A weighted inequality for trigonometric series

By applying Pitt’s inequality we prove a weighted Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} 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 supported in the same interval. The eventual aim is the comparison of such weights in view of possible refinements of our inequality for p=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document}.


Introduction
The trigonometric series considered here are of the type SðtÞ :¼ X m aðmÞeðmtÞ; where eðxÞ :¼ e 2pix and aðmÞ 2 C for a strictly increasing sequence m 2 R. For any given T 2 ð0; þ1Þ and p 2 ½1; þ1Þ, let us denote the L p -norm of f : R ! C by kf k p :¼ À R R jf ðtÞj p dt Á 1=p and set kf k p;T :¼ À R T ÀT jf ðtÞj p dt Á 1=p . The space of the then we say that (U,V) is a (p,q)-pair of Pitt weights.
In Sect. 2 the following result is proved. 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., 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 properties established in [5]. Let d; T 2 ð0; þ1Þ and let w 2 L 1 loc ðR; CÞ be even such that If SðtÞ ¼ P m aðmÞeðmtÞ and BðyÞ ¼ P m bðmÞeðmyÞ are absolutely convergent, and the first sum is a majorant for the second one, i.e., jaðmÞj bðmÞ for all m, then Z T ÀT jSðyÞj 2 dy 1 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 T ÀT jSðyÞj 2 dy 3 Z T ÀT jBðyÞj 2 dy: 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] 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 C d ðyÞ :¼ maxð1 À d À1 jyj; 0Þ; ð1:4Þ whose Fourier transform is c C d ðyÞ ¼ d sinc 2 ðdyÞ. In the literature, this is known as Fejer kernel [4]. For dT 2 ð0; 1Þ, the inequality (1.2) gives 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 1 loc ðR; CÞ let us define the normalized self-convolution of For example, the Cesàro weight (1.4) is the normalized self-convolution of the restriction to ½Àd=2; d=2 of 1: It is well-known that the iteration of the self-convolution gives rise to a process of smoothing (see [4]). Moreover, the support of f w d is doubled with respect to the support of w d in the sense that it is a subset of ½À2d; 2d. Because of the normalizing factor ð2dÞ À1 , that takes into account the length of the integration interval, the magnitude of w d is not altered much by the normalized self-convolution. More precisely, if one has w d 1, i.e., 1 ( w d ( 1, in an interval of length ) d, then there exists an interval of length ) d (not necessarily the same) where f w d 1. Recall that A ( h B stands for jAj cB, where c [ 0 is an unspecified constant depending on h.
From another well-known property of the convolution it follows that the Fourier The normalized self-convolution generates recursively the family of weights: In [5] we proved the following inductive formula for the Fourier transform of such weights: Consequently, d C hji d ðyÞ j d at least for 0 jyj 2 jÀ2 d À1 . Such a process of continuous smoothing through the self-convolution of a weight w d has a discrete counterpart given by the autocorrelation of w d (since no confusion can arise in the following, we will use the simpler term correlation): For example, since it turns out that the Cesàro weight (1.4) is the normalized correlation of the unit step weight u d . 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 parallel the self-convolution process to generate the whole family of weights C hji d , 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, A particularly well-known case is the Fejér kernel (compare the c 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 by assumption in order to avoid triviality. Thus, we define ð1:6Þ In order to take advantage of such an inequality one would require E w either with a small measure or free of peaks of S(y), if it does not support a sufficiently small L 2norm. For example, let us examine the possible scenario for the weights (1.5). To this aim, recall that The aforementioned triviality for (1.2) does not occur if we assume dT 2 ð0; 1=2Þ, for otherwise it would be dt 0 ¼ 1=2 for some t 0 2 ð0; T and consequently On the other hand, it is plain that if dT 2 ð0; 1=2Þ, then one has for some T 0 ! T . It is easily seen that analogous considerations hold for the whole family fC hji d g j ! 0 . However, in several applications of (1.2), like in the case of 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 trivially holds for any choice of w d such that m d;T 6 ¼ 0. Even for this reason, it is 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. are such that e 0 k ; e 00 k $ dT 2 =ð2 j kÞ À T =ð2kÞ as dT ! 2 jÀ1 , 8j; k 2 N.

Remark 1.2 For any fixed
tends to cover the whole interval ½ð2k þ 1ÞT ; ð2k þ 3ÞT as dT ! 2 jÀ1 . Somehow this means that (1.8) holds almost everywhere in R n ½ÀT ; T as long as dT ! 2 jÀ1 and it gives a further chance to get even a refinement of the second term on the left-hand side of (1.6) by replacing C hji d with C hjþ1i d . Furthermore, such a refinement might rely also on the value of the series Sðð2k þ 1ÞT Þ, 8k 2 Z.
Note that the support of C h2i d is ½Àd; 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 for all f such that U 1=p f 2 L p ðR; CÞ.
Proof First, note that from the hypotheses on S and w it follows that X As before, we can clearly assume that aHw 2 L p ðR; CÞ and apply (2.2) with d ¼ n 2 N to write kV 1=q S c w n k q CkU 1=p ðaHw n Þk p : ð2:4Þ Since w n ðtÞeðÀtyÞ converges to wðtÞeðÀtyÞ, with jw n ðtÞeðÀtyÞj jwðtÞj and w 2 L 1 ðR; CÞ, the dominated convergence theorem yields On the other hand, the same theorem implies that Hence, (2.3) follows from (2.4) after passage to the limit as n ! 1. Again, by taking any fixed T 2 ð0; þ1Þ, it is plain that (1.1) holds if w d is replaced by w 2 L 1 ðR; CÞ. h Remark 2.1 It is well konwn that, by taking V ðxÞ ¼ jxj Àaq , U ðxÞ ¼ jxj bp with maxf0; 1=p þ 1=q À 1g a\1=q and b :¼ a þ 1 À 1=p À 1=q; ð2:5Þ Pitt's inequality (2.1) yields three classical inequalities in Fourier analysis: Accordingly, for V ðxÞ ¼ jxj Àaq and U ðxÞ ¼ jxj bp the inequality (2.2) turns into where we have set vðxÞ ¼ jxj. Thus, (1.1) specializes to the following instances paralleling the aforementioned three properties: (in the latter C ¼ p 1=ð2pÞ =q 1=ð2qÞ is the so-called Beckner's constant); 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 where p; q; a; b are as in (2.5).

Proof of Theorem 1.2: comparison of weights
Proof Let us start with j ¼ 0 and show that C d is T-better than 1 d by assuming that dT \1=2, namely we prove that holds for all y 2 ½ÀT ; T . First, recall that dT \1=2 yields we can assume that y 6 ¼ 0 and set x ¼ y=T , h ¼ dT . Thus, (3.1) becomes Since it turns out that then we conclude that D 2d 0 T C 2d once dT \1=2.