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A weighted inequality for trigonometric series

A weighted inequality for trigonometric series By applying Pitt’s inequality we prove a weighted L version of Gallagher’s inequality for trigonometric series. Furthermore, we consider a family of weights generated by a smoothing process, via convolution operation, whose first steps are the indicator function of a compact interval and the so-called Cesàro weight sup- ported in the same interval. The eventual aim is the comparison of such weights in view of possible refinements of our inequality for p ¼ 2. Keywords Trigonometric series · Pitt’s inequality · Majorant properties Mathematics Subject Classification 42A05 · 26D15 · 42A38 1 Introduction The trigonometric series considered here are of the type SðtÞ :¼ aðmÞeðmtÞ; 2pix where eðxÞ :¼ e and aðmÞ2 C for a strictly increasing sequence m 2 R. For any given T 2ð0; þ1Þ and p 2½1; þ1Þ, let us denote the L -norm of f : R ! C by R  R 1=p 1=p p p kf k :¼ jf ðtÞj dt and set kf k :¼ jf ðtÞj dt . The space of the p p;T R T Communicated by S. Ponnusamy. & Maurizio Laporta mlaporta@unina.it Giovanni Coppola giovanni.coppola@unina.it Via Partenio 12, 83100 Avellino, AV, Italy Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli, Complesso di Monte S. Angelo, Via Cinthia, 80126 Naples, NA, Italy 123 G. Coppola, M. Laporta p p complex-valued functions defined on R with a finite L -norm is denoted by L ðR;CÞ. By f 2 L ðR; BÞ, with B  C, we mean that f : R ! B is summable on each loc compact subset of R. The Fourier transform of an integrable f : R ! C is denoted by f ðyÞ :¼ f ðtÞeðtyÞ dt: The convolution of f ; g 2 L ðR;CÞ is ðf  gÞðxÞ¼ f ðtÞgðx  tÞ dt: For any weight w : R ! C and any d[ 0, we set w ðxÞ¼ wðxÞ or 0 according as jxj d or not. Further, we denote the discrete convolution of aðmÞ and w ðxÞ as ðaHw ÞðyÞ :¼ aðmÞw ðy  mÞ: d d The function that is identically 1 we denote by 1, so that 1 is the characteristic function of the interval ðd; dÞ. Recall that 1 ðyÞ¼ 2dsincð2dyÞ, where sinðpxÞ if x 6¼ 0; sinc x :¼ px 1if x ¼ 0: Definition 1.1 Let 1\p  q\1 be fixed and p ¼ p=ðp  1Þ.If U ; V 2 L ðR;R Þ are even, U is non-decreasing on ð0; þ1Þ, V is non-increasing on loc ð0; þ1Þ and Z Z s 1=p 1=s 1=q 1p sup UðtÞ dt VðtÞ dt \ þ1; s[ 0 0 0 then we say that (U,V)isa(p,q)-pair of Pitt weights. In Sect. 2 the following result is proved. Theorem 1.1 Let d; T 2ð0; þ1Þ and 1\p  q\1 be fixed. If w 2 L ðR;CÞ is loc even,(U,V) is a (p,q)-pair of Pitt weights and SðtÞ¼ aðmÞeðmtÞ is absolutely convergent, then there exists C ¼ Cðp; q; U :VÞ[ 0 such that 1=q 1=p kV Sk  kU ðaHw Þk : ð1:1Þ q;T p min jwcðtÞj jtj T d More generally, the same inequality holds if w is replaced by w 2 L ðR;CÞ. Remark 1.1 Unless otherwise stated, hereafter the constant C is not necessarily the same at each occurrence. Further, let us assume henceforth that min jw ðtÞj 6¼ 0, for jtj T otherwise (1.1) is trivial. 123 A weighted inequality for... In order to prove Theorem 1.1, we apply a weighted Fourier transform norm inequality, called Pitt’s inequality, which is an old result in Fourier analysis ([15]). Such an inequality is quoted in Sect. 2, before the proof of Theorem 1.1, by referring to the version essentially given by Heinig et al. ([8]), [10, 14] in 1983–1984 (see also [2, 6] for advanced and more recent versions). It is well-known that Pitt’s inequality yields the Hardy-Littlewood inequality, the Hausdorff-Young inequality and the Plancherel identity. In the latter case, i.e., p ¼ q ¼ 2, U ¼ V ¼ 1, C ¼ 1, the inequality (1.1) becomes Z Z 2 2 jSðyÞj dy  jðaHw ÞðyÞj dy: ð1:2Þ min jwcðtÞj T R jtj T d In the present paper we continue our study of (1.2), that was started in [5]. We point out that most of the next considerations can be easily adapted to the general case (1.1). Let us recall that a first consequence of (1.2) is the following majorant prop- erties established in [5]. Let d; T 2ð0; þ1Þ and let w 2 L ðR;CÞ be even such that loc ðw  w ÞðxÞ :¼ w ðtÞw ðx  tÞ dt  0; 8x 2 R: d d d d P P If SðtÞ¼ aðmÞeðmtÞ and BðyÞ¼ bðmÞeðmyÞ are absolutely convergent, and the m m first sum is a majorant for the second one, i.e., jaðmÞj  bðmÞ for all m, then Z Z 2 2 jSðyÞj dy  jðbHw ÞðxÞj dx: ð1:3Þ T min jwcðtÞj R jtj T d Now, it is worthwhile to recall that by a straightforward application of the Hardy- Littlewood majorant principle (see [13, Ch.7] or [11]) one gets Z Z T T 2 2 jSðyÞj dy  3 jBðyÞj dy: T T The combination of such an inequality with (1.2) for B(y) gives a weaker inequality than (1.3) because of the factor 3 in the right-hand side. Moreover, we recall that such a factor is best possible in the Hardy-Littlewood majorant principle (see [12]). Example 1.1 A well-known instance of (1.2) is given by taking the weight u associated to the unit step function 1if t[ 0 uðtÞ :¼ 0 otherwise. It is easily seen that ub ðyÞ¼ eðdy=2Þ d sincðdyÞ. Therefore, for 0\dT\1, from (1.2)we find the Gallagher inequality [7] 123 G. Coppola, M. Laporta kSk  aðmÞ dx: 2;T 2 2 d sinc ðdTÞ x\m  xþd We refer the reader to [9] for a first introduction to the applications of such an inequality within the analytic number theory. Example 1.2 Another remarkable instance of (1.2) is the special case of the Cesàro weight ð1:4Þ C ðyÞ :¼ maxð1  d jyj; 0Þ; whose Fourier transform is C ðyÞ¼ d sinc ðdyÞ. In the literature, this is known as Fejer kernel [4]. For dT 2ð0; 1Þ, the inequality (1.2) gives 1 jm  xj kSk  1  aðmÞ dx: 2;T 2 4 d sinc ðdTÞ jmxj d In [5] we have applied the latter inequality to the Dirichlet polynomials and to the distribution of certain arithmetical functions in short intervals. In the present paper we focus on some further aspects of (1.2), mostly concerning the optimal choice of the weight w. To this aim and for the convenience of the reader, we need to quote some definitions and properties already introduced in [5]. First, for every w 2 L ðR;CÞ let us define the normalized self-convolution of w ,with d[ 0, loc as wfðxÞ :¼ ðw  w ÞðxÞ: d d d 2d For example, the Cesàro weight (1.4) is the normalized self-convolution of the restriction to ½d=2; d=2 of 1: 1 1 C ðxÞ¼ dt ¼ ð1  1 ÞðxÞ¼ 1 ðxÞ: d=2 d=2 d=2 d d jtj d=2 jx  tj d=2 It is well-known that the iteration of the self-convolution gives rise to a process of smoothing (see [4]). Moreover, the support of w is doubled with respect to the support of w in the sense that it is a subset of ½2d; 2d. Because of the normalizing factor ð2dÞ , that takes into account the length of the integration interval, the magnitude of w is not altered much by the normalized self-convolution. More precisely, if one has w  1, i.e., 1 w 1, in an interval of length d, then d d there exists an interval of length d (not necessarily the same) where wf  1. Recall that A B stands for jAj cB, where c[ 0 is an unspecified constant depending on h. 123 A weighted inequality for... From another well-known property of the convolution it follows that the Fourier 1 2 transform of wf is wfðyÞ¼ ð2dÞ wcðyÞ . In particular, from 1 ðyÞ¼ 2dsincð2dyÞ d d d d d 2 1 2 c g d we rediscover that C ðyÞ¼ 1 ðyÞ¼ d 1 ðyÞ ¼ dsinc ðdyÞ. d d=2 d=2 The normalized self-convolution generates recursively the family of weights: hji hj1i C ðxÞ :¼ C ðxÞ; j  1; d d=2 h0i h1i with the base steps C ðxÞ :¼ 1 ðxÞ and C ðxÞ :¼ C ðxÞ. d d d d In [5] we proved the following inductive formula for the Fourier transform of such weights: 4d dy d j hji ð1:5Þ C ðyÞ¼ sinc ; 8j  0: 2 j1 2 2 hji j2 1 Consequently, C ðyÞ d at least for 0jyj 2 d . Such a process of continuous smoothing through the self-convolution of a weight w has a discrete counterpart given by the autocorrelation of w (since no confusion d d can arise in the following, we will use the simpler term correlation): XX C ðaÞ :¼ w ðnÞw ðmÞ: w d d nm n  m ¼ a For example, since it turns out that X XX 1 1 C ðtÞ C ðtÞ¼ 1 ¼ 1 ¼ ; d d d a  djtj a; b  d b  a ¼ t the Cesàro weight (1.4) is the normalized correlation of the unit step weight u . Note that the Cesàro weight (1.4) is generated by both types of smoothing from the function 1. Moreover, through an iteration of the normalized correlation one might hji parallel the self-convolution process to generate the whole family of weights C , j  1. An important aspect is that if the coefficients of a trigonometric series are correlations of w, then such a series is non-negative. More precisely, X XXX X C ðhÞeðhmÞ¼ w ðnÞw ðmÞeðhmÞ¼ w ðnÞeðnmÞ : w d d d h h nm¼h n A particularly well-known case is the Fejér kernel (compare the C formula) 123 G. Coppola, M. Laporta X X X d C ðhÞeðhmÞ¼ C ðhÞeðhmÞ¼ eðnmÞ : d u h h 1  n  d Such a positivity property is the complete analogous of the aforementioned fact that the Fourier transform of a self-convolution is a square. Now, let us introduce our comparison argument for the weight w in view of possible refinements of (1.2). First, recall that m :¼ min jwcðtÞj 6¼ 0 d;T d jtj T by assumption in order to avoid triviality. Thus, we define E ¼ E ðT ; dÞ :¼fy 2 R : jwcðyÞj \m g w w d d;T and write Z Z 2 2 2 kSk  m jSðyÞwcðyÞj dy þ jSðyÞj dy 2;T d;T RnE E w w Z ð1:6Þ 2 2 m jSðyÞwcðyÞj dy þjE j sup jSðyÞj : d w d;T RnE y2E w w In order to take advantage of such an inequality one would require E either with a small measure or free of peaks of S(y), if it does not support a sufficiently small L - norm. For example, let us examine the possible scenario for the weights (1.5). To this aim, recall that sinð2pdyÞ if y 6¼ 0; h0i C ðyÞ¼ 1 ðyÞ¼ py 2d if y ¼ 0: The aforementioned triviality for (1.2) does not occur if we assume dT 2ð0; 1=2Þ, for otherwise it would be dt ¼ 1=2 for some t 2ð0; T and consequently 0 0 2 2 sin ð2pdtÞ sin ð2pdt Þ min 1 ðtÞ ¼ min ¼ ¼ 0: 2 2 jtj T jtj T ðptÞ ðpt Þ On the other hand, it is plain that if dT 2ð0; 1=2Þ, then one has sin ð2pdTÞ fy 2 R : jyj[ T g E ðT ; dÞ¼ y 2 R : 1 ðyÞ \ 0 1 d ðpTÞ for some T  T. It is easily seen that analogous considerations hold for the whole hji family fC g . However, in several applications of (1.2), like in the case of j  0 Dirichlet polynomials (see [5]), the contribution from the tail jyj[ T might amount to a remainder term. On the other hand, note that the inequality 123 A weighted inequality for... 2 2 kSk  m kwcSk 2;T d;T 2;T trivially holds for any choice of w such that m 6¼ 0. Even for this reason, it is d d;T worthwhile to compare weights in view of possible refinements of the first term on the right-hand side of (1.6). To this end, we give the following definition. Definition 1.2 Let us suppose that w ; v are such that wcðtÞ 6¼ 0, vb ðtÞ 6¼ 0, d d d d 8t 2½T ; T. We say that v is T-better than w if d d min jvb ðtÞj d 2 jtj T jvb ðyÞj 8y 2½T ; T: ð1:7Þ 2 2 min jwcðtÞj jwcðyÞj d d jtj T If so, we write v w . d d If v w , it is plain that v yields a refinement of the upper bound of kSk d d d T 2;T with respect to w , at least in ½T ; T. In §3 we prove the next result on the family of weights (1.5) and discuss the comparison involving also some weights as C  1 and d d the Lanczos weight. hjþ1i hji j1 Theorem 1.2 Let d; T 2ð0; þ1Þ. For every j  0, if dT\2 , then C C . d T d Moreover, the inequality d 2 hjþ1i min C ðtÞ d 2 hjþ1i C ðyÞ jtj T ð1:8Þ d 2 d 2 hji hji min C ðtÞ C ðyÞ d d jtj T j1 0 j1 00 still holds for all jyj2 2 ð2k þ 1Þ=d þ e ; 2 ð2k þ 3Þ=d  e , where k k k2N 0 0 00 00 j1 e ¼ e ðj; d; TÞ; e ¼ e ðj; d; TÞ2ð0; 2 =dÞ k k k k 0 00 2 j j1 are such that e ; e dT =ð2 kÞ T =ð2kÞ as dT ! 2 , 8j; k 2 N. k k j1 1 0 j1 Remark 1.2 For any fixed j; k 2 N, the interval 2 ð2k þ 1Þd þ e ; 2 ð2k þ 00 j1 3Þd  e tends to cover the whole interval ½ð2k þ 1ÞT ; ð2k þ 3ÞT as dT ! 2 . Somehow this means that (1.8) holds almost everywhere in R n½T ; T as long as j1 dT ! 2 and it gives a further chance to get even a refinement of the second term hji hjþ1i on the left-hand side of (1.6) by replacing C with C . Furthermore, such a d d refinement might rely also on the value of the series Sðð2k þ 1ÞTÞ, 8k 2 Z. hji Remark 1.3 An effective use of (1.2) with C requires finding explicit expressions of such weights. For example, the normalized self-convolution of C is the so- d=2 h2i called Jackson-de La Vallé Poussin weight C , that is the following cubic spline [4, Problem 5.1.2 (v)]: 123 G. Coppola, M. Laporta 6jtj  6dt þ d if jtj d=2; < 3d d ðC  C ÞðtÞ¼ d=2 d=2 2ðd jtjÞ if d=2\jtj d; > 3 3d 0if jtj[ d: h2i Note that the support of C is ½d; d, as expected. 2 Proof of Theorem 1.1 Here we recall Pitt’s inequality: Let 1\p  q\1 be fixed and (U, V) be a (p, q)-pair of Pitt weights. There exists C ¼ Cðp; q; U :VÞ[ 0 such that 1=q 1=p kV f k  CkU f k : ð2:1Þ q p 1=p p for all f such that U f 2 L ðR;CÞ. Proof First, note that from the hypotheses on S and w it follows that Z Z X X jaðmÞw ðx  mÞj dx ¼ jaðmÞj jwðx  mÞj dx R jxmj d m m ¼kwk jaðmÞj\1: 1;d Therefore, by applying Lebesgue’s dominated convergence theorem [16, Th.1.38] it turns out that aHw 2 L ðR;CÞ and ðaHw ÞðyÞ :¼ ðaHw ÞðxÞeðxyÞ dx d d ¼ aðmÞ w ðx  mÞeðxyÞ dx ¼ aðmÞeðmyÞ w ðtÞeðtyÞ dt ¼ SðyÞwcðyÞ: d d 1=p p Then, observe that we can assume U ðaHw Þ2 L ðR;CÞ for 1\p\1, for otherwise the inequality 1=q 1=p kV S wck  C kU ðaHw Þk : ð2:2Þ d d q p would be trivial. Hence, (2.2) follows by applying Pitt’s inequality (2.1), being plain that (recall that w, V are even functions) 123 A weighted inequality for... Z Z q q jSðyÞwcðyÞj VðyÞ dy ¼ jSðyÞwcðyÞj VðyÞ dy: d d R R By taking any fixed T 2ð0; þ1Þ, clearly (2.2) implies (1.1). Now, let us prove that 1=q 1=p kV S wbk  CkU ðaHwÞk : ð2:3Þ q p As before, we can clearly assume that aHw 2 L ðR;CÞ and apply (2.2) with d ¼ n 2 N to write 1=q 1=p kV S wck  CkU ðaHw Þk : ð2:4Þ n n q p Since w ðtÞeðtyÞ converges to wðtÞeðtyÞ, with jw ðtÞeðtyÞj  jwðtÞj and n n w 2 L ðR;CÞ, the dominated convergence theorem yields 1=q 1=q lim kV S wck ¼kV S wbk : q q On the other hand, the same theorem implies that Z Z p p lim jðaHw ÞðxÞj UðxÞ dx ¼ jðaHwÞðxÞj UðxÞ dx: R R Hence, (2.3) follows from (2.4) after passage to the limit as n !1. Again, by taking any fixed T 2ð0; þ1Þ, it is plain that (1.1) holds if w is replaced by w 2 L ðR;CÞ. aq bp Remark 2.1 It is well konwn that, by taking VðxÞ¼jxj , UðxÞ¼jxj with maxf0; 1=p þ 1=q  1g a\1=q and b :¼ a þ 1  1=p  1=q; ð2:5Þ Pitt’s inequality (2.1) yields three classical inequalities in Fourier analysis: ● the Hardy-Littlewood inequality if p ¼ q  2, a ¼ 0or 1\p ¼ q  2, b ¼ 0; ● the Hausdorff-Young inequality if q ¼ 1  1=p  2, a ¼ b ¼ 0; ● the Plancherel identity if p ¼ q ¼ 2, a ¼ b ¼ 0. aq bp Accordingly, for VðxÞ¼jxj and UðxÞ¼jxj the inequality (2.2) turns into a b kv wc Sk  Ckv ðaHw Þk ; d d q p where we have set vðxÞ¼ jxj. Thus, (1.1) specializes to the following instances paralleling the aforementioned three properties: 123 G. Coppola, M. Laporta 12=p kSk  kv ðaHw Þk ; with p ¼ q  2; p;T p min jwcðtÞj jtj T d 12=p kv Sk  kaHw k ; with 1\p ¼ q  2; p;T p min jwcðtÞj jtj T kSk  kaHw k ; with q ¼ 1  1=p  2; q;T p min jwcðtÞj jtj T 1=ð2pÞ 1=ð2qÞ (in the latter C ¼ p =q is the so-called Beckner’s constant); kSk  kaHw k : 2;T 2 min jwcðtÞj jtj T d Finally, we conclude this remark by recalling that the original version of Pitt’s inequality was tailored for the Fourier series and the power series [15]. In particular, by assuming that m 2 Z for our Fourier series S, Theorem 2 of [15] yields 1=q 1=p a q b p jaðmÞm j  C jt SðtÞj dt ; m2Z 1=q X 1=p q p a b jt SðtÞj dt  C jaðmÞm j ; m2Z where p; q; a; b are as in (2.5). 3 Proof of Theorem 1.2: comparison of weights Proof Let us start with j ¼ 0 and show that C is T-better than 1 by assuming that d d dT\1=2, namely we prove that min C ðtÞ 2 C ðyÞ jtj T ð3:1Þ 2 2 b   b min 1 ðtÞ 1 ðyÞ d d jtj T holds for all y 2½T ; T. First, recall that dT\1=2 yields 2 4 2 sin ð2pdTÞ 2 sin ðpdTÞ b  c min 1 ðtÞ ¼ 6¼ 0; min C ðtÞ ¼ 6¼ 0: d d 2 4 2 jtj T jtj T ðpTÞ ðpTÞ d Since min C ðtÞ 2 C ð0Þ jtj T tan ðpdTÞ 1 d ¼  ¼ ; 2 2 2 b  4  b 4ðpdTÞ min 1 ðtÞ 1 ð0Þ d d jtj T we can assume that y 6¼ 0 and set x ¼ y=T, h ¼ dT. Thus, (3.1) becomes 123 A weighted inequality for... C ðxTÞ tan ðphxÞ G ðxÞ :¼ ¼  G ð1Þ: h   h 2 2 ð2phxÞ 1 ðxTÞ It is easy to see that G ðxÞ satisfies the following properties: (i) G ðxÞ is even with respect to both x and h (ii) G ðxÞ is strictly increasing with respect to x 2ð0; 1 (iii) lim G ðxÞ¼ 1=4 8h 2ð0; 1=2Þ x!0 (iv) G ð1Þ is strictly increasing with respect to h 2ð0; 1=2Þ (v) lim G ð1Þ¼ 1=4, lim G ð1Þ¼þ1 h h h!0 h!1=2 (vi) G ðxÞ¼ 0 () x ¼ k=h 8k 2 Znf0g (note that 8k 2 Z nf0g and 8h 2 ð0; 1=2Þ one has jkj=h[ 2jkj 2) (vii) G ðxÞ! þ1 as x !ð2k þ 1Þð2hÞ 8k 2 Z. Note that j2k þ 1jð2hÞ [ j2k þ 1j 1, 8k 2 Z and 8h 2ð0; 1=2Þ. From properties (i)-(v) it follows that G ðxÞ G ð1Þ for all x 2½1; 1nf0g, that h h is to say that (3.1) holds for all jyj T, i.e., C 1 . d d In order to prove the second part of the theorem for j ¼ 0, observe that properties (i), (vi) and (vii) imply that the above inequality for G is true for all [ [ 1 0 1 00 jxj2 I :¼ ½ð2k þ 1Þð2hÞ þ D ; ð2k þ 3Þð2hÞ  D ; h;k k k k2N k2N 0 0 00 00 where 0\D ¼ D ðhÞ; D ¼ D ðhÞ\ð2hÞ are such that the endpoints of the k k k k intervals I are the solutions of the equation G ðxÞ¼ G ð1Þ. In particular, this h;k h h yields 0 0 tan ðp=2 þ phD Þ¼ G ð1Þ ð2k þ 1Þp þ 2phD ; k k 0 0 with 0\hD \1=2. From the latter equation and from (v) we deduce that D ðh k k 1=2Þk as h ¼ dT ! 1=2, for all k 2 N. An analogous property holds for D . 0 00 0 00 Hence, we set e ð0; d; TÞ¼ TD ; e ð0; d; TÞ¼ TD to get the theorem proved in the k k k k case j ¼ 0. For j[ 0 it sufficies to note that (see (1.5)) jþ1 d 2 hjþ1i 2 jþ1 C ðyÞ tanðpdy=2 Þ tanðphxÞ j ¼ ¼ ¼ G ðxÞ ; j1 d 2 dpy=2 2phx hji C ðyÞ j 2 where we have set h ¼ dT =2 , x ¼ y=T and G ðxÞ¼ð2phxÞ tan ðphxÞ as before. j1 Since 0\dT\2 if and only if 0\h\1=2, the conclusion follows from what we have showed in the case j ¼ 0. h Remark 3.1 Theorem 1.2 seems to indicate that one could generate more and more T-better weights by increasing the iteretad convolutions of 1 . However, the following examples suggest that one must exercise some caution about the choice of the weights to be compared. 123 G. Coppola, M. Laporta Example 3.1 Let us compare C and D :¼ C  1 . First, recall that 2d 2d d d 2 2 d d c b C ðyÞ¼ 2dsinc ð2dyÞ; D ðyÞ¼ C ðyÞ1 ðyÞ¼ 2d sinc ðdyÞsincð2dyÞ; 2d 2d d d so that > tan ðpdyÞ 2 4 D ðyÞ if y 6¼ 0; d sinc ðdyÞ 2d ¼ ¼ ðpyÞ 2 2 d sinc ð2dyÞ C ðyÞ : 2d 2 d if y ¼ 0: Further, for dT\1=2, one has 4 2 min D ðtÞ ¼ 4d min sinc ðdtÞsinc ð2dtÞ 2d jtj T jtj T 4 6 2 ¼ 4d sinc ðdTÞ cos ðpdTÞ 6¼ 0; 2 4 min C ðtÞ ¼ 4d sinc ð2dTÞ 6¼ 0: 2d jtj T Since it turns out that min D ðtÞ 2 2 2d 2 d  d jtj T tan ðpdTÞ D ðTÞ D ðyÞ 2d 2d ¼ ¼  ; 8y 2½T ; T; 2 2 2 2 d  d  d ðpTÞ min C ðtÞ C ðTÞ C ðyÞ 2d 2d 2d jtj T then we conclude that D C once dT\1=2. 2d 2d Note that both C and D have support in ½2d; 2d. On the other hand, by 2d 2d comparing C and D we see that d 2d > sin ð2pdyÞ 2 < d c b if y 6¼ 0; D ðyÞ 2d jC ðyÞ1 ðyÞj 2 d d ¼ ¼ 1 ðyÞ ¼ ðpyÞ 2 2 c  c  > C ðyÞ C ðyÞ : d d 4d if y ¼ 0; and, for dT\1=2, one has min D ðtÞ 2 2d jtj T sin ð2pdTÞ D ðyÞ 2 2d ¼  1 ðyÞ ¼ ; 8y 2½T ; T; 2 2 2 c  d ðpTÞ min C ðtÞ C ðyÞ d 2d jtj T that is to say, if dT\1=2, then C D . Analogously, it is easy to see that d 2d hji hji hji hj1i C C for all j  0 (and consequently C C for all j  1, because of d T 2d d T 2d j1 Theorem 1.2), as long as dT\2 . Example 3.2 Given real numbers d  c[ 0, the Lanczos weight is defined as [4, Problem 5.1.2 (v)] 123 A weighted inequality for... 1if jxj d  c; 1 d jxj L ðxÞ :¼ ð1  1 ÞðxÞ¼ if d  c\jxj d; d;c dc=2 c=2 c c 0if jxj[ d; whose Fourier transform is L ðyÞ¼ ð2d  cÞsincðcyÞsincðð2d  cÞyÞ d;c sinðpcyÞ sinð2pdy  pcyÞ if y 6¼ 0; ¼ ðpyÞ c 2d  c if y ¼ 0: The diagram of L is an isosceles trapezium. Note that 1  L  C ¼ L for d;c d d;c d d;d any d  c[ 0. Since L ¼ C , then we can assume that d[ c[ 0. Further, by d;d d b d taking cT\dT\ð2d  cÞT\1=2 one has 1 ðyÞ 6¼ 0 and L ðyÞ 6¼ 0 for all d d;c y 2½T ; T. Therefore, under such assumptions we can compare L and 1 by d;c d verifying the inequality 1 ðyÞ ð2dÞ sinc ð2dyÞ 2 2 2 2 ð2d  cÞ sinc ðcyÞsinc ðð2d  cÞyÞ L ðyÞ d;c min 1 ðtÞ 2 d ð2dÞ sinc ð2dTÞ jtj T ¼   : 2 2 2 2 ð2d  cÞ sinc ðcTÞsinc ðð2d  cÞTÞ min L ðtÞ d;c jtj T To this aim, let us write min 1 ðtÞ d 2 jtj T pcT sinð2pdTÞ sinðpcTÞ sinðpð2d  cÞTÞ min L ðtÞ d;c jtj T sinðpð2d  cÞT þ pcTÞ ¼ðpcTÞ sinðpcTÞ sinðpð2d  cÞTÞ ¼ðpcTÞ cotðpcTÞþ cot pð2d  cÞT 2 2 ¼FðcTÞ þ Fðð2d  cÞTÞ þ ð2d  cÞ 2c FðcTÞFðð2d  cÞTÞ; 2d  c where FðxÞ :¼ðpxÞ cotðpxÞ. Since FðxÞ 1 for jxj 1=2, we get 123 G. Coppola, M. Laporta min 1 ðtÞ 2 d 2 1 ð0Þ jtj T c 2c ð2dÞ 1 þ þ ¼ ¼ : 2 2 2 2 d  d 2d  c ð2d  cÞ ð2d  cÞ min L ðtÞ L ð0Þ d;c d;c jtj T Therefore, we can assume that y 6¼ 0 and write 1 ðyÞ c 2c 2 2 ¼ FðcyÞ þ Fðð2d  cÞyÞ þ FðcyÞFðð2d  cÞyÞ: 2 2 d  2d  c ð2d  cÞ L ðyÞ d;c Since it is readily seen that FðxÞ Fðx Þ for jxj x \1=2, we conclude that L 0 0 d;c 1 provided cT\dT\ð2d  cÞT\1=2. Finally, we show that C L , under the same conditions on c, d and T. Indeed, d d;c we prove that the inequality 2 4 C ðyÞ d sinc ðdyÞ 2 2 2 2 ð2d  cÞ sinc ðcyÞ sinc ðð2d  cÞyÞ L ðyÞ d;c min C ðtÞ 2 4 d sinc ðdTÞ jtj T 2 2 2 2 ð2d  cÞ sinc ðcTÞ sinc ðð2d  cÞTÞ min L ðtÞ d;c jtj T holds for all y 2½T ; T. To this aim, let us recall that, being concave in ðp; pÞ, the function sinc is also logarithmically concave in the same interval (see [3, Ch.3]), i.e., for all k 2½0; 1 and all x ; x 2ð1; 1Þ one has 1 2 k 1k sincðkx þð1  kÞx Þ sinc ðx Þsinc ðx Þ: 1 2 1 2 In particular, by taking k ¼ 1=2, x ¼ cT and x ¼ð2d  cÞT, with 1 2 cT\dT\ð2d  cÞT\1=2, this yields min C ðtÞ C ð0Þ jtj T d ¼ : 2 2 2 d  d ð2d  cÞ min L ðtÞ L ð0Þ d;c d;c jtj T Therefore, let assume that y 6¼ 0 and write c  2 C ðyÞ c sin ðpdyÞ 2 2 2 2 d sin ðpcyÞ sin ðpð2d  cÞyÞ L ðyÞ d;c c 1  cosðpcyÞ cosðpð2d  cÞyÞ ¼ 1 þ : sinðpcyÞ sinðpð2d  cÞyÞ ð2dÞ Thus, it is plain that we get the desired conclusion once we prove that, for any fixed real numbers a, b such that 0\a\b\p=ð2TÞ, the function 123 A weighted inequality for... 1  cosðayÞ cosðbyÞ 1 1 G ðyÞ :¼ ¼ a;b sinðayÞ sinðbyÞ sinðayÞ sinðbyÞ tanðayÞ tanðbyÞ is monotone increasing in (0, T]. Indeed, it turns out that d cosðayÞ cosðbyÞ G ðyÞ¼ b sinðayÞ a sinðbyÞ [ 0; a;b 2 2 dy sin ðayÞ sin ðbyÞ because 0\a\b\p=ð2TÞ and y 2ð0; T yield cosðayÞ cosðbyÞ[ 0 and 1 1 ðayÞ sinðayÞ[ ðbyÞ sinðbyÞ. Acknowledgements The authors wish to thank Laura De Carli for helpful comments on an early version of the present paper. Further, they wish to emphasize the excellent and constructive attention received from the referee. Author Contributions All the authors contributed equally for the preparation of the present paper. Funding Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI- CARE Agreement. Declarations Conflict of interest The authors declare that they do not have any competing interests. Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ References 1. Beckner, W. 1975. Inequalities in Fourier analysis. Annals of Mathematics (2) 102 (1): 159–182. 2. Benedetto, J.J., and H.P. Heinig. 2003. Weighted Fourier inequalities: new proofs and generalizations. Journal of Fourier Analysis and Applications 9: 1–37. 3. Boyd, S.P., and L. Vandenberghe. 2004. Convex Optimization. Cambridge: Cambridge University Press. 4. Butzer, P.L., and R.J. Nessel. 1971. Fourier Analysis and Approximation, vol. I. Basel und Stuttgart: Birkhäuser Verlag. 5. Coppola, G., and M. Laporta. 2015. A generalization of Gallagher’s Lemma for exponential sums. Šiauliai Maths Seminar 10 (18): 1–19. 6. De Carli, L., D. Gorbachev, and S. Tikhonov. 2017. Pitt inequalities and restriction theorems for the Fourier transform. Revista Matematica Iberoamericana 33 (3): 789–808. https://doi.org/10.4171/RMI/ 7. Gallagher, P.X. 1970. A large sieve density estimate near ¼ 1. Inventiones Mathematicae 11: 329–339. 123 G. Coppola, M. Laporta 8. Heinig, H.P. 1984. Weighted norm inequalities for classes of operators. Indiana University Mathe- matics Journal 33: 573–582. 9. Huxley, M.N. 1972. The Distribution of Prime Numbers. Oxford: Oxford University Press. 10. Jurkat, W.B., and G. Sampson. 1984. On rearrangement and weight inequalities for the Fourier transform. Indiana University Mathematics Journal 33: 257–270. 11. Laporta, M. 2014. Some remarks on the Hardy–Littlewood majorant principle for exponential sums. Ricerche di Matematica 63 (1): 179–186. https://doi.org/10.1007/s11587-014-0196-6. 12. Logan, B.F. 1988. An interference problem for exponentials. Michigan Mathematical Journal 35: 369–393. 13. Montgomery, H.L. 1994. Ten Lectures on the Interface Between Analytic Number Theory and Har- monic Analysis. CBMS Regional Conf. Ser. in Math. 84, Providence, RI: American Mathematical Society. 14. Muckenhoupt, B. 1983. Weighted norm inequalities for the Fourier transform. Transactions of the American Mathematical Society 276: 729–742. 15. Pitt, H.R. 1937. Theorems on Fourier series and power series. Duke Mathematical Journal 3: 747– 16. Rudin, W. 1987. Real and Complex Analysis, 3rd edn. New York, NY: Mc Graw Hill International Editions. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

A weighted inequality for trigonometric series

The Journal of Analysis , Volume OnlineFirst – Sep 22, 2022

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Abstract

By applying Pitt’s inequality we prove a weighted L version of Gallagher’s inequality for trigonometric series. Furthermore, we consider a family of weights generated by a smoothing process, via convolution operation, whose first steps are the indicator function of a compact interval and the so-called Cesàro weight sup- ported in the same interval. The eventual aim is the comparison of such weights in view of possible refinements of our inequality for p ¼ 2. Keywords Trigonometric series · Pitt’s inequality · Majorant properties Mathematics Subject Classification 42A05 · 26D15 · 42A38 1 Introduction The trigonometric series considered here are of the type SðtÞ :¼ aðmÞeðmtÞ; 2pix where eðxÞ :¼ e and aðmÞ2 C for a strictly increasing sequence m 2 R. For any given T 2ð0; þ1Þ and p 2½1; þ1Þ, let us denote the L -norm of f : R ! C by R  R 1=p 1=p p p kf k :¼ jf ðtÞj dt and set kf k :¼ jf ðtÞj dt . The space of the p p;T R T Communicated by S. Ponnusamy. & Maurizio Laporta mlaporta@unina.it Giovanni Coppola giovanni.coppola@unina.it Via Partenio 12, 83100 Avellino, AV, Italy Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli, Complesso di Monte S. Angelo, Via Cinthia, 80126 Naples, NA, Italy 123 G. Coppola, M. Laporta p p complex-valued functions defined on R with a finite L -norm is denoted by L ðR;CÞ. By f 2 L ðR; BÞ, with B  C, we mean that f : R ! B is summable on each loc compact subset of R. The Fourier transform of an integrable f : R ! C is denoted by f ðyÞ :¼ f ðtÞeðtyÞ dt: The convolution of f ; g 2 L ðR;CÞ is ðf  gÞðxÞ¼ f ðtÞgðx  tÞ dt: For any weight w : R ! C and any d[ 0, we set w ðxÞ¼ wðxÞ or 0 according as jxj d or not. Further, we denote the discrete convolution of aðmÞ and w ðxÞ as ðaHw ÞðyÞ :¼ aðmÞw ðy  mÞ: d d The function that is identically 1 we denote by 1, so that 1 is the characteristic function of the interval ðd; dÞ. Recall that 1 ðyÞ¼ 2dsincð2dyÞ, where sinðpxÞ if x 6¼ 0; sinc x :¼ px 1if x ¼ 0: Definition 1.1 Let 1\p  q\1 be fixed and p ¼ p=ðp  1Þ.If U ; V 2 L ðR;R Þ are even, U is non-decreasing on ð0; þ1Þ, V is non-increasing on loc ð0; þ1Þ and Z Z s 1=p 1=s 1=q 1p sup UðtÞ dt VðtÞ dt \ þ1; s[ 0 0 0 then we say that (U,V)isa(p,q)-pair of Pitt weights. In Sect. 2 the following result is proved. Theorem 1.1 Let d; T 2ð0; þ1Þ and 1\p  q\1 be fixed. If w 2 L ðR;CÞ is loc even,(U,V) is a (p,q)-pair of Pitt weights and SðtÞ¼ aðmÞeðmtÞ is absolutely convergent, then there exists C ¼ Cðp; q; U :VÞ[ 0 such that 1=q 1=p kV Sk  kU ðaHw Þk : ð1:1Þ q;T p min jwcðtÞj jtj T d More generally, the same inequality holds if w is replaced by w 2 L ðR;CÞ. Remark 1.1 Unless otherwise stated, hereafter the constant C is not necessarily the same at each occurrence. Further, let us assume henceforth that min jw ðtÞj 6¼ 0, for jtj T otherwise (1.1) is trivial. 123 A weighted inequality for... In order to prove Theorem 1.1, we apply a weighted Fourier transform norm inequality, called Pitt’s inequality, which is an old result in Fourier analysis ([15]). Such an inequality is quoted in Sect. 2, before the proof of Theorem 1.1, by referring to the version essentially given by Heinig et al. ([8]), [10, 14] in 1983–1984 (see also [2, 6] for advanced and more recent versions). It is well-known that Pitt’s inequality yields the Hardy-Littlewood inequality, the Hausdorff-Young inequality and the Plancherel identity. In the latter case, i.e., p ¼ q ¼ 2, U ¼ V ¼ 1, C ¼ 1, the inequality (1.1) becomes Z Z 2 2 jSðyÞj dy  jðaHw ÞðyÞj dy: ð1:2Þ min jwcðtÞj T R jtj T d In the present paper we continue our study of (1.2), that was started in [5]. We point out that most of the next considerations can be easily adapted to the general case (1.1). Let us recall that a first consequence of (1.2) is the following majorant prop- erties established in [5]. Let d; T 2ð0; þ1Þ and let w 2 L ðR;CÞ be even such that loc ðw  w ÞðxÞ :¼ w ðtÞw ðx  tÞ dt  0; 8x 2 R: d d d d P P If SðtÞ¼ aðmÞeðmtÞ and BðyÞ¼ bðmÞeðmyÞ are absolutely convergent, and the m m first sum is a majorant for the second one, i.e., jaðmÞj  bðmÞ for all m, then Z Z 2 2 jSðyÞj dy  jðbHw ÞðxÞj dx: ð1:3Þ T min jwcðtÞj R jtj T d Now, it is worthwhile to recall that by a straightforward application of the Hardy- Littlewood majorant principle (see [13, Ch.7] or [11]) one gets Z Z T T 2 2 jSðyÞj dy  3 jBðyÞj dy: T T The combination of such an inequality with (1.2) for B(y) gives a weaker inequality than (1.3) because of the factor 3 in the right-hand side. Moreover, we recall that such a factor is best possible in the Hardy-Littlewood majorant principle (see [12]). Example 1.1 A well-known instance of (1.2) is given by taking the weight u associated to the unit step function 1if t[ 0 uðtÞ :¼ 0 otherwise. It is easily seen that ub ðyÞ¼ eðdy=2Þ d sincðdyÞ. Therefore, for 0\dT\1, from (1.2)we find the Gallagher inequality [7] 123 G. Coppola, M. Laporta kSk  aðmÞ dx: 2;T 2 2 d sinc ðdTÞ x\m  xþd We refer the reader to [9] for a first introduction to the applications of such an inequality within the analytic number theory. Example 1.2 Another remarkable instance of (1.2) is the special case of the Cesàro weight ð1:4Þ C ðyÞ :¼ maxð1  d jyj; 0Þ; whose Fourier transform is C ðyÞ¼ d sinc ðdyÞ. In the literature, this is known as Fejer kernel [4]. For dT 2ð0; 1Þ, the inequality (1.2) gives 1 jm  xj kSk  1  aðmÞ dx: 2;T 2 4 d sinc ðdTÞ jmxj d In [5] we have applied the latter inequality to the Dirichlet polynomials and to the distribution of certain arithmetical functions in short intervals. In the present paper we focus on some further aspects of (1.2), mostly concerning the optimal choice of the weight w. To this aim and for the convenience of the reader, we need to quote some definitions and properties already introduced in [5]. First, for every w 2 L ðR;CÞ let us define the normalized self-convolution of w ,with d[ 0, loc as wfðxÞ :¼ ðw  w ÞðxÞ: d d d 2d For example, the Cesàro weight (1.4) is the normalized self-convolution of the restriction to ½d=2; d=2 of 1: 1 1 C ðxÞ¼ dt ¼ ð1  1 ÞðxÞ¼ 1 ðxÞ: d=2 d=2 d=2 d d jtj d=2 jx  tj d=2 It is well-known that the iteration of the self-convolution gives rise to a process of smoothing (see [4]). Moreover, the support of w is doubled with respect to the support of w in the sense that it is a subset of ½2d; 2d. Because of the normalizing factor ð2dÞ , that takes into account the length of the integration interval, the magnitude of w is not altered much by the normalized self-convolution. More precisely, if one has w  1, i.e., 1 w 1, in an interval of length d, then d d there exists an interval of length d (not necessarily the same) where wf  1. Recall that A B stands for jAj cB, where c[ 0 is an unspecified constant depending on h. 123 A weighted inequality for... From another well-known property of the convolution it follows that the Fourier 1 2 transform of wf is wfðyÞ¼ ð2dÞ wcðyÞ . In particular, from 1 ðyÞ¼ 2dsincð2dyÞ d d d d d 2 1 2 c g d we rediscover that C ðyÞ¼ 1 ðyÞ¼ d 1 ðyÞ ¼ dsinc ðdyÞ. d d=2 d=2 The normalized self-convolution generates recursively the family of weights: hji hj1i C ðxÞ :¼ C ðxÞ; j  1; d d=2 h0i h1i with the base steps C ðxÞ :¼ 1 ðxÞ and C ðxÞ :¼ C ðxÞ. d d d d In [5] we proved the following inductive formula for the Fourier transform of such weights: 4d dy d j hji ð1:5Þ C ðyÞ¼ sinc ; 8j  0: 2 j1 2 2 hji j2 1 Consequently, C ðyÞ d at least for 0jyj 2 d . Such a process of continuous smoothing through the self-convolution of a weight w has a discrete counterpart given by the autocorrelation of w (since no confusion d d can arise in the following, we will use the simpler term correlation): XX C ðaÞ :¼ w ðnÞw ðmÞ: w d d nm n  m ¼ a For example, since it turns out that X XX 1 1 C ðtÞ C ðtÞ¼ 1 ¼ 1 ¼ ; d d d a  djtj a; b  d b  a ¼ t the Cesàro weight (1.4) is the normalized correlation of the unit step weight u . Note that the Cesàro weight (1.4) is generated by both types of smoothing from the function 1. Moreover, through an iteration of the normalized correlation one might hji parallel the self-convolution process to generate the whole family of weights C , j  1. An important aspect is that if the coefficients of a trigonometric series are correlations of w, then such a series is non-negative. More precisely, X XXX X C ðhÞeðhmÞ¼ w ðnÞw ðmÞeðhmÞ¼ w ðnÞeðnmÞ : w d d d h h nm¼h n A particularly well-known case is the Fejér kernel (compare the C formula) 123 G. Coppola, M. Laporta X X X d C ðhÞeðhmÞ¼ C ðhÞeðhmÞ¼ eðnmÞ : d u h h 1  n  d Such a positivity property is the complete analogous of the aforementioned fact that the Fourier transform of a self-convolution is a square. Now, let us introduce our comparison argument for the weight w in view of possible refinements of (1.2). First, recall that m :¼ min jwcðtÞj 6¼ 0 d;T d jtj T by assumption in order to avoid triviality. Thus, we define E ¼ E ðT ; dÞ :¼fy 2 R : jwcðyÞj \m g w w d d;T and write Z Z 2 2 2 kSk  m jSðyÞwcðyÞj dy þ jSðyÞj dy 2;T d;T RnE E w w Z ð1:6Þ 2 2 m jSðyÞwcðyÞj dy þjE j sup jSðyÞj : d w d;T RnE y2E w w In order to take advantage of such an inequality one would require E either with a small measure or free of peaks of S(y), if it does not support a sufficiently small L - norm. For example, let us examine the possible scenario for the weights (1.5). To this aim, recall that sinð2pdyÞ if y 6¼ 0; h0i C ðyÞ¼ 1 ðyÞ¼ py 2d if y ¼ 0: The aforementioned triviality for (1.2) does not occur if we assume dT 2ð0; 1=2Þ, for otherwise it would be dt ¼ 1=2 for some t 2ð0; T and consequently 0 0 2 2 sin ð2pdtÞ sin ð2pdt Þ min 1 ðtÞ ¼ min ¼ ¼ 0: 2 2 jtj T jtj T ðptÞ ðpt Þ On the other hand, it is plain that if dT 2ð0; 1=2Þ, then one has sin ð2pdTÞ fy 2 R : jyj[ T g E ðT ; dÞ¼ y 2 R : 1 ðyÞ \ 0 1 d ðpTÞ for some T  T. It is easily seen that analogous considerations hold for the whole hji family fC g . However, in several applications of (1.2), like in the case of j  0 Dirichlet polynomials (see [5]), the contribution from the tail jyj[ T might amount to a remainder term. On the other hand, note that the inequality 123 A weighted inequality for... 2 2 kSk  m kwcSk 2;T d;T 2;T trivially holds for any choice of w such that m 6¼ 0. Even for this reason, it is d d;T worthwhile to compare weights in view of possible refinements of the first term on the right-hand side of (1.6). To this end, we give the following definition. Definition 1.2 Let us suppose that w ; v are such that wcðtÞ 6¼ 0, vb ðtÞ 6¼ 0, d d d d 8t 2½T ; T. We say that v is T-better than w if d d min jvb ðtÞj d 2 jtj T jvb ðyÞj 8y 2½T ; T: ð1:7Þ 2 2 min jwcðtÞj jwcðyÞj d d jtj T If so, we write v w . d d If v w , it is plain that v yields a refinement of the upper bound of kSk d d d T 2;T with respect to w , at least in ½T ; T. In §3 we prove the next result on the family of weights (1.5) and discuss the comparison involving also some weights as C  1 and d d the Lanczos weight. hjþ1i hji j1 Theorem 1.2 Let d; T 2ð0; þ1Þ. For every j  0, if dT\2 , then C C . d T d Moreover, the inequality d 2 hjþ1i min C ðtÞ d 2 hjþ1i C ðyÞ jtj T ð1:8Þ d 2 d 2 hji hji min C ðtÞ C ðyÞ d d jtj T j1 0 j1 00 still holds for all jyj2 2 ð2k þ 1Þ=d þ e ; 2 ð2k þ 3Þ=d  e , where k k k2N 0 0 00 00 j1 e ¼ e ðj; d; TÞ; e ¼ e ðj; d; TÞ2ð0; 2 =dÞ k k k k 0 00 2 j j1 are such that e ; e dT =ð2 kÞ T =ð2kÞ as dT ! 2 , 8j; k 2 N. k k j1 1 0 j1 Remark 1.2 For any fixed j; k 2 N, the interval 2 ð2k þ 1Þd þ e ; 2 ð2k þ 00 j1 3Þd  e tends to cover the whole interval ½ð2k þ 1ÞT ; ð2k þ 3ÞT as dT ! 2 . Somehow this means that (1.8) holds almost everywhere in R n½T ; T as long as j1 dT ! 2 and it gives a further chance to get even a refinement of the second term hji hjþ1i on the left-hand side of (1.6) by replacing C with C . Furthermore, such a d d refinement might rely also on the value of the series Sðð2k þ 1ÞTÞ, 8k 2 Z. hji Remark 1.3 An effective use of (1.2) with C requires finding explicit expressions of such weights. For example, the normalized self-convolution of C is the so- d=2 h2i called Jackson-de La Vallé Poussin weight C , that is the following cubic spline [4, Problem 5.1.2 (v)]: 123 G. Coppola, M. Laporta 6jtj  6dt þ d if jtj d=2; < 3d d ðC  C ÞðtÞ¼ d=2 d=2 2ðd jtjÞ if d=2\jtj d; > 3 3d 0if jtj[ d: h2i Note that the support of C is ½d; d, as expected. 2 Proof of Theorem 1.1 Here we recall Pitt’s inequality: Let 1\p  q\1 be fixed and (U, V) be a (p, q)-pair of Pitt weights. There exists C ¼ Cðp; q; U :VÞ[ 0 such that 1=q 1=p kV f k  CkU f k : ð2:1Þ q p 1=p p for all f such that U f 2 L ðR;CÞ. Proof First, note that from the hypotheses on S and w it follows that Z Z X X jaðmÞw ðx  mÞj dx ¼ jaðmÞj jwðx  mÞj dx R jxmj d m m ¼kwk jaðmÞj\1: 1;d Therefore, by applying Lebesgue’s dominated convergence theorem [16, Th.1.38] it turns out that aHw 2 L ðR;CÞ and ðaHw ÞðyÞ :¼ ðaHw ÞðxÞeðxyÞ dx d d ¼ aðmÞ w ðx  mÞeðxyÞ dx ¼ aðmÞeðmyÞ w ðtÞeðtyÞ dt ¼ SðyÞwcðyÞ: d d 1=p p Then, observe that we can assume U ðaHw Þ2 L ðR;CÞ for 1\p\1, for otherwise the inequality 1=q 1=p kV S wck  C kU ðaHw Þk : ð2:2Þ d d q p would be trivial. Hence, (2.2) follows by applying Pitt’s inequality (2.1), being plain that (recall that w, V are even functions) 123 A weighted inequality for... Z Z q q jSðyÞwcðyÞj VðyÞ dy ¼ jSðyÞwcðyÞj VðyÞ dy: d d R R By taking any fixed T 2ð0; þ1Þ, clearly (2.2) implies (1.1). Now, let us prove that 1=q 1=p kV S wbk  CkU ðaHwÞk : ð2:3Þ q p As before, we can clearly assume that aHw 2 L ðR;CÞ and apply (2.2) with d ¼ n 2 N to write 1=q 1=p kV S wck  CkU ðaHw Þk : ð2:4Þ n n q p Since w ðtÞeðtyÞ converges to wðtÞeðtyÞ, with jw ðtÞeðtyÞj  jwðtÞj and n n w 2 L ðR;CÞ, the dominated convergence theorem yields 1=q 1=q lim kV S wck ¼kV S wbk : q q On the other hand, the same theorem implies that Z Z p p lim jðaHw ÞðxÞj UðxÞ dx ¼ jðaHwÞðxÞj UðxÞ dx: R R Hence, (2.3) follows from (2.4) after passage to the limit as n !1. Again, by taking any fixed T 2ð0; þ1Þ, it is plain that (1.1) holds if w is replaced by w 2 L ðR;CÞ. aq bp Remark 2.1 It is well konwn that, by taking VðxÞ¼jxj , UðxÞ¼jxj with maxf0; 1=p þ 1=q  1g a\1=q and b :¼ a þ 1  1=p  1=q; ð2:5Þ Pitt’s inequality (2.1) yields three classical inequalities in Fourier analysis: ● the Hardy-Littlewood inequality if p ¼ q  2, a ¼ 0or 1\p ¼ q  2, b ¼ 0; ● the Hausdorff-Young inequality if q ¼ 1  1=p  2, a ¼ b ¼ 0; ● the Plancherel identity if p ¼ q ¼ 2, a ¼ b ¼ 0. aq bp Accordingly, for VðxÞ¼jxj and UðxÞ¼jxj the inequality (2.2) turns into a b kv wc Sk  Ckv ðaHw Þk ; d d q p where we have set vðxÞ¼ jxj. Thus, (1.1) specializes to the following instances paralleling the aforementioned three properties: 123 G. Coppola, M. Laporta 12=p kSk  kv ðaHw Þk ; with p ¼ q  2; p;T p min jwcðtÞj jtj T d 12=p kv Sk  kaHw k ; with 1\p ¼ q  2; p;T p min jwcðtÞj jtj T kSk  kaHw k ; with q ¼ 1  1=p  2; q;T p min jwcðtÞj jtj T 1=ð2pÞ 1=ð2qÞ (in the latter C ¼ p =q is the so-called Beckner’s constant); kSk  kaHw k : 2;T 2 min jwcðtÞj jtj T d Finally, we conclude this remark by recalling that the original version of Pitt’s inequality was tailored for the Fourier series and the power series [15]. In particular, by assuming that m 2 Z for our Fourier series S, Theorem 2 of [15] yields 1=q 1=p a q b p jaðmÞm j  C jt SðtÞj dt ; m2Z 1=q X 1=p q p a b jt SðtÞj dt  C jaðmÞm j ; m2Z where p; q; a; b are as in (2.5). 3 Proof of Theorem 1.2: comparison of weights Proof Let us start with j ¼ 0 and show that C is T-better than 1 by assuming that d d dT\1=2, namely we prove that min C ðtÞ 2 C ðyÞ jtj T ð3:1Þ 2 2 b   b min 1 ðtÞ 1 ðyÞ d d jtj T holds for all y 2½T ; T. First, recall that dT\1=2 yields 2 4 2 sin ð2pdTÞ 2 sin ðpdTÞ b  c min 1 ðtÞ ¼ 6¼ 0; min C ðtÞ ¼ 6¼ 0: d d 2 4 2 jtj T jtj T ðpTÞ ðpTÞ d Since min C ðtÞ 2 C ð0Þ jtj T tan ðpdTÞ 1 d ¼  ¼ ; 2 2 2 b  4  b 4ðpdTÞ min 1 ðtÞ 1 ð0Þ d d jtj T we can assume that y 6¼ 0 and set x ¼ y=T, h ¼ dT. Thus, (3.1) becomes 123 A weighted inequality for... C ðxTÞ tan ðphxÞ G ðxÞ :¼ ¼  G ð1Þ: h   h 2 2 ð2phxÞ 1 ðxTÞ It is easy to see that G ðxÞ satisfies the following properties: (i) G ðxÞ is even with respect to both x and h (ii) G ðxÞ is strictly increasing with respect to x 2ð0; 1 (iii) lim G ðxÞ¼ 1=4 8h 2ð0; 1=2Þ x!0 (iv) G ð1Þ is strictly increasing with respect to h 2ð0; 1=2Þ (v) lim G ð1Þ¼ 1=4, lim G ð1Þ¼þ1 h h h!0 h!1=2 (vi) G ðxÞ¼ 0 () x ¼ k=h 8k 2 Znf0g (note that 8k 2 Z nf0g and 8h 2 ð0; 1=2Þ one has jkj=h[ 2jkj 2) (vii) G ðxÞ! þ1 as x !ð2k þ 1Þð2hÞ 8k 2 Z. Note that j2k þ 1jð2hÞ [ j2k þ 1j 1, 8k 2 Z and 8h 2ð0; 1=2Þ. From properties (i)-(v) it follows that G ðxÞ G ð1Þ for all x 2½1; 1nf0g, that h h is to say that (3.1) holds for all jyj T, i.e., C 1 . d d In order to prove the second part of the theorem for j ¼ 0, observe that properties (i), (vi) and (vii) imply that the above inequality for G is true for all [ [ 1 0 1 00 jxj2 I :¼ ½ð2k þ 1Þð2hÞ þ D ; ð2k þ 3Þð2hÞ  D ; h;k k k k2N k2N 0 0 00 00 where 0\D ¼ D ðhÞ; D ¼ D ðhÞ\ð2hÞ are such that the endpoints of the k k k k intervals I are the solutions of the equation G ðxÞ¼ G ð1Þ. In particular, this h;k h h yields 0 0 tan ðp=2 þ phD Þ¼ G ð1Þ ð2k þ 1Þp þ 2phD ; k k 0 0 with 0\hD \1=2. From the latter equation and from (v) we deduce that D ðh k k 1=2Þk as h ¼ dT ! 1=2, for all k 2 N. An analogous property holds for D . 0 00 0 00 Hence, we set e ð0; d; TÞ¼ TD ; e ð0; d; TÞ¼ TD to get the theorem proved in the k k k k case j ¼ 0. For j[ 0 it sufficies to note that (see (1.5)) jþ1 d 2 hjþ1i 2 jþ1 C ðyÞ tanðpdy=2 Þ tanðphxÞ j ¼ ¼ ¼ G ðxÞ ; j1 d 2 dpy=2 2phx hji C ðyÞ j 2 where we have set h ¼ dT =2 , x ¼ y=T and G ðxÞ¼ð2phxÞ tan ðphxÞ as before. j1 Since 0\dT\2 if and only if 0\h\1=2, the conclusion follows from what we have showed in the case j ¼ 0. h Remark 3.1 Theorem 1.2 seems to indicate that one could generate more and more T-better weights by increasing the iteretad convolutions of 1 . However, the following examples suggest that one must exercise some caution about the choice of the weights to be compared. 123 G. Coppola, M. Laporta Example 3.1 Let us compare C and D :¼ C  1 . First, recall that 2d 2d d d 2 2 d d c b C ðyÞ¼ 2dsinc ð2dyÞ; D ðyÞ¼ C ðyÞ1 ðyÞ¼ 2d sinc ðdyÞsincð2dyÞ; 2d 2d d d so that > tan ðpdyÞ 2 4 D ðyÞ if y 6¼ 0; d sinc ðdyÞ 2d ¼ ¼ ðpyÞ 2 2 d sinc ð2dyÞ C ðyÞ : 2d 2 d if y ¼ 0: Further, for dT\1=2, one has 4 2 min D ðtÞ ¼ 4d min sinc ðdtÞsinc ð2dtÞ 2d jtj T jtj T 4 6 2 ¼ 4d sinc ðdTÞ cos ðpdTÞ 6¼ 0; 2 4 min C ðtÞ ¼ 4d sinc ð2dTÞ 6¼ 0: 2d jtj T Since it turns out that min D ðtÞ 2 2 2d 2 d  d jtj T tan ðpdTÞ D ðTÞ D ðyÞ 2d 2d ¼ ¼  ; 8y 2½T ; T; 2 2 2 2 d  d  d ðpTÞ min C ðtÞ C ðTÞ C ðyÞ 2d 2d 2d jtj T then we conclude that D C once dT\1=2. 2d 2d Note that both C and D have support in ½2d; 2d. On the other hand, by 2d 2d comparing C and D we see that d 2d > sin ð2pdyÞ 2 < d c b if y 6¼ 0; D ðyÞ 2d jC ðyÞ1 ðyÞj 2 d d ¼ ¼ 1 ðyÞ ¼ ðpyÞ 2 2 c  c  > C ðyÞ C ðyÞ : d d 4d if y ¼ 0; and, for dT\1=2, one has min D ðtÞ 2 2d jtj T sin ð2pdTÞ D ðyÞ 2 2d ¼  1 ðyÞ ¼ ; 8y 2½T ; T; 2 2 2 c  d ðpTÞ min C ðtÞ C ðyÞ d 2d jtj T that is to say, if dT\1=2, then C D . Analogously, it is easy to see that d 2d hji hji hji hj1i C C for all j  0 (and consequently C C for all j  1, because of d T 2d d T 2d j1 Theorem 1.2), as long as dT\2 . Example 3.2 Given real numbers d  c[ 0, the Lanczos weight is defined as [4, Problem 5.1.2 (v)] 123 A weighted inequality for... 1if jxj d  c; 1 d jxj L ðxÞ :¼ ð1  1 ÞðxÞ¼ if d  c\jxj d; d;c dc=2 c=2 c c 0if jxj[ d; whose Fourier transform is L ðyÞ¼ ð2d  cÞsincðcyÞsincðð2d  cÞyÞ d;c sinðpcyÞ sinð2pdy  pcyÞ if y 6¼ 0; ¼ ðpyÞ c 2d  c if y ¼ 0: The diagram of L is an isosceles trapezium. Note that 1  L  C ¼ L for d;c d d;c d d;d any d  c[ 0. Since L ¼ C , then we can assume that d[ c[ 0. Further, by d;d d b d taking cT\dT\ð2d  cÞT\1=2 one has 1 ðyÞ 6¼ 0 and L ðyÞ 6¼ 0 for all d d;c y 2½T ; T. Therefore, under such assumptions we can compare L and 1 by d;c d verifying the inequality 1 ðyÞ ð2dÞ sinc ð2dyÞ 2 2 2 2 ð2d  cÞ sinc ðcyÞsinc ðð2d  cÞyÞ L ðyÞ d;c min 1 ðtÞ 2 d ð2dÞ sinc ð2dTÞ jtj T ¼   : 2 2 2 2 ð2d  cÞ sinc ðcTÞsinc ðð2d  cÞTÞ min L ðtÞ d;c jtj T To this aim, let us write min 1 ðtÞ d 2 jtj T pcT sinð2pdTÞ sinðpcTÞ sinðpð2d  cÞTÞ min L ðtÞ d;c jtj T sinðpð2d  cÞT þ pcTÞ ¼ðpcTÞ sinðpcTÞ sinðpð2d  cÞTÞ ¼ðpcTÞ cotðpcTÞþ cot pð2d  cÞT 2 2 ¼FðcTÞ þ Fðð2d  cÞTÞ þ ð2d  cÞ 2c FðcTÞFðð2d  cÞTÞ; 2d  c where FðxÞ :¼ðpxÞ cotðpxÞ. Since FðxÞ 1 for jxj 1=2, we get 123 G. Coppola, M. Laporta min 1 ðtÞ 2 d 2 1 ð0Þ jtj T c 2c ð2dÞ 1 þ þ ¼ ¼ : 2 2 2 2 d  d 2d  c ð2d  cÞ ð2d  cÞ min L ðtÞ L ð0Þ d;c d;c jtj T Therefore, we can assume that y 6¼ 0 and write 1 ðyÞ c 2c 2 2 ¼ FðcyÞ þ Fðð2d  cÞyÞ þ FðcyÞFðð2d  cÞyÞ: 2 2 d  2d  c ð2d  cÞ L ðyÞ d;c Since it is readily seen that FðxÞ Fðx Þ for jxj x \1=2, we conclude that L 0 0 d;c 1 provided cT\dT\ð2d  cÞT\1=2. Finally, we show that C L , under the same conditions on c, d and T. Indeed, d d;c we prove that the inequality 2 4 C ðyÞ d sinc ðdyÞ 2 2 2 2 ð2d  cÞ sinc ðcyÞ sinc ðð2d  cÞyÞ L ðyÞ d;c min C ðtÞ 2 4 d sinc ðdTÞ jtj T 2 2 2 2 ð2d  cÞ sinc ðcTÞ sinc ðð2d  cÞTÞ min L ðtÞ d;c jtj T holds for all y 2½T ; T. To this aim, let us recall that, being concave in ðp; pÞ, the function sinc is also logarithmically concave in the same interval (see [3, Ch.3]), i.e., for all k 2½0; 1 and all x ; x 2ð1; 1Þ one has 1 2 k 1k sincðkx þð1  kÞx Þ sinc ðx Þsinc ðx Þ: 1 2 1 2 In particular, by taking k ¼ 1=2, x ¼ cT and x ¼ð2d  cÞT, with 1 2 cT\dT\ð2d  cÞT\1=2, this yields min C ðtÞ C ð0Þ jtj T d ¼ : 2 2 2 d  d ð2d  cÞ min L ðtÞ L ð0Þ d;c d;c jtj T Therefore, let assume that y 6¼ 0 and write c  2 C ðyÞ c sin ðpdyÞ 2 2 2 2 d sin ðpcyÞ sin ðpð2d  cÞyÞ L ðyÞ d;c c 1  cosðpcyÞ cosðpð2d  cÞyÞ ¼ 1 þ : sinðpcyÞ sinðpð2d  cÞyÞ ð2dÞ Thus, it is plain that we get the desired conclusion once we prove that, for any fixed real numbers a, b such that 0\a\b\p=ð2TÞ, the function 123 A weighted inequality for... 1  cosðayÞ cosðbyÞ 1 1 G ðyÞ :¼ ¼ a;b sinðayÞ sinðbyÞ sinðayÞ sinðbyÞ tanðayÞ tanðbyÞ is monotone increasing in (0, T]. Indeed, it turns out that d cosðayÞ cosðbyÞ G ðyÞ¼ b sinðayÞ a sinðbyÞ [ 0; a;b 2 2 dy sin ðayÞ sin ðbyÞ because 0\a\b\p=ð2TÞ and y 2ð0; T yield cosðayÞ cosðbyÞ[ 0 and 1 1 ðayÞ sinðayÞ[ ðbyÞ sinðbyÞ. Acknowledgements The authors wish to thank Laura De Carli for helpful comments on an early version of the present paper. Further, they wish to emphasize the excellent and constructive attention received from the referee. Author Contributions All the authors contributed equally for the preparation of the present paper. Funding Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI- CARE Agreement. Declarations Conflict of interest The authors declare that they do not have any competing interests. Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors. 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Real and Complex Analysis, 3rd edn. New York, NY: Mc Graw Hill International Editions. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Journal

The Journal of AnalysisSpringer Journals

Published: Sep 22, 2022

Keywords: Trigonometric series; Pitt’s inequality; Majorant properties; 42A05; 26D15; 42A38

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