Strong and Weak Weighted Convergence of Jacobi Series

Transcription

1 journal of approximation theory 88, 127 (1997) article no. AT9635 Strong and Weak Weighted Convergence of Jacobi Series R. A. Kerman* Department of Mathematics, Brock University, St. Catharines, Ontario, L2S 3A1, Canada Communicated by Paul evai Received May 21, 1993; accepted in revised form January 2, 1996 (:, ;) Given :, ;> &1, let p n (x)=p n (x), n=, 1, 2,... be the sequence of Jacobi polynomials orthonormal on (&1, 1) with respect to the weight u(x)= (1&x) : (1+x) ;. Denote by (S f )(x) the th partial sum of the FourierJacobi series of the function f on (&1, 1), so that (S f )(x)= n= a np n (x), with a n = 1 &1 f(x) p n(x) u(x) dx. For fixed p #(1,), we characterize the weights w such that lim 1 &1 [(S f )(x)&f(x)] w(x) p u(x) dx= whenever 1 &1 f(x) w(x) p dx <, the weights w such that lim sup *> *[ * w(x) p u(x) dx] = whenever 1 &1 f(x) w(x) p dx<, and the weights w such that lim sup *> *[ * w(x) p u(x) dx] = whenever [ F* w(x) dx] d*<; here, * =[x #(&1,1); (S f(x)& f(x) >*] and F * =[x # (&1, 1): f(x) >*] Academic Press I. ITRODUCTIO Let u(x)=u :, ; (x)=(1&x) : (1+x) ;, :, ;>&1, be a Jacobi weight and (:, ;) let p n (x)=p n (x)=# n x n +..., # n >, n=, 1, 2,..., be the corresponding sequence of (orthonormal) Jacobi polynomials 1 &1 p m (x) p n (x) u(x) dx=$ mn m, n=,1,2,... (1) (&12, &12) For example, p n (x)=(2-?) cos(n cos &1 x), n=1, 2,..., and the (, ) p n (x) are the (normalized) Legendre polynomials. An extended treatment of Jacobi polynomials can be found in [2]; in particular, one finds on p. 198 the asymptotic formula p n (cos %)= 2? u(cos %)&12 [cos(m%+#)+(n sin %) &1 (1)] (2) * Research supported in part by SRC Grant A Copyright 1997 by Academic Press All rights of reproduction in any form reserved.

2 2 R. A. KRMA for cn &1 %?&cn &1, c> a fixed constant, M=n+(:+;+1)2, and #=&(:+12)?2. Given a function satisfying 1 &1 f(x) u(x) dx<, denote by (S f )(x) the th partial sum of the FourierJacobi series of f, namely n= a n p n (x), with a n = 1 &1 f(x) p n(x) u(x) dx. The purpose of this paper is to characterize the nonnegative, measurable (weight) functions on (&1, 1) such that for fixed p #(1,) S f converges to f in one of the following senses: 1. weighted strong sense whenever lim 1 [(S f )(x)&f(x)] w(x) p u(x) dx=, (3) &1 1 &1 f(x) w(x) p u(x) dx<; 2. weighted weak sense lim sup * _ w(x) p u(x) dx & =, (4) *> * * =[x #(&1,1): (S f)(x)&f(x) >*], whenever 1 &1 f(x) w(x) p u(x) dx<; 3. weighted restricted weak sense lim sup * _ w(x) p u(x) dx & =, (5) *> * * =[x #(&1,1): (S f)(x)&f(x) >*], whenever F * =[x # (&1, 1): f(x) >*]. w(x) _ p u(x) dx & d*<, F* Any strong weight is a weak weight and any weak weight is a restricted weak weight. This is readily seen from the following relations between the

3 WIGHTD IQUALITIS FOR JACOBI SRIS 3 Lorentz L p (w p u) and L p1 (w p u) norms and the usual Lebesgue L p (w p u) norm: &g& L p (w p u)=sup * _ w(x) p u(x) dx & _ *> G* 1 g(x) w(x) p u(x) dx & &1 w(x) _ p u(x) dx & G* dx=&g& L p1 (w p u), G * =[x # (&1, 1): g(x) >*]. We will show that, in fact, the weak weights are the same as the strong weights. It will be convenient to work with the trigonometric form, p n (cos %), of the Jacobi polynomials for which (cos %) p n (cos %) u(%) d%=$ mm m, n=,1,2,..., where u(%)=2 :+; sin 2:+1 (%2) cos 2;+1 (%2) and a n =? f(,) p n(cos,)u(,) d,, f(,)= f(cos,). With this notation, it is seen we have the usual Fourier cosine series when :=;=&12. Moreover, (3), for example, becomes lim [(S f )(%)&f(%)] w(%) p u(%) d%=, whenever? f(,) w(,) p u(,) d,<. Here, w(%)=w(cos %). arlier weighted convergence theorems dealt with Jacobi weights w(x)= c(1+x) A (1&x) B or, equivalently, w(%)=c sin 2A+1 (%2) cos 2B+1 (%2). Thus, the classical 1927 result of Riesz [19] asserts (4) holds for 1<p< when :=;=&12 (so that u(%)=12) and w(%)=1. The pioneering 1948 paper of Pollard [18] characterized the range of indices p that work in (4) for :, ;&12 and w(%)=1; for instance, in the Legendre case :=;=, the range is 43<p<4. Muckenhoupt proved the definitive theorem on Jacobi weights for :, ;> &1 in [1]. Askey [1] used such results to investigate the weighted strong convergence of Lagrange interpolation polynomials based on the zeros of Jacobi polynomials. Badkov [3] studied the generalized Jacobi polynomials satisfying (1) with respect to u of the form m u(x)=h(x)(1+x) : (1&x) ; ` k=1 x&x k # k ; here, :, ;, # k > &1, &1<x 1 <}}}<x m <1, H(x)> on (&1, 1), and 2 (H, $) d$$ <, (H, $) being the usual modulus of continuity of H in C([&1, 1]). He extended Muckenhoupt's results to Fourier

4 4 R. A. KRMA series in such polynomials for the generalized Jacobi weights w(x)= C(1+x) A (1&x) B > m x&x k=1 k 1 k ; the x k are the same as in u(x) and A, B, 1 k >&1. evai in a series of papers, [16], completed the abovementioned work of Askey for generalized Jacobi polynomials. Chanillo [4] proved the first restricted weak convergence theorem for FourierLegendre series when p=43 orp=4 and w(%)=1. The result for this weight was extended to all FourierJacobi series by Guadalupe, Perez and Varona [5]; for example, when :;&12 and :>&12, they obtained restricted weak convergence when p=4(:+1)(2:+3) or p=4(:+1)(2:+1), values of p for which they show weak convergence doesn't hold. The general weighted strong convergence problem for Fourier series (:=;=&12) was solved by Hunt, Muckenhoupt, and Wheeden [7]. A weight w yields (4) in this case, for fixed p #(1,), if and only if w # A p (,?): _ 1 I w(%) p d% & I _ 1 I w(%) &p$ d% & $ C, I where, as usual, p$=p( p&1) and the constant C> is independent of the interval I/(,?) with length I. A standard argument involving the BanachSteinhaus theorem shows (3) is equivalent to (X f )(%) w(%) p u(%) d%c f(,) w(,) p u(,) d(,), (6) (4) is equivalent to sup * _ w(%) p u(%) d% & C *> * f(,) w(,) p u(,) d,, (7) and (5) is equivalent sup * _ w(%) p u(%) d% & C *> * w(,) _ p u(,) d, & d*; (8) F* here, * =[% #(,?): (S f)(%) >*], F * =[, #(,?): f(,) >*] and C> is independent of f and. The converse of Ho lder's inequality for Lebesgue and Lorentz spaces (see [6] for the latter) readily yields

5 WIGHTD IQUALITIS FOR JACOBI SRIS 5 Lemma 1. The condition w(,) &p$ u(,) d,< (9) is necessary and sufficient to guarantee f(,) u(,) d,<, (1) whenever f(,) w(,) p u(,) d,<. Similarly, &w &p & L p$ (w p u)=sup * _ w(,) p u(,) d, & $ <, (11) *> * * = [, #(,?):w(,) &p >*], is necessary and sufficient to have (1) whenever F * =[, #(,?): f(,) >*]. w(,) _ p u(,) d, & d*<, F* The condition (1) is required, of course, in order that the Fourier Jacobi series of f be defined. Further, the constant function p (%) (and hence S f for all ) belongs to L p (w p u) (equivalently to L p (w p u)) if and only if w(%) p u(%) d%<. (12) With these facts in mind, we call weights satisfying (9) and (12) S-admissible, while weights for which (11) and (12) hold will be said to be RW-admissible. We can now state our main results. Theorem 2. Fix :, ;>&1 and p #(1,). Let u(%)=u :, ; (%)=2 :+; sin 2:+1 (%2) cos 2;+1 (%2) and, given f with? f(,) u(,) d,<, let (S f )(%)= n= a n p n (cos %), a n =? f(,) p n(,) u(,) d,, be the th partial sum of the FourierJacobi series of f. Then, given an S-admissible weight w on(,?), the following are equivalent:

6 6 R. A. KRMA (a) lim [(S f )(%)&f(%)] w(%) p u(%) d%=, whenever? f(,) w(,) p u(,) d,<; (b) lim sup * _ w(%) p u(%) d% & =, *> * * =[, #(,?): (S f)(,)&f(,) >*], whenever? f(,) w(,) p u(,)d,<; (c) There exists C> such that and _ 1 I w(%) p u(%) d% & I _ 1 I w(%) &p$ u(%) d% & $ C (13) I _ 1 I w(%) p u(%) 1&p2 d% & 12 I _ 1 I w(%) &p$ u(%) 1&p$2 d% & $ C (14) I for all intervals I/(,?). When :, ;&12 (14) implies (13), so (14) alone is required in that case. Theorem 3. Let :, ;, u, f, and S f be as in Theorem 2. Then, in order that the RW-admissible weight w on (,?) satisfy lim sup * _ w(%) p u(%) d% & =, *> * * =[% #(,?): (S f)(%)&f(%) >*], whenever [ F * w(,) p u(,) d,] d*<, F * =[, #(,?): f(,) >*], it is necessary and sufficient that there exist C>, independent of all intervals I/(,?) and measurable sets /(,?), related to I as specified below, such that u(,) d, I u(,) d, C _ w(,) p u(,) I w(,) p u(,) d,& \/I (15) and u(,) 12 d, % u(,)12 d, C _ w(,) p u(,) % w(,) p u(,) d,& u(,) 12 d, u(%) 12 sin(,2) C _ w(,)p u(,) % w(,) p u(,) d,& \/(, %) (16) \/(%,?2), (17)

7 WIGHTD IQUALITIS FOR JACOBI SRIS 7 when <%<,2, and u(,) 12 d,? u(,)12 d, C _ w(,) p u(,)? w(,) p % u(,) d,& u(,) 12 d, u(%) 12 cos(,2) C _ w(,)p u(,)? w(,) p % u(,) ud,& p \/(%,?) (18) \/ \? 2, % +, (19) when?2<%<?. A crucial element in the proof of a weighted convergence theorem is an alternative to the formula of ChristoffelDarboux for the Dirichlet kernel, K, due to Pollard [18]: K (%,,)= : n= p n (cos %) p n (cos,) =: h 1 (%,,, )&; [h 2 (%,,, )+h 3 (%,,, )], (2) in which lim : =lim ; =1, h 1 (%,,, )=P +1 (cos %) p +1 (cos,), and h 2 (%,,, )= P +1(cos %) q (cos,) 2 sin((%+,)2) sin((%&,)2) h 3 (%,,, )=h 2 (,, %, ), with q (cos,)=sin 2 (:+1, ;+1),p (cos,). To use (2) we will rely on the fact that [2, p. 169] p (cos )=b () s () &(:+12) c () &;+12), (21) where s ()=sin(2)+1, c ()=cos(2)+1 and b () C for all and. We also require a new estimate, proved in [9], for the kernel, P (:, ;) (r, %,,), of the Poisson integral of f f(r, %)= : a n r n p n (cos %)= n= p (:, ;) (r, %,,) f(,) u(,) d,, a n =? f() p n() u(,) d.

8 8 R. A. KRMA Theorem 4. Let \=r 12. Then, for 12<r<1, P (:, ;) (r, %,,)rp(\, %,,) _ 1 (1&2\ cos(%+,)+\ 2 ) : , (22) (1&2\ cos(%&,)+\ 2 ) ;+12& in the sense that each side is dominated by a constant multiple of the other. Here, P(\, %,,)=(1&\ 2 )(1&2\ cos(%&,)+\ 2 ) is the classical Poisson kernel. The proofs of the necessity of the conditions in Theorems 3 and 4 use the following special case of Theorem 2 in [14]; see also [15]. Lemma 5. Let f be a Lebesgue-measurable function on (,?). Then, u(%) 12 f(%) d%- 2? lim n p n (cos %) f(%) u(%) d%. (23) II. AUXILIARY OPRATORS The proof of the sufficiency of (13) and (14) involves a singular integral operator related to the Hilbert transformation, namely (Hf )(%)=(P) as well as the Stieltjes operators and (S 1 f )(%)= (S 2 f )(%)= f(,) sin((%&,)2) d,, f(,) sin(%2)+sin(,2) d, f(,) cos(%2)+cos(,2) d,. It is well-known [7] that for fixed p #(1,) (Hf )(%) w(%) p d%c f(,) w(,) p d, (24) if and only if w # A p (,?). Moreover,

9 WIGHTD IQUALITIS FOR JACOBI SRIS 9 Lemma 6. The condition w # A p (,?) guarantees [(S 1 +S 2 ) f](%) w(%) p d%c f(,) w(,) p d,. (25) Proof. Given f, there holds?(s 1 f )(%) 1 % % f(,) d,+ % f(,) d, =(Pf )(%)+(Qf)(%)., ow, (Pf )(%)(Mf )(%)= sup % # I/(o,?) 1 I I f(,) d,, so, in view of [11], (Pf )(%) w(%) p d%c f(,) w(,) p d, when w # A p (,?). Again, the operator Q being the dual of P, we'll have (Qf)(%) w(%) p d%c f(,) w(,) p d, if and only if (Pf )(%) w(%) &1 p$ d%c f(,) w(,) &1 p$ d,. But, the latter is true if w &1 # A p$ (,?), which is equivalent to w # A p (,?). Hence, when w # A p (,?), (S 1 f )(%) w(%) p d%c f(,) w(,) p d,. Letting g(,)= f(?&,) and observing that w(?&%)#a p (,?) if and only if w(%)#a p (,?), we get (S 2 f )(%) w(%) p d%= (S 1 g)(?&%) w(?&%) p d% C g(,) w(?&,) p d, C f(,) w(,) p d,. K

10 1 R. A. KRMA The proof of Theorem 3 needs mapping properties of the maximal operator and the Hardy operators (M u f )(%)= sup = I f(,) u(,) d, % # I/(,?) I u(,) d, (P 1 f )(%)=(sin(%2)) &1 u(%) &12 % f(,) u(,) 12 d, (Q 1 f )(%)=u(%) &12 2 % f(,) u(,) 12 d% sin(,2) <%<?2 and (P 2 f )(%)=(cos(%2)) &1 u(%) &12 % f(,) u(,) 12 d, (Q 2 f )(%)=u(%) &12 %?2 f(,) u(,) 12 d, cos(,2)?2<%<?. The methods of [18, Proposition 1] can be easily adapted to show there exists C>, independent of f, such that sup * _ w(%) p u(%) d% & C *> * w(,) _ p u(,) d, & d*, F* * =[% #(,?): (M u f)(%)>*], F * =[, #(,?): f(,) >*], if and only if (15) holds. Concerning the Hardy operators we have the following result obtained in a collaboration with Bloom and Stepanov. Lemma 7. Suppose the weight w satisfies (12) and % w(,) p u(,) d,r 2% w(,) p u(,) d, <%<?2. (26) % Then, there exists C>, independent of f, such that sup * _ w(%) p u(%) d% & C *> * w(,) _ p u(,) d, & d*, (27) F*

11 WIGHTD IQUALITIS FOR JACOBI SRIS 11 * =[% #(,?2): (Tf )(%) >*], F * =[, #(,?2): f(,) >*], if and only if (16) holds when T=P 1 and (17) holds when T=Q 1. Again, if w satisfies (12) and w(,) p u(,) d,r % 2%&? w(,) p u(,) d,?2<%<?, then, one has (27) if and only if (18) when T=P 2 and (27) if and only if (19) when T=Q 2 ; in * and F * we have %,, #(?2,?), in this case. Proof. The proofs of the four criteria are quite similar. We give the details for Q 1. To begin, observe that (27) is equivalent to the existence of C>, independent of measurable /(,?2) and *>, such that * _ w(%) p u(%) d% & C _ F* w(,) p u(,) d, &, (28) F * =[% #(,?2): (Q 1 / )(%)>*]. See [6]. Since w, <?2 w(,) p?4 u(,) d,<, because of (26). Thus, for?4<%<?2, (17) amount to d% u(%) 12 sin (%2) C w(,) _ p u(,) d, & /(%,?2). (29) But, when?4<%<?2 and /(%,?2), (Q 1 / )(%)c u(,) 12 d, sin(,2), for some c> independent of. Thus, taking *=c u(,) 12 d,sin(,2) in (28) gives c u(,) 12 d, sin(,2)_2?4 w(%) p u(%) d% & C _ w(,) p u(,) d, & and (29) follows. Suppose, then, <%<?4. If %<,<2% and /(%,?2), (Q 1 / )(,)C &1 u(,) &12 u() 12 d sin(2),

12 12 R. A. KRMA for some C> independent of, so(q 1 / )(,)>* whenever C* u(,) &12 > u() 12 dsin(2). (3) But, there exists a constant C 1 > so that (3) always holds with *=( u() 12 dsin(2))c 1 u(%) 12 and %<,<2%. Hence, from (28), u(%) &12 u() 12 d sin(2)_ 2% C _ w(,) p u(,) d, & % w(,) p u(,) d% & and we obtain (17) by (26). We next show (17) implies (28). ow, for some C>, independent of, so, (Q 1 / )(%)C &1 u(%) &12 & (%,?2) u(,) 12 d, sin(,2), F * =[% #(,?2): (Q 1 / )(%)>*] / {% #(,?2): u(%)&12 & (%,?2) u(,) 12 d, sin(,2) >C* = / {% #(,?2): _ &(%,?2) w(,) p u(,) d, < % by (17) / {% #(,?2): * _ % Thus, w(,) p u(,) d, & >C 1 * = w(,) p u(,) d, & C _ w(,) p u(,) d, & = =G *. * _ w(%) p u(%) d% & * _ F* w(,) p u(,) d, & G* * _ lub G * w(,) p u(,) d, & C _ w(,) p u(,) d, &. K

13 WIGHTD IQUALITIS FOR JACOBI SRIS 13 III. PROOF OF THORM 2 Since (7) holds whenever (8) does, we need only show that (13) and (14) imply (7) and that (8) yields (13) and (14). To prove the former it is enough to consider f such that? f(,) w(,) p u(,) d,=1. By (2), 3 (S f )(%) C : k=1} 3 h k (%,,, ) f(,) u(,) d, }=C : J k (f, %, ). (31) k=1 The estimate (21) for p +1 and Ho lder's inequality gives J 1 ( f, %, )= }p +1(cos %) p +1 (cos,) f(,) u(,) d, } Cs (%) &(:+12) c (%) &(;+12) s (,) &(+12) _c (,) &(;+12) f(,) u(,) d, C [1+u(%) &12 ] C[1+u(%) &12 ] _ [1+u(,) &12 ] f(,) u(,) d, w(%) &p$ [1+w(%) &12 ] p$ u(%) d% & $, so, [J 1 ( f, %, ) w(%)] p u(%) d% C \ C, _ \ w(%) p [1+u(%) &12 ] p u(%) d% w(%) &p$ [1+u(%) &12 ] p$ u(%) d% + p&1 + by (13) and (14). ext, from (21), J 2 ( f, %, ) is less than a constant times s (%) &(:+12) c (%) &(;+12) _ } b +1 (,) sin 2,f(,) s (,) &(:+32) c (,) &(;+32) u(,) d, sin((%+,)2) sin((%&,)2) },

14 14 R. A. KRMA where b +1 (,) C, <,<?. But, with sin, sin((%+,)2) sin((%&,)2) 1 = sin((%&,)2) + sin, _ sin((%+,)2) &1 & 1 _ sin((%&,)2), 1 = +R(%,,), sin((%&,)2) R(%,,)= _ 1 sin(%2)+sin(,2)& 1 cos(%2)+cos(,2)& <%<?2?2<%<?. Thus, setting g(,)=b +1 (,) sin,f(,)s (,) &(:+32) c (,) &(;+32) u(,), we have J 2 ( f, %, )C[ (Hg)(%) +[(S 1 +S 2 )( g )](%)] s (%) &(:+12) _c (%) &(;+12). (32) When :, ;&12, it follows from (32) that [J 2 ( f, %, ) w(%)] p u(%) d% C [ (Hg)(%) +[(S 1 +S 2 )( g )](%) w(%)] p u(%) 1& p2 d% C g(,) w(,) p u(,) 1&p2 d, C f(,) w(,) p u(,) d,, by (13), in view of (24) and (25).

15 WIGHTD IQUALITIS FOR JACOBI SRIS 15 We illustrate the case min[:, ;]<&12 by :<&12;. Here, the preceding considerations yield? [J 2( f, %, ) w(%)] p u(%) d% dominated by a constant multiple of [ (Hg)(%)+[(S 1 +S 2 )( g )](%) w(%) u(%) S (%) &(:+12) _c (%) &(;+12) ] p d% C f(,) w(,) p u(,) d,, provided w(%) u(%) s (%) &(:+12) c (%) &(;+12) # A p (,?). (33) In proving (13) and (14) imply (33) for a given (with constant independent of ), only intervals of the form I=(, b), 1<b<?2, offer any difficulty. For such I, (33) is equivalent to _ 1 J w(%) p u(%) s (%) &(:+12) p d% & J 1 I w(%) &p$ u(%) 1&p$ s (%) &(:+12) p$ d% & $ C, I with J=(, 1), (1, b). This inequality holds when J=(, 1) by (14) and when J=(1, b) by(13). The proof that [J 3 ( f, %, ) w(%))] p u(%) d%c f(,) w(,) p u(,) d, is similar to the one for J 2 ( f, %, ). We now show (8) implies (13) and (14). A simple argument reduces considerations to intervals I/(,?) of the forms (i) I=(, b) b<c?2 (ii) I=(a, b) b&a<min[a,?&b] (iii) I=(a,?) a>?&c?2, for any fixed c, <c<1, a value of which will be specified later. We only look at cases (i) and (ii).

16 16 R. A. KRMA Given (8), there holds sup * _ w(%) p u(%) d% & C _ r *> * f(,) w(,) p u(,) d, &, r * =[% #(,?): f(r, %) >*], with C> independent of f and r #(,1), since f(r, %)= : = Letting 1&r 12 = I 6, (22) yields (r &r +1 )(S f )(%) <r<1. f(r, %) c I f(,)u(,)d, I u(,) d, f, % # I, (34) whence (8) implies (13) by a standard argument, [12]. But (14) (as well as (13)) is equivalent to A p (,?), if I satisfies (ii). We note in passing that a simple consequence of this is that for fixed A and B 3I w(%) p sin A % 2 cosb % 2 d%d I w(%) p sin A % 2 cosb % 2 d%, (35) whenever 3I=3(x I & I 2, x I + I 2=(x I &3 I 2, x I +3 I 2) is of type (ii). To obtain (14) in case (i), we show (8) implies _ 2% % w(,) p u(,) 1&p2 (sin,) &p d, & c% w(,) &p$ u(,) 1&p$2 d, & $ C (36) and observe that, by a similar argument, (8) also ensures _ c% % w(,) p u(,) 1&p2 d, & 1r 2% % w(,) &p$ u(,) 1&p$2 (sin,) &p$ d, & $ C. (37) Proceeding as in [2, pp. 2122], we then multiply (36) by (37) and use Ho lder's inequality to dominate % &1 r 2% % (sin,) &2 d, by the product of the integrals over (%, 2%) and thus obtain (14).

17 WIGHTD IQUALITIS FOR JACOBI SRIS 17 Fix % #(,?4) and consider functions f =f } / (, c%), where c #(,12) will be determined below. Since p +1 (cos,) p +1 (cos ) is the kernel of S +1 &S, the inequality (8) gives sup * _ w(,) p u(,) d, & C _ * *> c% f() w() p u() d &, (38) * =[, #(,?): (T f)(,) >*], where Set Then, for %,?2, (T f )(,)= c% (h 2 +h 3 )(,,, ) ()u()d. (39) (T f )(,) c% We recall here (2) f ()=sgn[ p +1 (cos )] f(). (4) & c% q (cos,) P +1 (cos ) f() u() sin((,+)2) sin((,&)2) d P +1 (cos,) q (cos ) f() u() d sin((,+)2) sin((,&)2) A (,)&B (,). (:+1, ;+1) p (cos,)= 2? u(,)&12 (sin,) &1 _[cos(m,+#)+( sin,) &1 (1)], c &1,?&c &1, with M=+(:+;+3)2, #=&(:+32)?2. When is large (in particular, >1%) it is possible to get pairwise disjoint open intervals I 1,..., I l, I l =I l (), l =l (), contained in (%, 2%), such that and l cos(m,+#)+( sin,) &1 (1) >13 on J=. I l l=1 l. l=1 3I l #(%,2%) 3I l satisfies (ii).

18 18 R. A. KRMA Indeed, for sufficiently large, (2) together with (23), yields, in addition, A (,)k c% (sin,) &1 u(,) &12 f() u() d k=14? for, # J. Again, by (21), we always have B (,)K c% s (,) &(:+12) c (,) &(;+12) sin f() u() 12 d, sin 2, %<,<?2. On J=J(), for fixed large, then (T f )(,)A (,)&B (,) k(sin,) &1 u(,) &12 c% _ sin 1&K k sin,& f() u()12 d k2(sin,) &1 u(,) &12 c% f() u() 12 d, where c is chosen so that 1&(K sin )(k sin,)1&(k sin %)(k sin %) 12. ow, there exists C 1 > such that, on <,<?2, the decreasing function h(,)=(sin(,2)) &(:+32) satisfies This means 1 C 1 h(,)(sin,) &1 u(,) &12 C 1 h(,). J [(sin,) &1 u(,) &12 w(,)] p u(,) d, C p 1 J [h(,) w(,)] p u(,) d, [C 1 h(%)] p J w(,) p u(,) d, [2 :+32 C 1 inf h(,)] p w(,) p u(,) d,, # J J [2 :+32 C 1 ] p &h } / J & p L p (w p u) [2 :+32 C 2 1 ]p &(sin( } )) &1 u &12 / J & L P (w p u).

19 WIGHTD IQUALITIS FOR JACOBI SRIS 19 Thus, C c% f() w() p u() d &T f & p L p (w p u) (k2) p &(sin( } )) &1 u &12 / J & p L p (w p u)_ c% f() u() 12 d & p (k2) p [2 :+32 C 2 1 ]&p J [(sin,) &1 u(,) &12 w(,)] p u(,) d, c% f() u() 12 d & p D &1 (k2) p [2 :+32 C 2 1 2% ]&p w(,) p u(,) 1& p2 (sin,) &p d, c% f() u() 12 d & p. Finally, taking the supremum over f such that c% f() w() p u() d=1, we get (36). % IV. PROOF OF THORM 3 Sufficiency. Using the estimate (31) for S f we need only show &J k ( f, },)& L p (w p u)c &f& L p1 (w p u), k=1, 2, 3, (41) where C> is independent of f and. The argument in the proof of Theorem 2, though with Ho lder's inequality for Lorentz spaces, gives J 1 ( f, %, )C[1+u(%) &12 ] [1+u(,) &12 ] f(,) u(,) d, C[1+u(%) &12 ] &(1+u &12 ) w &p & L p$ (w p u) & f & L p1 (w p u). In view of Lemma 7, P 1 +P 2 and Q 1 +Q 2 are bounded from L p 1 (w p u)to L p (w p u). The former means? f(,) u(,)12 d,< for all f # L p 1 (w p u) or, equivalently, &u &12 w &p & L p$ (w p u)<; (42)

20 2 R. A. KRMA the latter means &u &12 & L p (w p u)<. (43) Thus, &J 1 ( f; )& L p (w p u) C &1+u &12 & L p (w p u) &(1+u &12 ) w &p & L p$ (w p u) &f& L p1 (w p u) C & f & L p1 (w p u), by (11), (12), (42), and (43). We next show that, with g(,)=b +1 (,) sin,f(,) s (,) &(:+12) c (,) &(;+12) u(,), J 2 ( f, %, ) is dominated by a constant multiple of s (%) &(:+12) } 3%2 %2 g(,) sin((%&,)2) d, } +(M u f )(%)+[(P 1 +Q 1 )( f )](%) +(1+u(%) &12 ) f(,) u(,) 12 d,, (44) when <%<?2, and by c (%) &(;+12) } (%+?)2 (3%&?)2 +[(P 2 +Q 2 )( f )](%) g(,) sin((%&,)2) d, } +(M u f )(%) +(1+u(%) &12 ) f(,) u(,) 12 d,, (45) when?2<%<?. Suppose <%<?2. As in the proof of Theorem 2, J 2 ( f, %, )Cs (%) &(:+12) [ (Hg)(%) +[(S 1 +S 2 )( g )](%)] Cs (%) &(:+12) _} 3%2 %2 g(,) sin((%&,)2) d, } +[S 1( g )](%) &.

21 WIGHTD IQUALITIS FOR JACOBI SRIS 21 If :&12, s (%) &(:+12) [S 1 ( g )](%) &12_\ Cu(%) sin 2+ % &1 % f(,) u(,) 12 d, +2 % d, f(,) u(,) 12 sin(,2) + f(,) u(,) 12 d, &?2 C _[(P 1+Q 1 )( f )](%)+u(%) &12 f(,) u(,) 12 d, &. Again, if &1<:<&12, [S 1 ( g )](%)C _: f(,) sin 2:+2 (,2) sin(%2)+sin(,2) d, f(,) u(,) 12 sin(%2)+sin(,2) d, + f(,) u(,) 12 d, &. But, s (%) &(:+12) f(,) u(,) 12 d,c[1+u(%) &12 ] f(,) u(,) 12 d,. Moreover, when <%<1, and S (%) &(:+12) :+32 1 C 2:+2 1 f(,) u(,) d, C(M u f )(%) S (%) &(:+12) 2 1 C :+12 2 f(,) u(,) 12 sin(%2)+sin(,2) d, 1 f(,) sin 2:+2 (,2) sin(%2)+sin(,2) d, f(,) u(,) 12 d, sin(,2)

22 22 R. A. KRMA when 1<%<?2, and log 2 (?2) C :+12 : 2k k=1 2 k&1 log 2 (?2) C : k=1 C(M u f )(%); 2 (:+12) k\ S (%) &(:+12) :+32 1 f(,) u(,) 12 d, sin(,2) k+ 2:+2 2 2k f(,) u(,) d, % f(,) sin 2:+2 (,2) sin(%2)+sin(,2) d, C % &1 2+ \sin u(%) &12 % f(,) (,) :+32 u(,) 12 d, C[P 1 ( f )](%), s (%) &(:+12) 2 1 f(,) u(,) 12 sin(%2)+sin(,2) d, C % &1 2+ _\sin u(%) &12 % f(,) u(,) 12 d, 1 +u(%) &12 2 C[(P 1 +Q 1 )( f )](%). d, f(,) u(,) sin(,2)& 12 The asserted upper bound for J 2 ( f, %, ),?2<%<?, is obtained in the same way. We now prove (41) for k=2 using (44) and (45); the proof for k=3 is similar. We have by (15)(19), while &M u f+(p 1 +Q 1 +P 2 +Q 2 ) f& L p (w p u)c &f& L p1 (w p u), " (1+u&12 ) f(,) u(,) 12 d, "L p (w p u)c &f& L p1 (w p u) has been shown in the course of proving (4) for k=1.

23 WIGHTD IQUALITIS FOR JACOBI SRIS 23 To obtain " s (%) &(:+12) } 3%2 %2 g(,) sin((%&,)2) d, } / (,?2)(,) C & f & L p1 "L (w p u), p (w p u) we again take advantage of the fact that it suffices to consider f =/, /(,?2). We write g k = g } / Ik, I k =(?2 k+2,3?2 k+1 ), k=1, 2,..., whence 3%2 %2 g(,) sin((%&,)2) d, } / (,?2)(%)= : (Hg k )(%)}/ Jk (%), J k =(?2 k+1,?2 k ). Therefore, * p w(%) p u(%) d%= : H* k=1 k=1 * p H* k w(%) p u(%) d%, where H * = {% #(,?2): s 3%2 (%) &(:+12)} %2 g(,) sin((%&,)2) d, } >* = and H k *=[% # J k : s (%) &(:+12) (Hg k )(%) >*]. At this point we observe that w p u satisfies the A condition I C _ w(%) p r u(%), I w(%) p u(%) d%& with C> independent of the interval I/(,?) and the measurable set /I, in view of (15) and the fact that u satisfies A. The arguments of [21, Chapter XIII] then yield C> for which k=1, 2,.... Letting &Hg k & L p (w p u)c &Mg k & L p (w p u) C &M u g k & L p (w p u), (46) G k * =[% #(,?2): (Hg k)(%)>* k ],

24 24 R. A. KRMA * k =min[s (?2 k ), s (?2 k+1 )] :+12 *=c k *, we have * p w(%) p u(%) d%* p k H * w(%) p u(%) d% k G * c &p k &Hg k & p L p (w p u) Cc &p k &M u g k & p L p (w p u) by (46) Cc &p k &g k & p L p1 (w p u) by (15) C & Ik w(%) p u(%) d%, k=1, 2,.... Thus, " s (%) &(:+12) } 3%2 %2 g(,) sin((%&,)2) d, } * (,?2)(%) " p L p (w p u) C : w(%) p u(%) d%=c k=1 &I k w(%) p u(%) d%=c &/ & p L p1 (w p u) and we are done on taking pth roots. ecessity. First we observe (15) follows from the consequence & f(r, })& L p (w p u)c &f& L p1 (w p u) of (8), together with (34), on using the argument of Proposition 1 in [8]. ext, we show (8) implies (17) to illustrate the method of proving the necessity of the remaining conditions. To this end, fix % #(,?2B) and /(%,?2), a finite disjoint union of intervals, with B>32 to be specified below. Consider functions f =f } / and define T and f as in (39) and (4), respectively. By (8), we have &T g& L p (w p u)c &g& L p1 (w p u), g # L p1 (w p u); further (T f )(,) p +1 (cos,) & q (cos %) C (,)&D (,),#(,%). q (cos ) f() u() sin((+,)2) sin((&,)2) d p +1 (cos ) f() u() sin((+,)2) sin((&,)2) d

25 WIGHTD IQUALITIS FOR JACOBI SRIS 25 Arguing as in the proof of Theorem 2, we obtain, for a sufficiently large (in particular, >2B%) two sets of pairwise disjoint intervals I l =I l ()/(1B, %), l=1, 2,..., l =l () and K m =K m ()/, m=1, 2,..., m =m (), such that l 3I #(1B, %), l=1 l m 3K m=1 m#, 3I l, 3K m intervals of type (ii) 1 ; moreover, when, # F= l I, l=1 l # G= m K m=1 m, and p +1 (cos,) 1 3? 2 2:+; sin &(:+12), 2 cos&(;+12), 2 (:+1, ;+1) p +1 (cos ) 3 1? 2 2:+;+2 sin &(:+32) 2 cos &(;+32) 2. Restricting attention to f =f } / G, the latter yield c (,)k sin, &(:+12) 2 sin 2 (:+1, ;+1) p (cos ) f()(sin ) &2 G _sin 2(:+1)+1 2 cos2(;+1) 2 d k sin &(:+12), 2 G f() sin :&12 2 which, together with d,, # F, D (,)K sin &(:+12), 2 G f() sin :&12 2 d, gives (T f )(%)k sin &(:+12), 2 G f() sin :&12, 2_ 1&K k k2 sin &(:+12), 2 G f() sin :&12 2, & d d # F, when (Kk)}(1B)2 or B2Kk. Thus, choosing B=2Kk+32, C & f } / G & L p1 (w p u)&t f } / F & L p (w p u) k 2", sin&(:+12) 2 } / F(,) f() sin :&12 "Lp (w p u) G 2 d. 1 See page 19.

26 26 R. A. KRMA Taking the supremum over f with & f } / G & L p1 (w p u)=1, we obtain which means &u(,) &12 / F (,)& L p (w u)" u(,)&12 p sin(,2) w(,)&p / G (,) C, "Lp$ (w p u) w(,) _ p u(,) d, & &1 H u(,) 12 w(,) p d, _ H w(,) p u(,) d, & &1 G d, _ u(,) 12 C (47) G sin(,2) for all H/F. In particular, set H=(%2, %) & F and observe that by (15), whence (47) implies % w(,) p u(,) d,c w(,) p u(,) d,, %2 H d, u(%) &12 u(,) 12 G sin(,2) C _ w(,) p G u(,) % w(,) p %2 u(,) d,& and so (17), since (15) ensures w p u is doubling, as is u(,) 12 sin(,2). RFRCS 1. R. Askey, Mean convergence of orthogonal series and Lagrange interpolation, Acta Math. Acad. Hungar 23 (1972), K. F. Andersen and R. A. Kerman, Weighted norm inequalities for generalized Hankel conjugate transformations, Studia Math. 71 (1981), V. M. Badkov, Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval, Math. USSR Sb. 24 (1974), S. Chanillo, On the weak behaviour of partial sums of Legendre series, Trans. Amer. Math. Soc. 268 (1981), J. Guadalupe, M. Perez, and J. Varona, Weak behaviour of Fourier-Jacobi series, J. Approx. Theory 61 (199), R. Hunt, On L( p, q) spaces, nseign. Math. 12 (1966), R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math. 71 (1982), R. Kerman, A sharp estimate for the generating function of Jacobi polynomials, preprint, Math. Inst. Czech Acad. Sci., Prague, to appear. 1. B. Muckenhoupt, Mean convergence of Jacobi series, Proc. Amer. Math. Soc. 23 (1969), B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 3138.

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