Sufficient conditions

In this section we give some sufficient conditions for a functionθ, which ensures that θˆ∈L1(R^d), resp. θ∈S0(R^d). As mentioned beforeθ∈S0(R^d)implies also thatθˆ∈L1(R^d). Recall thatS0(R^d)contains all Schwartz functions. Ifθ∈L1(R^d) andθˆhas compact support or ifθ∈L1(R^d)has compact support andθˆ∈L1(R^d) thenθ∈S0(R^d).

Sufficient conditions can be given with the help of Sobolev, fractional Sobolev and Besov spaces, too. For a detailed description of these spaces see Triebel [31], Runst and Sickel [21], Stein [26] and Grafakos [11].

A function θ ∈Lp(R^d)is in the Sobolev space W_p^k(R^d) (16p6∞, k ∈N) if D^αθ∈Lp(R^d)for all|α|6kand

kθk_Wk

p := X

|α|6k

kD^αθkp<∞,

whereD denotes the distributional derivative.

This definition is extended to every realsin the following way. Thefractional Sobolev space L^s_p(R^d) (16p6∞, s∈ R) consists of all tempered distribution θ for which

kθkL^s_p:=kF⁻¹((1 +| · |²)^s/2θ)kˆ p<∞.

It is known that L^s_p(R^d) =W_p^k(R^d)if s=k ∈Nand1 < p <∞with equivalent norms.

176 F. Weisz In order to define the Besov spaces take a non-negative Schwartz functionψ∈ S(R)with support [1/2,2]which satisfiesP_∞

k=−∞ψ(2^−ks) = 1 for alls∈R\ {0}.

Forx∈R^d let

φk(x) :=ψ(2^−k|x|) for k>1 and φ0(x) = 1− X∞

k=1

φk(x).

The Besov space B_p,r^s (R^d) (0 < p, r 6 ∞, s ∈ R) is the space of all tempered distributionsf for which

kfkB_p,r^s :=

³X^∞

k=0

2^ksrk(F⁻¹φk)∗fk^r_p

´_1/r

<∞.

The Sobolev, fractional Sobolev and Besov spaces are all quasi Banach spaces and if16p, r6∞then they are Banach spaces. All these spaces contain the Schwartz functions. The following facts are known: in case16p, r6∞one has

W_p^m(R^d), B_p,r^s (R^d),→Lp(R^d) if s >0, m∈N, W_p^m+1(R^d),→B_p,r^s (R^d),→W_p^m(R^d) if m < s < m+ 1, B_p,r^s (R^d),→B_p,r+²^s (R^d), B_p,∞^s+²(R^d),→B_p,r^s (R^d) if ² >0, B_p^d/p_1,1¹(R^d),→B_p^d/p_2,1²(R^d),→C(R^d) if 16p16p2<∞.

Theorem 6.1.

(i) If16p62andθ∈B_p,1^d/p(R^d)thenθˆ∈L1(R^d)and kθkˆ 16Ckθk_Bd/p

p,1. (ii) If s > dthenL^s₁(R^d),→S0(R^d).

(iii) Ifd⁰ denotes the smallest even integer which is larger thandands > d⁰ then

B^s_1,∞(R^d),→W₁^d0(R^d),→S0(R^d).

The embedding W₁²(R),→S0(R)follows from (iii). With the help of the usual derivative we give another useful sufficient condition for a function to be inS0(R^d).

A functionθis inV₁^k(R) (k>2, k∈N), if there are numbers−∞=a0< a1<

. . . < an< an+1=∞such thatn=n(θ)is depending onθand θ∈C^k−2(R), θ∈C^k(ai, ai+1), θ^(j)∈L1(R)

for alli= 0, . . . , nandj= 0, . . . , k. HereC^kdenotes the set ofk-times continuously differentiable functions. The norm of this space is introduced by

kθk_V^k

1 :=

Xk

j=0

kθ^(j)k1+ Xn

i=1

|θ^(k−1)(ai+ 0)−θ^(k−1)(ai−0)|

Wiener amalgams and summability of Fourier series 177 whereθ^(k−1)(ai±0)denote the right and left limits ofθ^(k−1). These limits do exist and are finite becauseθ^(k)∈C(ai, ai+1)∩L1(R)implies

θ^(k−1)(x) =θ^(k−1)(a) + Z _x

a

θ^(k)(t)dt

for some a ∈ (ai, ai+1). Since θ^(k−1) ∈ L1(R) we establish that lim−∞θ^(k−1) = lim∞θ^(k−1)= 0. Similarly,θ^(j)∈C0(R)forj= 0, . . . , k−2.

Of course,W₁²(R)andV₁²(R)are not identical. Forθ∈V₁²(R)we haveθ⁰=Dθ, however,θ⁰⁰=D²θonly if limai+0θ⁰ = limai−0θ⁰ (i= 1, . . . , n).

We generalize the previous definition for thed-dimensional case as follows. For d >1andk>2 letθ∈V₁^k(R^d)ifθ is even in each variable and

θ∈C^k−2(R^d), θ∈C^k([0,∞)^d\ {(0, . . . ,0)}), ∂₁ⁱ¹· · ·∂_d^idθ(t)∈L1([0,∞)^l) for eachij = 0, . . . , k (j = 1, . . . , d)and fixed0< tm1, . . . , tmd−l <∞ (16m1 <

m2<· · ·< md−l6d)and16l6d.

Theorem 6.2. If θ∈V₁²(R^d)thenθ∈S0(R^d).

The next Corollary follows from the definition ofS0(R^d).

Corollary 6.3. If each θj∈V₁²(R) (j= 1, . . . , d)thenθ:=Q_d

j=1θj∈S0(R^d).

7. A.e. convergence of the θ-means of Fourier trans-forms

For the a.e. convergence we will investigate first Fourier transforms rather than Fourier series, because the theorems for Fourier transforms are more complicated.

The proofs of the results can be found in Feichtinger and Weisz [4].

L^loc_p (X^d) (16p6∞) denotes the space of measurable functions f for which

|f|^p is locally integrable, resp. f is locally bounded if p = ∞. For 1 6 p 6 ∞ and f ∈ L^loc_p (X^d) let us define a generalization of the Hardy-Littlewood maximal function by

Mpf(x) := sup

x∈I

³ 1

|I|

Z

I

|f|^pdλ

´_1/p

(x∈X^d)

with the usual modification forp=∞, where the supremum is taken over all cubes with sides parallel to the axes. Ifp= 1, this is the usual Hardy-Littlewood maximal function. The following inequalities follow easily from the case p= 1, which can be found in Stein [27] or Weisz [37]:

kMpfkLp,∞ 6Cpkfkp (f ∈Lp(X^d)) (7.1) and

kMpfkr6Crkfkr (f ∈Lr(X^d), p < r6∞). (7.2)

178 F. Weisz The first inequality holds also ifp=∞.

The spaceE˙q(R^d)contains all functionsf ∈L^loc_q (R^d)for which kfkE˙q:=

X∞

k=−∞

2^kd(1−1/q)kf1Pkkq <∞,

where Pk := {2^k−1 6 |x| <2^k}, (k ∈ Z). These spaces are special cases of the Herz spaces [15] (see also Garcia-Cuerva and Herrero [9]). The non-homogeneous version of the spaceE˙q(R^d)was used by Feichtinger [8] to prove some Tauberian theorems. It is easy to see that

L1(R^d) = ˙E1(R^d)←-E˙q(R^d)←-E˙q⁰(R^d)←-E˙∞(R^d), 1< q < q⁰<∞.

To prove pointwise convergence of theθ-means we will investigate themaximal operator

σ_¤^θf := sup

T >0|σ_T^θf|.

Ifθˆ∈L1(R^d)then (5.1) implies

kσ^θ_¤fk∞6kθkˆ 1kfk∞ (f ∈L∞(R^d)).

In the one-dimensional case Torchinsky [30] proved that if there exists an even functionη such thatη is non-increasing onR+,|θ|ˆ 6η,η∈L1thenσ^θ_¤is of weak type(1,1) and a.e. convergence holds. Under similar conditions we will generalize this result for the multi-dimensional setting. First we introduce an equivalent condition.

Theorem 7.1. Forθ∈L1(R^d)letη(x) := sup_ktkr>kxkr|θ(t)|ˆ for some16r6∞.

Thenθˆ∈E˙∞(R^d)if and only ifη∈L1(R^d)and C⁻¹kηk₁6kθkˆ E˙∞ 6Ckηk₁.

Theorem 7.2. Letθ∈L1(R^d),16p6∞and1/p+ 1/q= 1. Ifθˆ∈E˙q(R^d)then kσ_¤^θfkLp,∞6Cpkθkˆ E˙qkfkp

for allf ∈Lp(R^d). Moreover, for everyp < r6∞,

kσ^θ_¤fkr6Crkθkˆ E˙qkfkr (f ∈Lr(R^d)).

The proof of this theorem follows from the pointwise inequality

σ_¤^θf(x)6Ckθkˆ E˙qMpf(x) (7.3) and from (7.1) and (7.2). Inequality (7.3) is proved in Feichtinger and Weisz [4].

Theorem 7.2 and the usual density argument due to Marcinkiewicz and Zyg-mund [17] imply

Wiener amalgams and summability of Fourier series 179 Corollary 7.3. Ifθ∈L1(R^d),θ(0) = 1,16p6∞,1/p+1/q= 1andθˆ∈E˙q(R^d) then

Tlim→∞σ_T^θf =f a.e.

iff ∈Lr(R^d)forp6r <∞ orf ∈C0(R^d).

Note that E˙q(R^d)⊃E˙q⁰(R^d) wheneverq < q⁰. If θˆis in a smaller space (say in E˙∞(R^d)) then we get convergence for a wider class of functions (namely for f ∈Lr(R^d),16r6∞).

In order to generalize the last theorem and corollary for the larger space W(L1, `∞)(R^d), we have to define the local Hardy-Littlewood maximal function by

mpf(x) := sup

0<r61

³ 1

|B(x, r)|

Z

B(x,r)

|f|^pdλ

´_1/p

(x∈R^d),

where f ∈ L^loc_p (R^d), 1 6p6 ∞ and B(x, r) denotes the ball with center x and radiusr. It is easy to see that inequalities (7.1) and (7.2) imply

kmpfk_{W(Lp,∞,`s)</sub>6Cpkfk<sub>W(Lp,`s)} (f ∈W(Lp, `s)(R^d)) (7.4) and

kmpfk_{W(Lr,`s)</sub>6Crkfk<sub>W(Lr,`s)} (f ∈W(Lr, `s)(R^d)) (7.5) for allp < r6∞and16s6∞. Recall that

kfk_{W(Lp,∞,`∞)}= sup

k∈Z^d

sup

ρ>0ρ λ(|f|> ρ,[k, k+ 1))^1/p.

Theorem 7.4. Letθ∈L1(R^d),16p6∞and1/p+ 1/q= 1. Ifθˆ∈E˙q(R^d)then kσ_¤^θfk_{W(Lp,∞,`∞)</sub>6Cpkθkˆ E˙qkfk<sub>W(Lp,`∞)}

for allf ∈W(Lp, `∞)(R^d). Moreover, for everyp < r6∞,

kσ^θ_¤fkW(Lr,`∞)6Crkθkˆ E˙qkfkW(Lr,`∞) (f ∈W(Lr, `∞)(R^d)).

It is easy to see that

Mpf 6Cmpf+CpkfkW(Lp,`∞) (16p6∞).

The proof of Theorem 7.4 follows from (7.3)–(7.5).

Corollary 7.5. Ifθ∈L1(R^d),θ(0) = 1,16p <∞,1/p+1/q= 1andθˆ∈E˙q(R^d) then

T→∞lim σ_T^θf =f a.e.

iff ∈W(Lp, c0)(R^d).

180 F. Weisz Note that W(Lp, c0)(R^d)contains allW(Lr, c0)(R^d)spaces forp6r6∞.

We can characterize the set of convergence in the following way. Lebesgue differentiation theorem says that

h→0lim 1

|B(0, h)|

Z

B(0,h)

f(x+u)du=f(x)

for a.e.x∈X^d, wheref ∈L^loc₁ (X^d), X=Tor X=R. A point x∈X^d is called a p-Lebesgue point (or a Lebesgue point of orderp) off ∈L^loc_p (X^d)if

h→0lim

³ 1

|B(0, h)|

Z

B(0,h)

|f(x+u)−f(x)|^pdu

´_1/p

= 0 (16p <∞) resp.

h→0lim sup

u∈B(0,h)

|f(x+u)−f(x)|= 0 (p=∞).

Usually the 1-Lebesgue points, called simply Lebesgue points are considered (cf.

Stein and Weiss [25] or Butzer and Nessel [3]). One can show that almost every pointx∈X^dis ap-Lebesgue point off ∈L^loc_p (X^d)if16p <∞, which means that almost every pointx∈R^d is ap-Lebesgue point off ∈W(Lp, `∞)(R^d). x∈X^d is an∞-Lebesgue point off ∈L^loc_∞(X^d)if and only iff is continuous atx. Moreover, allr-Lebesgue points arep-Lebesgue points, wheneverp < r.

Stein and Weiss [25, p. 13] (see also Butzer and Nessel [3, pp. 132-134]) proved that if η(x) := sup_|t|>|x||θ(t)|ˆ and η ∈L1(R^d)then one has convergence at each Lebesgue point off ∈Lp(R^d) (16p6∞). Using theE˙q spaces we generalize this result.

Theorem 7.6. Let θ ∈ L1(R^d), θ(0) = 1, 1 6 p 6 ∞ and 1/p+ 1/q = 1. If θˆ∈E˙q(R^d)then

Tlim→∞σ_T^θf(x) =f(x) for allp-Lebesgue points off ∈W(Lp, `∞)(R^d).

Note that W(L1, `∞)(R<sup>d</sup>) contains all Lp(R<sup>d</sup>) spaces and amalgam spaces W(Lp, `q)(R^d)for the full range16p, q6∞.

If f is continuous at a point x then x is a p-Lebesgue point of f for every 16p6∞.

Corollary 7.7. Let θ ∈ L1(R^d), θ(0) = 1, 1 6 p 6 ∞ and 1/p+ 1/q = 1. If θˆ∈E˙q(R^d)andf ∈W(Lp, `∞)(R^d)is continuous at a pointxthen

T→∞lim σ_T^θf(x) =f(x).

Recall thatE˙1(R^d) =L1(R^d)andW(L∞, `∞)(R^d) =L∞(R^d). Iff is uniformly continuous then we have uniform convergence (see Corollary 5.4).

Let us consider converse-type problems. The partial converse of Theorem 7.2 is given in the next result.

Wiener amalgams and summability of Fourier series 181 Theorem 7.8. Let θ∈L1(R^d),θˆ∈L1(R^d),16p <∞ and1/p+ 1/q= 1. If

σ_¤^θf(x)6CMpf(x) (7.6)

for allf ∈Lp(R^d)andx∈R^d thenθˆ∈E˙q(R^d).

The converse of Theorem 7.6 reads as follows.

Theorem 7.9. Suppose that θ∈L1(R^d),θ(0) = 1, θˆ∈L1(R^d), 1 6p < ∞ and 1/p+ 1/q= 1. If

Tlim→∞σ_T^θf(x) =f(x) (7.7) for allp-Lebesgue points off ∈Lp(R^d)thenθˆ∈E˙q(R^d).

Corollary 7.10. Suppose that θ∈L1(R^d),θ(0) = 1,θˆ∈L1(R^d),16p <∞and 1/p+ 1/q= 1. Then

Tlim→∞σ_T^θf(x) =f(x)

for all p-Lebesgue points off ∈Lp(R^d) (resp. off ∈W(Lp, `∞)(R^d)) if and only ifθˆ∈E˙q(R^d).

If we take the supremum in the maximalθ-operator over a cone, say over{T ∈ R^d₊: 2^−τ6Ti/Tj 62^τ;i, j= 1, . . . , d} for some fixedτ>0:

σ_c^θf := sup

2−τ6Ti/Tj62τ i,j=1,...,d

|σ_T^θf|,

then all the results above can be shown for σ_c^θ. In this case, under the conditions above we obtain the convergenceσ_T^θf →f a.e. as T → ∞and 2^−τ 6Ti/Tj 62^τ (i, j = 1, . . . , d). This convergence has been investigated in a great number of papers (e.g. in Marcinkiewicz and Zygmund [17], Zygmund [39], Weisz [34, 36, 37]).

For more details see Feichtinger and Weisz [4]. The unrestricted convergence of σ_T^θf, i.e. asTj→ ∞for eachj= 1, . . . , d, is also investigated in that paper.

8. A.e. convergence of the θ-means of Fourier series

All the results of Section 7 holds also for Fourier series. In this case we define themaximal operatorof theθ-means by

σ^θ_¤f := sup

n∈N|σ^θ_nf|.

Similarly to Theorem 7.4 we have

182 F. Weisz Theorem 8.1. Let θ ∈ W(C, `1)(R^d), 1 6 p 6 ∞ and 1/p+ 1/q = 1. If θˆ ∈ E˙q(R^d)then

kσ_¤^θfk_Lp,∞6C_pkθkˆ E˙qkfk_p for allf ∈Lp(T^d). Moreover, for everyp < r6∞,

kσ_¤^θfkr6Crkθkˆ E˙qkfkr (f ∈Lr(T^d)).

The analogue of Theorems 7.6, 7.9 and Corollary 7.10 reads as follows.

Theorem 8.2. Let θ ∈ W(C, `1)(R^d), θ(0) = 1, θˆ ∈ L1(R^d), 1 6 p < ∞ and 1/p+ 1/q= 1. Then

n→∞lim σ^θ_nf(x) =f(x)

for allp-Lebesgue points off ∈L_p(T^d)if and only if θˆ∈E˙_q(R^d).

Corollary 8.3. Let θ ∈ L1(R^d), θ(0) = 1, 1 6 p 6 ∞ and 1/p+ 1/q = 1. If θˆ∈E˙q(R^d)andf ∈Lp(T^d)is continuous at a pointx∈T^d then

n→∞lim σ^θ_nf(x) =f(x).

In document Annales Mathematicae et Informaticae (32.) (Pldal 176-183)