-adjoint extension for which the spectral decomposition is given by the l α k. Hence the associated heat semigroup T t = exp tl is defined for f L 2 dµ α by T t fx = e tn+ α +d/2 f, l α k l α k x, t > 0, x R d +. n=0 k =n The above series provides also a definition of T t on L 1 dµ α, but for our purposes it is appropriate to use an integral representation instead see [No]: T t fx = e t α +d/2 G α e tx, yfy dµ αy, f L 1 dµ α,

9 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 5 where the kernel is given by G α r x, y = d i=1 Gα i r x i, y i with G a rv, w = 1 exp r rvw a 2 rvw v + w Ia 2 for v, w > 0, 0 < r < 1 and a > 1. Here I a is the modified Bessel function of the first kind and of order a, cf. [Le]. Notice that G α r x, y is strictly positive for x, y R d +, 0 < r < 1. Also, by using the asymptotics I a x x a, x 0 +, I a x x 1/2 e x, x, and the fact that I a is continuous on 0,, we see that G a rv, w D a r v, w + E a r v, w, v, w > 0, 0 < r < 1, a > 1, where Dr a v, w = a+1 exp r 2 a+1/2 rvw Er a v, w = exp 1 2 v + w χ {rvw<1 r 2 }, 1 + rv + w 4 rvw χ {rvw 1 r 2 }. Note that the argument of the exponential function in the last expression is equal to r 1 v w r 2 v + w. 2 Our task is to show that T is of weak type 1, 1. In fact, we are going to show a slightly stronger result, namely that the operator 2 G α fx = sup G α r x, yfy dµ α y 0<r<1 is of weak type 1, 1. The plan of proof is the following cf. [Di]. If the supremum in 2 is restricted to 0 < r 1/2, the operator obtained is easily seen to be bounded on L 1 dµ α, see Lemma 2.1 below. For 1/2 < r < 1 we consider for each i {1,..., d} the three cases a y i 2x i, b x i /2 < y i < 2x i and c y i x i /2. The maximal operator is then split into 3 d operators. If case a occurs for all i, the corresponding operator is easily seen to be bounded on L 1, as verified in Lemma 2.3. Moreover, a simple argument involving the product structure and the symmetry of the kernel makes it possible to assume that a occurs for no i. When case c occurs for all i, a straightforward estimate of the kernel, by an expression independent of r, gives the required control of the corresponding operator; see Lemma 2.4. Next, if one has case b for all i, the space is split into dyadic rectangles. In each rectangle, the kernel is controlled by an anisotropic dilation of the standard heat kernel, and the operator is easy to handle. The remaining possibility, that both b and c occur, is the most complicated, and the argument combines the last two methods just described. The final stroke uses the technique of a piece hierarchy developed in [Sj2].

10 6 A.NOWAK AND P.SJÖGREN Lemma 2.1. The operator f sup G α r, y fy dµ α y 0<r<1/2 is bounded on L 1 dµ α. Proof. Assuming that 0 < r < 1/2, we get D a r v, w exp v + w/2, E a r v, w rvw a+1/2 exp rv + w 4 rvw χ {rvw 1 r 2 }. Taking 1 into account, we obtain This gives E a r v, w rvw a+1/2 exp cv + w χ {rvw 1 r 2 } = rvw 1/2 rvw a+1 exp cv + w χ{rvw 1 r 2 } v + w 1/2 exp cv + w a+1 exp cv + w exp cv + w. G α r x, y exp c d x i, 0 < r < 1/2. i=1 The right-hand side above is independent of r, and it corresponds to an integral operator which is obviously bounded on L 1 dµ α. We next give estimates for the one-dimensional kernel in the three cases a, b, c introduced above. Lemma 2.2. Let a > 1. The one-dimensional kernel G a rv, w satisfies the following estimates, uniformly for 1/2 r < 1. a For w 2v, b For 1/2 < w/v < 2, G a rv, w v a 1 v exp G a rv, w v a+1/2 w a+1/2. c v w2 v + v a+1/2 w a+1/2. c For w v/2, G a rv, w a+1 exp v c.

11 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 7 Proof. Assume first that rvw < 2, so that G a rv, w D a r v, w. In the cases a and b we get D a r v, w a+1 rvw a+1/2 v a+1/2 w a+1/2. Considering c, we obtain Dr a v, w a+1 exp c v + w a+1 exp v c Assume next rvw 1 r 2, so that G a rv, w E a r v, w. In the case a we see that the first term in 1 is, up to a multiplicative constant, less than w/1 r vw/1 r. This implies that E a r v, w vw a+1/2 vw a+1/2 exp = v a+1/2 w a+1/2. In the case b, the first term of 1 is at most vw c vw 1/2 v w 2 c v + w2 v w 2 v. Here vw v, and we conclude that a+1/2 vw Er a v w2 v, w exp c v v a 1 exp c v. v w2 v In the case c, the first term of 1 is not greater than cv/. We first assume that a 1/2. Since rvw 2, we have a+1/2 rvw Er a v v, w exp c a+1 v exp c. When instead a < 1/2, we use the fact that rvw v to write a+1/2 rvw Er a v v, w exp c v a+1/2 v a+1/2 v exp c = a+1 v exp c. Altogether, this proves the lemma.. Lemma 2.3. The operator A α fx = sup G α r x, yχ {yi 2x i,i=1,...,d} fy dµ α y 1/2 r<1

12 8 A.NOWAK AND P.SJÖGREN is bounded on L 1 dµ α. Proof. By Lemma 2.2 a, for 1/2 r < 1 we have d G α r x, y x α i+1/2 i y α i+1/2 i. Consequently, A α fx dµ α x d i=1 i=1 y α i+1/2 i fy dµ α y. yi /2 0 x α i+1/2 i x α i i dx i fy dµ α y Lemma 2.4. The operator fx sup 1/2 r<1 is bounded from L 1 dµ α to L 1, dµ α. Proof. For n {0, 1,..., d} define G α r x, yχ {yi 2x i,i=1,...,d} fy dµ α y W n = { x, y : x i /2 < y i < 2x i for i = 1,..., n and y i x i /2 for i = n + 1,..., d }, with the natural interpretation when n = 0 or n = d. In view of the product structure and the symmetry of the kernel, it is sufficient to prove that for each n the operator B α fx = G α r x, yχ Wn x, y fy dµ α y sup 1/2 r<1 is of weak type 1, 1 with respect to the measure µ α for all α 1, d. In what follows points in R d will be written x = x, x R n R d n, and and will denote sums over 1 i n and n + 1 j d, respectively, and similarly for products. Further, for m = m 1,..., m n Z n we denote Q m = {x : 2 m i < x i 2 m i+1, i = 1,..., n}, Q m = {x : 2 m i 1 < x i 2 m i+2, i = 1,..., n}. Notice that if x, y W n, then the condition x Q m forces y to stay in Q m. Thus we have B α fx = Bmfx, α x Q m R d n +, where the restricted operators Bm α are defined by Bmf α = B α fχ Qm R d n + Observe that by Lemma 2.2 b,c, the kernel can be estimated, up to a multiplicative constant, by the tensor product of the factors 3 x α i i 1 xi exp c x i y i 2 x i. + x α i+1/2 i y α i+1/2 i, i n,

13 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 9 and 4 αj+1 exp c x j, j > n. Moreover, the second term in 3 may be neglected in further reasoning. This is a consequence of [Di, Proposition 1] and the fact that the operator with the integral kernel d i=1 x α i+1/2 i y α i+1/2 i χ {xi /2<y i <2x i } is independent of r and bounded on L 1 dµ α in any dimension d, cf. the proof of Lemma 2.3. We can thus consider the kernel [ x α 5 i 1 i exp c x ] [ i y i 2 1 r α j +1 exp c x ] j xi x i in place of G α r x, y. Let S = α j + 1. Case 1: n = 0. The product in 5 is xj S S x j x j xj S. exp c Hence the value at x of the operator B α applied to a function f is not greater than C xj S f L 1 dµ α. It is now enough to verify that the function x x j S is in L 1, dµ α, which is straightforward. Case 2: n = d. We shall consider each Q m, m Z n, separately. Observe that in each Q m the measure µ α is essentially proportional to Lebesgue measure. The constant of proportionality will cancel against the factors x α i i in 5. This means that Bmf α can be controlled on Q m, uniformly in m, by the supremum in r < 1 of the convolution with the Euclidean heat kernel, dilated by the factor C 2 m i in the ith variable. It is enough to verify that the maximal operator thus defined is of weak type 1, 1 with respect to Lebesgue measure, uniformly in m. To this end, fix δ = δ 1,..., δ n R n +, write δx = δ1 1 x 1,..., δn 1 x n and let Wt δ x = δ 1 δ n 1 W t δx be the δ-anisotropic dilation of the standard heat kernel W t in R n. Then it is easily verified that Wt δ fx = W t f δ 1 δx for reasonable functions f. Consequently, W δ fx = sup W δ t fx M f δ 1 δx, t>0 where M is the Hardy-Littlewood maximal operator. It follows that W δ is of weak type 1, 1, uniformly in δ. Therefore, with the notation h m y = χ Qmyfy y α i i, we have, uniformly in m, µ α {x Q m : B α fx > λ} = µ α {x Q m : B α mfx > λ}

14 10 A.NOWAK AND P.SJÖGREN µ α {x Q m : C x α i i } W δ h m x > λ 2 m i α i { x : W δ h m x > cλ2 m i α i } λ 1 h m L 1, where δ R n + satisfies δ i 2 mi/2. Clearly, h m L 1 = fy dµ Qm α y and since the Q m have bounded overlap, the estimates obtained can be summed over m Z n. This finishes Case 2. Case 3: 0 < n < d. Here we combine the methods of the two preceding cases. First, we shall see that it is enough to let = x j in 5. Estimating the first exponential in 5 by a negative power, we see that for any ε > 0 the kernel 5 is not greater than [ x α 6 C i 1/2 i S/n 1/2 1 + x ] i y i 2 1/2 ε x j exp c. x i Now fix ε = S/n and observe that the expression in the square bracket here is decreasing in. On the other hand, since x 1/2 εn j x j exp c, the entire expression in 6 is not greater than [ x α C i 1/2 xj 1/2 ε i 1 + x ] i y i 2 1/2 ε, x i which in turn is increasing in. Using these two monotonicity properties for > x j and < x j, respectively, we see that the whole quantity in 6 is at most C times its value at = x j here we neglect the restriction < 1, which would only make the kernel smaller. This means that the kernel is dominated by 7 C xj S x α i i 1 x i x j 1 + x i y i 2 1/2 ε, x i x j an expression which does not contain r. It remains to prove that the integral operator defined by 7 satisfies a weak type 1, 1 inequality with respect to the measure µ α. With y in Q m, we carry out the integration in y and define g m y = y α i i R d n + fy y α j j dy. We observe that the L 1 norm of g m, written g m L 1 Q m and taken with respect to the n-dimensional Lebesgue measure in Q m, is equal to that of f with respect to µ α in

15 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 11 Q m R d n +. Moreover, at a point x with x in Q m, the restricted operator B α m is controlled by the integral with respect to dy of g m against the expression in 7. Hence B α mfx 2 m i α i H m x g mx, x Q m R d n +, uniformly in m Z n ; here the power of 2 comes from the factors x α i i in 7, the convolution is taken in R n, and the kernel Hx m is given by xj S Hx m 1 x 2 1/2 ε x i = m i x 2 m i. x j j Now it follows that for 4 k 1 < x j 4 k written x j 4 k we have B α mfx T m k g m x, uniformly in m Z n and k Z, with and x Q m R d n +, T m k g m x = 4 Sk 2 m i α i H m k g m x Hk m x = 1/2 ε x2 i. 4 k 2 m i 4 k 2 m i Notice that Hk m arises as an anisotropic dilation of the function 1 + x 2 i 1/2 ε. Therefore, given an arbitrary λ > 0, we have { µ α x Q m R d n + : Bmfx α > λ } { µ α x Q m R d n + : Tk m g m x χ { } } λ x j 4 k k Z = { µ α x Q m R+ d n : Tk m g m x χ { } } λ x j 4 k k Z k Z 2 m i α i 4 Sk { x Q m : T m k g m x λ }. In the above, all relations and are uniform in m, k and λ, and the factor 2 m i α i 4 Sk comes from the density of µ α. It is now enough to prove that for all λ > 0 m 8 i α i 4 Sk { x Q m : Tk m g m x > λ } C λ g m L 1 Q m, k=k 0 2 with C independent of the negative integer k 0 and of m Z n. Let z = 4 S, so that 0 < z < 1, and define λ = 2 m i α i λ. Then 8 is equivalent to 9 { x Q m : Hk m g m x > z k λ } C λ g m L 1 Q m. k=k 0 z k Consider the one-dimensional kernel s 1 4 k 2 m i 1 + s2 4 k 2 m i 1/2 ε

16 12 A.NOWAK AND P.SJÖGREN occurring in the definition of Hk m. For ν = 0, 1,... and 2ν 1 2 k 2 mi/2 s < 2 ν 2 k 2 m i/2 the value of this kernel is of the order of magnitude 2 2εν 2 ν 2 k 2 mi/2, and the kernel can be estimated from above by C ν=0 2 2εν 1 2 ν 2 k 2 m i/2 χ { s <2 ν 2 k 2 m i /2 }. To write a similar estimate for the n-dimensional kernel H m k, we let now ν = ν 1,..., ν n be a multi-index. Denote by R k ν the rectangle in R n with center 0 and sides 2 ν i 2 k 2 m i/2. The one-dimensional estimate given above immediately leads to 10 H m k x ν N n 2 2ε ν 1 R k ν χ R k ν x, uniformly in k and m. We shall now fix ν N n. In 9 we shall replace H m k by Rk ν 1 χ R k ν and prove that 11 k=k 0 z k { x Q m : R k ν 1 χ R k ν g m x > z k λ } C λ g m L 1 Q m, uniformly in ν, m and k 0. Because of the rapidly decreasing factor 2 2ε ν in 10, these estimates can then be added in ν in L 1, by means of the summation theorem due to E.M. Stein and N. Weiss, cf. [SW, Lemma 2.3], see also [Di, Theorem B] for a slightly restated version. This will imply 9 and end the proof of Lemma 2.4. For the sake of clarity and the reader s convenience, we start with the case ν = 0,..., 0, and afterwards indicate the modifications needed for an arbitrary ν. Write R k = R0,...,0 k. The convolution Rk 1 χ R k g m x is simply the mean value of g m in the rectangle x + R k, which is centered at x. For each k {k 0, k 0 + 1,...} we now split R n into rectangles of sides 2 k 2 mi/2 and thus congruent to R k. These rectangles will be called k-pieces, and the splittings shall be successive refinements of each other, so that each k + 1-piece is contained in some k-piece. We then have a piece hierarchy in the sense of [Sj2]. Now if instead of R k 1 χ R k g m x we had the mean of g m in the k-piece containing x, [Sj2, Lemma 1] would imply 11 for ν = 0,..., 0 even though this lemma is stated for a sum starting at k = 1, our case follows immediately, since we can multiply λ by a suitable power of z. In our situation, however, we have to modify this argument and we use a variant of [Sj2, Lemma 1] due to Dinger. Clearly, R k 1 χ R k g m x can be estimated by the sum of the means of g m in the k-piece containing x and its 3 n 1 adjacent neighbors. Proposition 2 of [Di] can be applied here to give an estimate for the mean in each adjacent k-piece, so using it 3 n times, we obtain 11 with ν = 0,..., 0. The case of an arbitrary ν is analogous. Now the k-pieces will have sides 2 ν i 2 k 2 mi/2, but otherwise the argument is the same. Lemma 2.4 is proved, and this ends the case of the system {l α k } in Theorem 1.1.

17 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 13 Remark. In fact we proved a slightly stronger result than weak type 1, 1 of B α. Indeed, excluding only the case n = d above, which is the local part, the operator f sup 1/2 r<1 G α r, yχ Wn, y fy dµ α y, with the supremum inside the integral, was shown to be bounded from L 1 dµ α to L 1, dµ α. 3. Proof of Theorem 1.1; the systems {ψ α k } and {Lα k } 3.1. The system {ψk α}. Here, as in the previous section, we assume that α 1, d. The system of Laguerre functions {ψk α : k Nd } is an orthonormal basis in L 2 R d +, dη α with dη α x = x 2α x 2α d+1 dx. Each ψk α is an eigenfunction of d L = d 2αi + 1 i=1 x i x2 i x i 4 a differential operator which is symmetric and positive in L 2 dη α, and the corresponding eigenvalue is 2 k + α + d. Moreover, L has a self-adjoint extension for which the spectral decomposition is given by the ψk α. Thus the associated heat semigroup is defined for f L 2 dη α by T t fx =, e t2n+ α +d f, ψk α ψk α x, t > 0, x R d +. n=0 k =n This definition extends to more general functions f, in particular to f L 1 dη α, either by means of the above series, or by the integral representation T t fx = e t α +d H α e 2tx, yfy dη αy, where, in view of the connection with the system {l α k } see Section 1, the kernel is given by Hr α x, y = d i=1 Hα i r x i, y i with one-dimensional kernels Hr a v, w = 2 a G a rv 2 /2, w 2 / 2 for v, w > 0 and a > 1. Consequently, given α 1, d, for 0 < r < 1, we have 12 H α r x, y = 2 α G α r φx, φy, x, y R d +, where φ is the one to one mapping of R d + onto itself defined by φx 1,..., x d = x 2 1/2,..., x 2 d/2. We claim that the operator H α fx = sup 0<r<1 Hr α x, yfydη α y

18 14 A.NOWAK AND P.SJÖGREN is of weak type 1, 1, which is slightly stronger than needed for Theorem 1.1. This can be easily justified in view of the result already obtained for the system {l α k }. Using the notation from Section 2, observe that 12 implies H α f φx = G α f φx, f L 1 dµ α, x R d +. Therefore, since G α is of weak type 1, 1 with respect to the measure µ α, we may write { η α x R d + : H α fx > λ } { = η α x R d + : H α f φ 1 φ x > λ } = χ {G α f φ 1 φx>λ} dη α x = 2 α χ {G α f φ 1 y>λ} dµ α y 2 α Cλ 1 f φ 1 y dµ α y = Cλ 1 fx dη α x. The assertion of Theorem 1.1 for the system {ψk α } is proved The system {L α k }. In this subsection we assume that α [0, d. This restriction is motivated by the fact that only for these α the functions L α k belong to all Lp spaces, 1 p <. The case when some of α i are negative is substantially different, and in the one-dimensional setting it was recently investigated in detail by Macías, Segovia and Torrea [MST]. In particular, if d = 1 and α 1, 0, there is no weak type 1, 1 inequality for the heat semigroup maximal operator. A simple reasoning shows that a similar negative result holds in higher dimensions as soon as α / [0, d. Recall that the system of Laguerre functions {L α k : k Nd } is an orthonormal basis in L 2 R d +, dx. Each L α k is an eigenfunction of the differential operator d L = x 2 i + x i x i 4 α2 i, 4x i i=1 x 2 i which is symmetric and positive in L 2 dx, and the corresponding eigenvalue is k + α + d/2. Moreover, L has a self-adjoint extension for which the spectral decomposition is given by the L α k. Then the associated heat semigroup is defined for f L2 dx by T t fx = e tn+ α +d/2 f, L α k L α k x, t > 0, x R d +, n=0 k =n and the integral representation, valid also for f L 1 dx, is T t fx = e t α +d/2 H α e tx, yfy dy.

19 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 15 Here the kernel is given by H α r x, y = d i=1 Hα i r x i, y i with H a rv, w = v a/2 w a/2 G a rv, w for v, w > 0 and a 0. It follows that for α [0, d and 0 < r < 1, 13 H α r x, y = G α r x, y d i=1 x α i/2 i y α i/2 i, x, y R d +. Our aim is to prove the weak type 1, 1 of the operator H α fx = sup Hr α x, yfy dy 0<r<1 again, this is slightly more than we need. By means of 13, this can be obtained in a straightforward manner from the results we already proved in Section 2. Here are the details. In the proof of Lemma 2.1, we obtained the estimate G a rv, w exp cv + w valid for 0 < r < 1/2 and v, w > 0. Since now a 0, this implies a similar estimate for H a rv, w and therefore Hr α x, y exp c d x i, 0 < r < 1/2. Consequently, we get the boundedness on L 1 dx of the operator f sup Hr α, y fy dy. 0<r<1/2 To deal with the range 1/2 r < 1, we need the following estimates. i=1 Lemma 3.1. Let a 0. The one-dimensional kernel H a rv, w satisfies the inequalities given below, uniformly in 1/2 r < 1. a For w 2v, H a rv, w v 1/2 w 1/2. b For 1/2 < w/v < 2, Hrv, a 1 v w2 w exp c + v 1/2 w 1/2. v v c For w v/2, H a rv, w 1 exp v c Proof. Items a and b are immediate consequences of Lemma 2.2 and 13. For item c we again invoke Lemma 2.2 and write Hrv, a w a+1 v a/2 w a/2 v exp c a v 1 exp v c 1 exp v c,.

20 16 A.NOWAK AND P.SJÖGREN since now w < v and a 0. Observe that the right-hand sides of the estimates in this lemma do not contain a, and that the estimates coincide with those of the case a = 0 of Lemma 2.2. On the other hand, only Lemma 2.2 together with the positivity, symmetry and product structure of the kernel were needed to prove Lemmas 2.3 and 2.4. Since for α = 0,..., 0 the measure µ α is Lebesgue measure, the proof of the weak type 1, 1 of the operator f sup 1/2 r<1 Hr α, yfy dy is already contained in the case α = 0,..., 0 of Lemmas 2.3 and 2.4. The proof of Theorem 1.1 is now complete. References [Di] U. Dinger, Weak-type 1, 1 Estimates of the Maximal Function for the Laguerre Semigroup in Finite Dimensions, Rev. Mat. Iberoamericana , [Du] J. Duoandikoetxea, Fourier Analysis, American Mathematical Society, Providence, [Le] N.N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, [MST] R. Macías, C. Segovia and J.L. Torrea, Heat-diffusion maximal operators for Laguerre semigroups with negative parameters, J. Funct. Anal., to appear. [Mu] B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc , [No] A. Nowak, Heat-diffusion and Poisson integrals for Laguerre and special Hermite expansions on weighted L p spaces, Studia Math , [NoSt] A. Nowak and K. Stempak, Riesz transforms and conjugacy for Laguerre function expansions, preprint 2004, p [Sj1] P. Sjögren, On the maximal function for the Mehler kernel, Harmonic analysis Cortona, 1982, 73 82, Lecture Notes in Math., 992, Springer, Berlin, [Sj2] P. Sjögren, Fatou theorems and maximal functions for eigenfunctions of the Laplace-Beltrami operator in a bidisk, J. Reine Angew. Math , [St] E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Math. Studies, Vol. 63, Princeton Univ. Press, Princeton, NJ, [Stem] K. Stempak, Heat-diffusion and Poisson integrals for Laguerre expansions, Tohoku Math. J , [StTo] K. Stempak and J.L. Torrea, Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal , [SW] E.M. Stein and N.J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc , [Th1] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes 42, Princeton University Press, Princeton, [Th2] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkhäuser, Boston, 1998.

21 WEAK TYPE 1, 1 OF MAXIMAL FUNCTIONS FOR LAGUERRE SEMIGROUPS 17 Adam Nowak, Institute of Mathematics and Computer Science, Wroc law University of Technology, Wyb. Wyspiańskiego 27, PL Wroc law, Poland address: anowak@pwr.wroc.pl Peter Sjögren, Department of Mathematics, Chalmers University of Technology and Göteborg University, S Göteborg, Sweden address: peters@math.chalmers.se