LINGXIAN (ROSE) ZHANG

Transcription

1 HE L p CONVERGENCE OF FOURIER SERIES LINGXIAN ROSE ZHANG Abstract. In this expository paper, we show the L p convergence of Fourier series of functions on the one-dimensional torus. We will first turn the question of convergence into a question regarding the uniform boundedness of the partial-sum operators of Fourier series, and then bound these partial-sum operators with the help of Hilbert transform. Contents. Preliminaries.. Basics about Fourier series.2. he L 2 convergence of Fourier series 3.3. Some other useful facts 7 2. Poisson Kernel and Hardy Spaces Poisson kernel Correspondence between h D and M 9 3. wo Maximal Operators 3 4. Conjugate Function and Hilbert ransform Conjugate function Harmonic measures and conjugate radial maximal operator Hilbert transform 8 5. he L p Convergence of Fourier Series 2 6. Generalization: Multiplier Operators 2 Appendix 22 References 25. Preliminaries Let s begin with some definitions and conventions, followed by the motivating example of L 2 -convergence of Fourier series, and some propositions that will be handy later in this paper... Basics about Fourier series. Definitions.. Let = R/Z denote the one-dimensional torus, that is, the circle. We will occasionally use the interval 2, 2] to represent for the sake of simplicity and of symmetry, especially when it comes to measuring subsets, integrating over a subset, or talking about the monotonicity of functions defined on. 2 We mean by e the function from to C given by eθ := e 2πiθ, and we define, for each n Z, the function e n : C by e n θ := enθ.

2 2 LINGXIAN ZHANG 3 he notation, when applied to a measurable subset of, will denote its Lebesgue measure. For instance, =. 4 Let D = z C : z < } = reθ : r <, θ } denote the open unit disk, with its boundary D identified with. Observe that the set e n } n Z L 2 is orthonormal with respect to the usual inner product f, g := fḡdm, since e ndm = χ } n. Definitions.2. We denote by M the space of complex Borel measures of bounded variation on, and we mean by µ the total variation of a complex measure µ M. 2 he convolution of a function g L with a complex measure µ M, written g µ, is defined by g µθ := gθ τ dµτ. In the case that dµ = fdm for some f L, we may write g f in place of g µ. By a change of variables, we can see that convolution, as a binary operation on L, is commutative. By the Fubini-onelli theorem, convolution is associative. And the bilinearity of convolution follows from the linearity of integration. It is worth noting that every µ M is regular, since is a second-countable compact Hausdorff space. Additionally, by the Riesz-Markov theorem, there is a canonical isometric isomorphism between M and C, namely µ f fdµ, which allows us to identify these two spaces and to equip the former with the weak* topology. hen, as an immediate result of the Banach-Alaoglu theorem, the unit ball of M is compact in the weak* topology. Moreover, since C is a separable Banach space, the closed unit ball of its dual C = M is metrizable and therefore sequentially compact with respect to the weak* topology. For detailed proof, see Corollary 7.6 and Corollary 7.8 in [2], and heorem 3.29 in []. Proposition.3. Young s inequality If f L p, with p [, ], and if µ M, then f µ p f p µ. In particular, if f L p and g L, then f g p f p g. he above inequality essentially follows from Minkowski s inequality for integrals. Proof is provided in the appendix. Proposition.4. If f L and if g L, then f g C. he idea of the proof is to first show that f g C in the case that g C, and then generalize to the case where g L, using a density argument. Details of the proof are provided in the appendix. Remark.5. While no integrable function can serve as an identity for the convolution, there is the Dirac measure δ at such that g δ = g for every g L.

3 HE L p CONVERGENCE OF FOURIER SERIES 3 Definitions.6. For each f L, we define its Fourier coefficients ˆfn := fe n dm = fe n dm for every n Z. 2 A trigonometric polynomial is a function of the form f = n= a ne n, where all but finitely many coefficients a n are zero. By the orthonormality of the collection e n } n Z, if f : C is a trigonometric polynomial, then ˆfn = a n for every n Z, and f = ˆfne n. n= So, it is natural to ask if the right-hand side of the above equality is well-defined for other integrable functions on, and if the equality will still hold in that case. From now on, we will refer to the formal sum ˆfne n= n as the Fourier series associated to a general f L, despite that the actual series may be ill-defined. Definition.7. For each f L, we define its symmetric partial sums N S N f := ˆfne n for every N N. n= N Expanding the formula in the above definition, we obtain that for every N Z + and every f L N S N fθ = enθ fτe nτdτ n= N N N = fτ e n θ τ dτ = f e n θ, n= N which leads us to define the following. n= N Definition.8. For each N N, we define a Dirichlet kernel D N := N n= N e n. While the formulae for the Dirichlet kernels seem rather simple, these kernels are not so easy to analyze as one would like. For this reason, we will turn to the betterbehaved Cesàro sums of the Dirichlet kernels, known as the Fejér kernels, in order to prove that the trigonometric polynomials are dense in L 2, and consequently the L 2 convergence of the symmetric partial sums of Fourier series..2. he L 2 convergence of Fourier series. Definitions.9. For each N Z +, we define the Fejér kernel K N := N 2 For each f L, we define its Cesàro sums σ N f := N N n= S n f = f K N. N n= D n.

4 4 LINGXIAN ZHANG 3 A family Φ N } N= L is said to form an approximate identity if it satisfies the following: a Φ Ndm = for every index N; b sup N Φ N dm < ; c given any ɛ >, one has that x >ɛ Φ N dm as N. he following proposition justify the appellation approximate identity. Proposition.. Let Φ N } N= be an approximate identity. If f C, then Φ N f f as N. 2 If f L p, with p [,, then Φ N f f p as N. 3 For every µ M, Φ N µ µ in the weak* sense as N. Proof is provided in the appendix. Examples.. N } he box kernels 2 χ [ N, N ] form an approximate identity. N= 2 he Fejér kernels K N } N= form an approximate identity. 3 he Dirichlet kernels D N } N= do not form an approximate identity. Let s justify briefly the second and the third examples. Expanding the formulae given in the definitions, we can show that for every positive integer N and every x \ }, D N x = sin 2N + πx sinπx and K N x = N It then follows from the nonnegativity of each K N that K N = K N xdx =. In addition, for every ɛ >, lim sup x >ɛ K N x dx lim sup N x >ɛ 2 sinnπx. sinπx sin 2 dx =. πx Hence, the Fejér kernels K N } N= form an approximate identity. On the other hand, for every positive integer N, D N = 2 2 π 2 π /2 N k= N k= sin2n + πx dx 2 sinπx k+ 2N+ k 2N+ 2 /2 sin2n + πx dx x 2N + 2N + k π N k= sin2n + πx dx πx and whence that sup N D N =. herefore, the Dirichlet kernels D N } N= do not form an approximate identity. Corollary.2. If f L and if ˆfn = for every integer n, then f = a.e. on. 2 he trigonometric polynomials are dense in L p for every p [,, and are dense in C with respect to the uniform norm. k,

5 HE L p CONVERGENCE OF FOURIER SERIES 5 Proof. Suppose that f L and that ˆfn = for every integer n. Since the Fejér kernels K N } N= form an approximate identity, it follows that = lim σ Nf f = lim f = f, which implies that f = a.e. 2 he second corollary follows from the fact that the Cesàro sums of any L p function are trigonometric polynomials, and that the Fejér kernels form an approximate identity. he L 2 convergence of Fourier series now follows from the first collorary. heorem.3. he collection e n } n Z forms an orthonormal basis for L 2, and the following holds. For every f L 2, S N f f in L 2 as N. 2 For any two f, g L 2, f, g = n Z ˆfnĝn. In particular, one has Parseval s identity f 2 2 = ˆfn 2. n Z Proof. Suppose that f is an L 2 function with f, e n = for every n N. Since L 2 L, and since ˆfn = f, e n = for every n, it follows that f = a.e. Hence, the orthonormal system e n } n= is complete, i.e. e n } n= is an orthonormal basis for L 2. he remainder of this theorem follows immediately from basic properties of an orthonormal basis for a Hilbert space, but can also be proven using the ensuing proposition, which states an equivalent but more accessible condition for the L p convergence of Fourier series. Proposition.4. For each p [, ], the following statements are equivalent: sup N N S N p p < ; 2 For every f L p or for every f C if p =, S N f f p as N. Proof. Fix an arbitrary p [, ]. Firstly, let s suppose sup N S N p p <. hen, for every f L p or for every f C if p = and for every trigonometric polynomial g, lim sup S N f f p lim sup S N f S N g p + g f p + S N g g p lim sup S N p p f g p + f g p + sup S N p p + N N f g p Since the trigonometric polynomials are dense in L p and in C with respect to the uniform norm, taking the infimum over all trigonometric polynomials g, we obtain that lim sup S N f f p for every f L p, which implies that S N f f p as N.

6 6 LINGXIAN ZHANG On the other hand, suppose that sup N S N p p =. Since L p is complete and C is complete with respect to the uniform norm, by the uniform boundedness principle, there exists f L p or f C if p = such that sup N S N f p =, and consequently S N f diverges in L p as N. An alternative proof of heorem.3. For every f L 2 and every N N, we have f S N f 2 2 = f Re fxs N fx dx N = f Re ˆfn 2 + = f 2 2 N n= N n= N ˆfn 2, + S N fx 2 2 N n= N ˆfn which implies that S N f 2 2 = N n= N ˆfn 2 f 2 2. hus, sup S N 2 2 <, N N and it follows that lim S N f f 2 = for every f L 2. Furthermore, we deduce from the continuity of the inner product that 2 f, g = lim = lim lim S N f, S M g M lim minn,m} M n= minn,m} ˆfnĝn = n= ˆfnĝn. In the end, we will show that sup N N S N p p < for every p,, and then invoke Proposition.4 to complete the proof of the L p convergence of Fourier series. But before doing so, let s show that the Fourier series does not always converge to the original function in L, nor in C. Examples.5. here exist L functions whose Fourier series do not converge to the original function. 2 here exist C functions whose Fourier series do not converge to the original function with respect to the uniform norm. Proof. By the previous proposition, it suffices to show that sup N N S N and sup N N S N are both infinite. Since every Fejér kernel has L norm K M =, and since the family of Fejér kernels form an approximate identity, we have that sup S N = sup N N N N sup sup f: f = sup N N M N D N f D N K M sup D N =. N N

7 HE L p CONVERGENCE OF FOURIER SERIES 7 2 Observe that sgn D N Id x = sgnd N x for every x and every N N. Hence, sup S N sup N N N N sup f: f = D N f sup D N sgn D N Id N N = sup D N =. N N.3. Some other useful facts. o generalize the L 2 convergence of Fourier series to L p convergence for every p,, we will utilize the Marcinkiewicz interpolation theorem, which provides us with boundedness properties on intermediate spaces. Definition.6. A complex valued measurable function g on a measure space X, Σ, µ is said to belong to weak-l p, with p [,, if there exists a finite constant C > such that for every λ >, i.e. if the following quantity is finite: µ x X : gx > λ} Cλ p, [g] p := sup µ x X : gx > λ} λ p. λ> Note that by Chebyshev s inequality, every L p function, with p [,, belongs to weak-l p. Definitions.7. Let be a map from some vector space D of complex valued measurable functions on a measure space X, Σ, µ to the space of all complex valued measurable functions on the space X, Σ, µ. is said to be sublinear if for all f, g D and c >, f + g f + g and cf = c f. 2 A sublinear operator is said to be of strong-type p, p, with p [, ], if L p X D, and if is bounded on L p X, meaning that there exists a constant C > such that for every f L p X, f p C f p. 3 A sublinear operator is said to be of weak-type p, p, with p [,, if L p X D and is bounded from L p X to weak-l p X, meaning that there exists a constant C > such that for every f L p X and every λ >, µ x X : fx > λ} C f p λ p. We also define weak-type, to mean the same as strong-type,. heorem.8. A simplified version of the Marcinkiewicz interpolation theorem If a sublinear operator is simultaneously of weak-type p, p and of weak-type q, q, with p < q, then is of strong-type r, r for every p < r < q. Proof of this theorem can be found in [2], [3], and [4].

8 8 LINGXIAN ZHANG 2. Poisson Kernel and Hardy Spaces It is known that Cesàro summability does not imply the usual summability which we are seeking to prove. For this reason, we now introduce a different derivate of the Dirichlet kernels, called the Poisson kernel, which will allow us to invoke some nice properties of functions of complex variables, such as the mean-value property and the maximal principle of harmonic functions. 2.. Poisson kernel. Observe that for each f L, the formal sum ˆfne n θ n= n= can also be written as ˆf nr n e n θ + ˆfnr n e n θ, with r =. Moreover, when r [,, the two series, ˆf nr n= n e n θ and ˆfnr n= n e n θ, are both absolutely convergent since ˆfn f for every n Z, whence ˆf nr n e n θ + ˆfnr n e n θ n= = = n= ft n= ˆfnr n e n θ = n= n= r n e n θ t n= ftr n e n θ tdt dt = f n= r n e n θ where the third equality holds by the Fubini-onelli theorem, considering the Lebesgue measure on and the counting measure on Z. his observation leads us to define the following. Definition 2.. For each r [,, we define the Poisson kernel P r : C by P r θ := n= r n e n θ. From now on, whenever F is a function on D and r is in [,, the notation F r will refer to the function on defined by F r θ := F reθ. Conversely, if a family F r } r [, of functions on is defined, with F constant, then we will mean by F the function on D defined by F reθ := F r θ. Remark 2.2. Rearranging the formula for the Poisson kernel, we obtain that P z = P r θ = r n e nθ + r n enθ n= n= = re θ re θ + reθ = r 2 + z 2r cos2πθ + r 2 = Re z for every z = reθ D. Since +z z is analytic on D, the function P is harmonic on D. Note also that for every r [,, P r r2 + 2r + r 2 = r + r >

9 HE L p CONVERGENCE OF FOURIER SERIES 9 and P r = r n e n = r e =. n= Proposition 2.3. For every µ M, the function F : D C defined by F r := P r µ for every r [, is harmonic on D. Proof. Fix an arbitrary µ M. hen, for every z = reθ D, we have F z = F r θ = ˆµ nr n e n θ+ ˆµnr n e n θ = ˆµ nz n + ˆµnz n, with n= n= ˆµ n z n µ z z < and ˆµn z n µ z <. n= From the absolute convergence of the two series n= ˆµ nzn and n= ˆµnzn, where z D, it follows that the functions n= ˆµ nzn and n= ˆµnzn are both analytic in D. Hence, F is a sum of two harmonic functions, and is therefore itself harmonic. n= n= n= Proposition 2.4. he Poisson kernels P r } r, form an approximate identity, with the index set [, in place of Z +. Moreover, all the convergence properties mentioned in Proposition. still hold, with the limit r in place of N. he proof of the above proposition is omitted, as the first half of this proposition can be shown via a simple calculation, and the proof of the second half is parallel to that of Proposition., which can be found in the appendix Correspondence between h D and M. Definition 2.5. For each harmonic function u : D C and each p [, ], we define u p := sup u re Lp which may attain the value. Further, for each p [, ], we define the little Hardy space } h p D := u : D C harmonic : u p <, and equip it with the norm p.

10 LINGXIAN ZHANG Examples 2.6. Every non-negative harmonic function on D is in h D. In particular, the Poisson kernel P is in h D. 2 If f L p, with p [, ], then the function F : D C defined by F r = P r f is in h p D, with F p f p. Proof. Let F : D [, be any non-negative harmonic function. hen, by the mean-value theorem, F r = F reθ dθ = F for every r,, and thus that F = F <. 2 If f L p, then by Proposition 2.3 is the function F : D C harmonic, and by Young s inequality, Lemma 2.7. F p = sup P r f p sup P r f p = f p. If F C D, and if F = in D, then F r = P r F for every r [,. 2 If F = in D, then F rs = P r F s for any r, s [,. 3 If F = in D, then for any p [, ], the norm F r p is non-decreasing as a function of r [,. Proof. Define a function u : D C by ureθ := P r F θ. Since P r } r< forms an approximate identity as r, the uniform norm ure F as r. So the harmonic function u can be extended to a continuous function on D with the same boundary value as that of F. Since the continuous function F is also harmonic in D, by the maximum principle, F D = u, which is to say that F r = P r F for every r [,. 2 he idea is to rescale the unit disc so that part applies. Fix an arbitrary s [,, and define a function G : D C by G reθ = F sreθ. It follows that G C D, G = in D, and that G r = F rs for every r [,. Hence, F rs = G r = P r G = P r F s for every r [,. 3 Fix any p [, ], and let r < r 2 < be given arbitrarily. Since r /r 2 [,, we have that F r p = Pr/r 2 F r2 p Pr/r 2 F r2 p = F r2 p. hus, the norm F r p, as a function of r [,, is non-decreasing. Lemma 2.8. For each F h D, there exists a unique complex measure µ M such that F r = P r µ for every r [,. Proof. Let F h D be given arbitrarily. If F =, then there is obviously the measure µ null : B } such that F r = = P r µ null for every r [,. So, we may assume that F >. Since the closed unit ball } of M is sequentially compact in the weak* topology, the sequence F F /n has a subsequence, } n= say F F /nk that converges in the weak* sense to some µ M. Set k=,

11 HE L p CONVERGENCE OF FOURIER SERIES µ := F µ M. hen for every r [, and every θ, since P r C, we obtain that P r µ θ = lim Pr F /nk θ = lim F r /n k θ = F r θ. k k Moreover, if there are two complex measures µ, ν M such that P r µ = F r = P r ν for every r [,, then fdµ = lim fθp r µθdθ = lim fθp r νθdθ = fdν r r for every f C, whence µ = ν. From the above, we conclude that there is a unique complex measure µ M such that F r = P r µ for every r [,. heorem 2.9. here is a one-to-one correspondence between M and h D, given by µ F r θ := P r µθ. Moreover, for every µ M, In addition, the following holds. µ = sup F r = lim F r. r A measure µ M is absolutely continuous with respect to Lebesgue measure iff F r } converges in L as r. If this is the case, then dµ = fdθ, where f is the L limit of F r. 2 For every p, ] and every µ M, the following are equivalent: a dµ = fdθ for some f L p ; b F r } converges in L p if p, and converges in the weak* sense if p = as r ; c F r } is L p -bounded. 3 For every µ M, the following statements are equivalent: a dµ = fdθ for some f C; b F r } converges uniformly as r ; c F extends to a continuous function on D. Proof. Recall from Proposition 2.3 that for each µ M, the function F : D C defined by F r := P r µ is harmonic. Since sup P r µ sup P r µ < for each µ M, the function given by µ F r θ := P r µθ is indeed a function from M to h D. Additionally, given any F h D, by Lemma 2.8, there is a unique µ M such that F r θ := P r µθ r [,. Hence, the mapping µ F r θ := P r µθ gives a one-to-one correspondence between M and h D. Fix any µ M. Since fdµ = lim ff r dθ f sup F r r for every f C, we have µ V = µ C sup F r. On the other hand, we also have sup F r = sup P r µ sup P r µ = µ.

12 2 LINGXIAN ZHANG hus, µ = sup F r = lim r F r, with the last equality following from Proposition We now proceed to prove the remainder of the theorem. Suppose that µ m. hen, by Radon-Nikodym theorem, there exists a function f L such that dµ = fdθ. By Proposition 2.4, lim r F r f = lim r P r f f =, which is to say that F r } converges in L to f as r. Conversely, suppose that F r } converges in L to some function f as r. hen, for every g C, lim gθf r θdθ gθfθdθ lim g F r θ fθ dθ r r = g lim F r f r =, that is, lim r gθf rθdθ = gθfθdθ. In other words, F rdθ} converges to fdθ in the weak* sense as r. By uniqueness, dµ = fdθ, and consequently µ m. 2 Let q denote the conjugate exponent of p, that is, the number such that p + q =. Since Lp is isometrically isomorphic to L q via the map ψ ϕ ϕψ dθ, it makes sense to speak of weak* convergence in L p. Suppose that dµ = fdθ for some f L p. hen it follows from Proposition 2.4 that, as r, F r = P r f converges in L p to f if p, and converges in the weak* sense to f if p =. Next, suppose that p, and that F r } converges in L p to some f as r. hen, by Lemma 2.73, sup F r p = lim F r p = f p <. r If p = and F r converges in the weak* sense to some f L as r, then it follows from the inclusion C L and the uniqueness mentioned in Lemma 2.8 that dµ = f dx. Hence, sup F r sup P r f = f <. Finally, suppose that the collection F r } is L p -bounded. hen, the sequence } F /n n= Lq has a subsequence, say } F, that /nk k= converges in the weak* sense to some f L p, because the closed unit ball of L p is compact and metrizable with respect to the weak* topology. As a result, we have that for every g C, gθdµθ = lim gθ P /nk µ θ dθ = gθfθ dθ, k and whence dµ = fdθ, with f L p. 3 Suppose that dµ = fdθ for some f C. hen, by Proposition 2.4, the functions F r = P r f f uniformly as r. Next, suppose that the functions F r converge uniformly to some function f : C as r. Define a function G : D C by G D := F and G := f. Since the harmonic function F is continuous on D, the continuity of f follows from uniform convergence. In order to prove that the function G is continuous on D, it suffices to show that G is continuous on D \ D, that is, on. Fix

13 HE L p CONVERGENCE OF FOURIER SERIES 3 an arbitrary eθ, and an arbitrary ɛ >. hen, there exist δ, such that F r f < ɛ/2 whenever r δ,, and δ 2, δ such that fτ fθ < ɛ/2 whenever τ θ < δ 2. Consequently, for every reτ D with reτ eθ < δ 2, we have r > δ 2 > δ and τ θ < 2 δ 2 < δ 2, whence Greτ Geθ F r τ fτ + fτ fθ < ɛ if r <, Greτ Geθ = fτ fθ < ɛ/2 < ɛ if r =. herefore, the function G : D C is a continuous extension of F. Lastly, suppose that F extends to a continuous function, say G, on D. Define a continuous function f := G. hen, the continuity of G implies that F r f pointwise as r. In addition, the maximum principle tells us that sup F r = sup D F = sup f = f <. hus, by the compactness of and by the dominated convergence theorem, for every g C, lim gf r dθ = gfdθ, r which is to say that fdθ is the weak* limit of F r as r. From the uniqueness of such limit, it follows that dµ = fdθ, with f C. 3. wo Maximal Operators Definition 3.. For each µ M, we define its Hardy-Littlewood maximal function Mµ : [, ] by µ I Mµθ := sup, I θ I where I is an open interval. When, dµ = fdm for some f L, we may write Mf in place of Mµ. We will refer to the map µ Mµ as the Hardy- Littlewood maximal operator. Proposition 3.2. For every µ M and every λ >, one has that θ : Mµθ > λ} 3λ µ. Proof. Let µ M and λ > be given arbitrarily. hen, for each η θ : Mµθ > λ}, there exists some open interval I η containing η such that µ I η > λ I η. Fix any compact K θ : Mµθ > λ}. hen, there is a finite subset η,..., η N } K such that K N n= I η n. By the finite Vitali covering lemma, there exist I ηn,..., I ηnj I ηn n =,..., N} pairwise disjoint such that N n= I η n J j= 3I η nj, where 3I denotes the open interval with the same centre as I but triple the length. As a result, N J J K I ηn 3 I ηnj 3λ µ I ηnj 3λ µ = 3λ µ. n= j= j= hus, by the inner regularity of Lebesgue measure, θ : Mµθ > λ} 3λ µ.

14 4 LINGXIAN ZHANG Definition 3.3. For each function F : D C, we define its radial maximal function F : [, ] by F θ := sup F reθ. In order to relate the radial maximal operator to the Hardy-Littlewood maximal operator, we will need the following lemma. Lemma 3.4. Suppose that a function K : [ 2, 2] R is nonnegative, continuous, even and decreasing, that is, Kθ Kη whenever θ > η. hen for every µ M and every θ [ 2, 2], one has that K µ θ K Mµθ. he idea of the proof is to write K as an average of box kernels, for which the above inequality is easier to establish. Details can be found in the appendix. Proposition 3.5. For every u h D, one has u Mµ, where µ is the complex Borel measure corresponding to u. Proof. Fix an arbitrary u h D, and let µ M be the complex measure corresponding to u, as mentioned in Lemma 2.8. Observe from the formula P r θ = r 2 2r cos2πθ + r 2 that every P r is nonnegative, continuous, even and decreasing. So it follows from the previous lemma that for every θ, u θ = sup P r µθ sup P r Mµθ = Mµθ. Corollary 3.6. If u h D, then, for every λ >, θ : u θ > λ} 3λ u. Proof. Fix any u h D, and let µ be the corresponding complex measure. hen, for every λ >, θ : u θ > λ} θ : Mµθ > λ} 3λ µ = 3λ u. Corollary 3.7. If f L, then P r f f a.e. as r. Proof. Let ɛ > be given arbitrarily. hen, there exists some g C such that f g < ɛ 2. Set h := f g, and define a function u : D C by u r := P r h. Since lim r P r gθ gθ = for every θ, we have } θ : lim sup P r fθ fθ > ɛ r θ : lim sup P r hθ > ɛ } + θ : hθ > ɛ r 2 2} θ : u θ > ɛ } + θ : hθ > ɛ 2 2} 3 2 ɛ u + 2 ɛ h 6 ɛ h + 2 ɛ h 8ɛ.

15 HE L p CONVERGENCE OF FOURIER SERIES 5 Since ɛ > is arbitrary, the above inequality implies that for every δ,, } θ : lim sup P r fθ fθ > r } θ : lim sup P r fθ fθ > δ n 8δ r δ. n= aking δ +, we obtain that } θ : lim sup P r fθ fθ > =. r herefore, P r fθ fθ as r for a.e. θ. he above corollaries will help us show that the Hilbert transform, to be defined in the next section, meets the premise of the Marcinkiewicz interpolation theorem. 4. Conjugate Function and Hilbert ransform 4.. Conjugate function. Definition 4.. For each real-valued and harmonic function u on D, we define its conjugate function ũ to be the unique real-valued and harmonic function on D such that u + iũ is analytic and ũ =. And, for each complex-valued and harmonic function u on D, we define its conjugate function ũ := Reu + iĩmu. Proposition 4.2. Let u : D C be a harmonic function. holds. hen the following ū = ũ 2 If u is constant, then ũ =. 3 If u is analytic in D, and if u =, then ũ = iu. If u is co-analytic, meaning that ū is analytic, and if u =, then ũ = iu. 4 he function u can be written uniquely as u = c + f + ḡ with c a constant, f, g analytic, and f = g =. Proof. By the definition of conjugate function, ū = Reū + iĩmū = Reu + i Imu = Reu + i Ĩmu = ũ. 2 Suppose that u is constant. hen, it is clear from the Cauchy-Riemann equations that ũ is constant. Since ũ =, we deduce that ũ =. 3 Suppose that u is analytic, and that u =. hen, ũ = Reu + iĩmu = Imu + i Reu = iu. Next, suppose that u is co-analytic, and that u =. hen, it follows from above that ũ = ũ = ū = iū = iu. 4 Set f := 2 u + iũ u, g := 2 u iũ u = 2 ū + i ū u, and c := u. Since u + iũ and ū + i ū are analytic, the functions f and g are both analytic. hus, we can write u as a sum u = c + f + ḡ, with c a constant, f, g analytic, and f = g =.

16 6 LINGXIAN ZHANG Now suppose that there exist a constant γ and two analytic functions ϕ and ψ such that ϕ = ψ = and u = γ + ϕ + ψ. hen, γ = u ϕ ψ = u = c, thus ϕ + ψ = u u = f + ḡ, and consequently ϕ f = g ψ. Since ϕ f is analytic, while g ψ is co-analytic, ϕ f = g ψ =, or in other words, ϕ = f and ψ = g. his proves the uniqueness of the decomposition. Notation 4.3. Hereinafter, we will sometimes write Ffn in place of ˆfn for the sake of clarity. Proposition 4.4. Let u : D C be a harmonic function. hen, for every r, and every n Z, Fũ r n = i sgnnû r n Proof. Since the Fourier transform is a linear operator, by the last part of the previous proposition, it suffices to prove the above equality in the cases that u is constant, analytic, or co-analytic. o begin with, let s suppose that u is constant. hen, ũ =. So, for every r,, u r and ũ r are both constant. Hence, for every n Z, Fũ r n = = i sgnn = i sgnnû r n. Next, suppose that u is analytic. hen, ũ = iu is also analytic. hus, for every r,, we have that Fũ r n = = i sgnn = i sgnnû r n n Z, since u and ũ are both analytic on the simply connected region D, and that Fũ r n = F iu r n = ifu r n = i sgnnû r n n Z +. Finally, suppose u is co-analytic. hen, for every r, and every n Z, Fũ r n = F ū r n = i sgn n ū r n = i sgnnû r n. Proposition 4.5. Let u h 2 D. hen, ũ r 2 2 = u r 2 2 u 2 for every r,, and thus ũ h 2 D. Proof. By heorem.3, for every r,, ũ r 2 2 = F ũ r n 2 = i sgnnû r n 2 n Z n Z = û r n 2 û r 2 = u r 2 2 u r 2. n Z Hence, ũ 2 = sup ũ r 2 = sup u r 2 2 u r 2 /2 sup u r 2 = u 2 <, implying that ũ h 2 D. Corollary 4.6. If u h 2 D, then the functions ũ r converges in L 2 as r, and thus lim r ũ r θ exists for a.e. θ as an L 2 function. Proof. Since the collection ũ r } is L 2 -bounded, by heorem 2.9, there exists g L 2 such that ũ r g in L 2 as r, and ũ r = P r g for every r. herefore, by Corollary 3.7, lim r ũ r θ exists and equals gθ for a.e. θ.

17 HE L p CONVERGENCE OF FOURIER SERIES Harmonic measures and conjugate radial maximal operator. Definition 4.7. For each λ >, we define a function ω λ : R R + R by ω λ x, y := y π x t 2 + y 2 dt,, λ] [λ, which some may recognize as the harmonic measure of, λ] [λ, centred at x, y with respect to the open upper half-plane. Proposition 4.8. For every λ >, the function ω λ is harmonic in the open upper half-plane H := z C : Imz > } = R R +. Proof. Fix any λ >. hen, for every z = x + iy H, ω λ z = π 2 y dt + = π, λ] [λ, arctan t x y x + λ y arctan And a routine calculation shows that ω λ = in H. x λ y. Proposition 4.9. Let u h D. hen, for every λ >, one has that θ : ũ θ > λ } Cλ u λ, with C some absolute constant. Proof. Fix some λ >. o start with, suppose that u is real-valued and nonnegative. hen, by the maximum principle, u is either constantly zero or strictly positive on D. he case that u is constantly zero is trivial, so let s assume in the rest of the paragraph that u is strictly positive. Set F := iu + iũ = ũ + iu. hen, F is an analytic function from C to H, and thus the composition ω λ F is harmonic in D. In addition, for every θ with ũ θ > λ, we have that sup ω λ F reθ = + π sup arctan + π sup = 2 + π arctan arctan sup ũreθ λ ureθ ũreθ λ ureθ ũreθ λ ureθ arctan π 2 > 2. Hence, θ : ũ θ > λ } θ : ω λ F θ > 2} ũreθ + λ ureθ 6 ω λ F = 6 ω λ F = 6 ω λ, u, where the second inequality follows from Corollary 3.6, and the first equality follows from the mean-value property of harmonic functions. Furthermore, since ω λ, u = 2 π λ/u t 2 + dt 2 π λ/u t 2 dt = 2 π u λ = 2 π λ u,

18 8 LINGXIAN ZHANG we deduce that θ : ũ θ > λ } 2 π λ u. Now, allow u to be any function in h D. By Lemma 2.8, there is a unique µ M such that u r = P r µ for every r,. Decompose the complex measure µ into µ = µ + Re µ Re +iµ+ Im iµ Im, with µ+ Re, µ Re, µ+ Im, µ Im being the unique positive measures for which the above equality holds. Let v +, v, w +, and w be the harmonic functions on D corresponding to µ + Re, µ Re, µ+ Im, and µ Im, respectively, as specified in heorem 2.9. hen, u = v + v + iw + iw, and thus ũ = ṽ + ṽ + i w + i w. Consequently, for every θ, ũ θ = sup ṽ+ reθ ṽ reθ + i w + reθ i w reθ sup ṽ reθ + + ṽ reθ + w + reθ + w reθ ṽ+ θ + ṽ θ + w+ θ + w θ. Furthermore, since µ + Re, µ Re, µ+ Im, µ Im are positive measures, and since P r > for every r,, the functions v +, v, w +, w are obviously real-valued and nonnegative. herefore, θ : ũ θ > λ } } λ θ : ṽ+ θ > + } λ 4 θ : ṽ θ > 4 + } λ θ : w+ θ > + } λ 4 θ : w θ > 4 2 π 4 v + λ + v + w + + w = 48 π λ µ + Re + µ Re + µ+ Im + µ Im = 48 π λ µ = 48 π λ u Hilbert transform. Definition 4.. Given any f L, we define the Hilbert transform of f by Hfθ := lim r ũ f r θ, where u f is the h D function defined by u f r := P r f. It is not obvious that the Hilbert transform of each f L is well-defined, so we will justify the above definition in the following proposition. Proposition 4.. Given any f L, the limit lim r ũ f r θ exists for a.e. θ. Proof. Fix an f L, and let ɛ > be given arbitrarily. Since C is a dense subset of L, there exists some g C such that f g < ɛ. Put

19 HE L p CONVERGENCE OF FOURIER SERIES 9 h := f g L, and for each δ >, define E δ := θ : lim sup ũf r θ ũ f s θ > δ, r,s } F ɛ,δ := Since ũ f = every δ >, and consequently E δ = F ɛ,δ θ : lim sup r,s ũ h r θ ũ h s θ > δ u h + u g = ũ h + ũ g, and since g L 2, by Corollary 4.6, E δ = F ɛ,δ for θ : ũh θ > δ 2 } Cδ u h = Cδ h < Cδ ɛ. As ɛ > may be arbitrarily small, we deduce from the above inequality that E δ = for every δ >. hus, for a.e. θ, the net ũ f r θ } is Cauchy. It then r, follows from the completeness of C that lim r ũ f r θ exists a.e. on. Corollary 4.2. he Hilbert transform is bounded from L to weak-l. Proof. For any f L and any λ >, θ : Hfθ > λ} θ : uf θ } > λ C uf λ = C f λ, where C is the same constant as in Proposition 4.9. Proposition 4.3. For any p,, the Hilbert transform is bounded on L p. Proof. It is clear from Proposition 4.5 that for any f L 2, Hf 2 ũ f 2 u f 2 = f 2, which implies that the Hilbert transform H is bounded on L 2. hen, by Marcinkiewicz interpolation theorem, H is bounded on L p for every p, 2]. Further, by the dominated convergence theorem and by Proposition 4.4, for every f L 2 and every n Z, Ĥfn = e nx lim ũ f r θdθ = lim e nx ũ f r θdθ r r = i sgnn lim r ûf n = i sgnn ˆfn. Hence, by heorem.3, for any two f, g L 2, Hf, g = n Z Ĥfnĝn = n Z ˆfn } i sgnnĝn. = n Z ˆfnĤgn = f, Hg, which implies that the adjoint H : L 2 L 2 satisfies H = H. Now, fix any p 2,, and let q denote its conjugate exponent. Since L p L 2, we may consider the Hilbert transform on L p as a linear map from L p to L 2. Because L 2 is a dense subset of L q, the adjoint H : L q L 2 L q satisfies H = H : L q L q, and is thus bounded on L q. Consequently, the Hilbert transform H = H : L p L p is bounded.

20 2 LINGXIAN ZHANG 5. he L p Convergence of Fourier Series heorem 5.. For every p,, the partial sum operators S N are uniformly bounded on L p, and thus S N f f in L p for every L p function f. Proof. Fix any p,, and define a linear operator : L p L p by f := 2 f + i 2 Hf ˆf. 2 hen, p p H p p + 2 <. Moreover, using the same argument as in the proof of Proposition 4.3, we can show that for every f L p and every n Z, Ĥfn = i sgnn ˆfn, and consequently fn = χ [, ˆfn. hus, for every f L p and every N Z +, and S N f = n= χ [ N,N] ˆfnen = e N+ e N+ f e N e N f, S N f p p p e N+ f p + e N f p = 2 p p f p. It follows that sup N Z + S N p p 2 p p <. herefore, by Proposition.4, S N f f in L P for every L p function f. 6. Generalization: Multiplier Operators Fix an arbitrary p,. Since the Hilbert transform is bounded on L p, repeating the argument in the proof of Proposition 4.3, we can show that Ĥfn = i sgnn ˆfn for every n Z and every f L p. Let m H : Z C denote the sequence i sgnn} n=. hen, Hf = lim S NHf = lim N n= N m H n ˆfne n in L p for every f L p, and we call m H the multiplier associated to the Hilbert transform H. Now, consider an arbitrary sequence m : Z C, and associate to it, formally, an operator m on L p given by 6. m fn = mn ˆfn n Z f L p. Under what condition is there an actual bounded operator m with the above property? Notation 6.2. We will mean by l the space of all bounded functions from Z to C, equipped with the supremum norm. Let s first look for some necessary conditions. Assume that there is an operator m such that 6. holds. hen, since e k p = and m e k = mke k for every k Z, we have that m p p sup m e k p = sup mk = m. k Z k Z Hence, in order for there to exist a bounded operator m on L p such that 6. holds, it is necessary that m l. In the special case that p = 2, if m l, then

21 HE L p CONVERGENCE OF FOURIER SERIES 2 for every f L 2, mn 2 2 ˆfn m 2 n= N n= ˆfn 2 = m 2 f 2 2 <, which implies that the sequence n= N mn ˆfne } n is Cauchy and whence N= convergent. herefore, for each multiplier m l, there is the bounded operator m : L 2 L 2, defined by N m f := lim mn ˆfne n n= N for each f L 2, such that 6. holds. So, we have found an equivalent condition, namely that m l, for the existence of an operator m that is bounded on L 2. Next, observe that for every g L, µ M and n Z, the equality g µn = ĝn ˆµn holds as an easy consequence of the Fubini-onelli theorem. hus, if m = Fµ for some µ M, then there is the bounded operator m : L p L p, defined by m f := f µ for each f L 2, such that 6. holds, and m p p µ <. he converse is also true in the special cases that p = or p =, though I will only give a proof of the former in the appendix. heorem 6.3. If m : Z C is a multiplier for which there exists a bounded operator m : L L such that 6. holds, then m = ˆµ for some µ M. For other value of p, the condition m = Fµ is clearly not necessary for the existence of the multiplier operator m, because there exists the bounded multiplier operator H whose associated multiplier m H is not of the form Fµ. he equivalent condition for the existence of a bounded operator m satisfying 6. remains largely an open question in these cases. But why do we even care about these multipliers and multiplier operators, apart from the fact that they seem fun? Since Fourier coefficients are preserved under translation, every bounded multiplier operator is translation-invariant; and it turns out that these multiplier operators are exactly the translation-invariant operators. Moreover, the theory of singular integrals, when initially developed by Calderón and Zygmund, was built upon Mikhlin s work on multipliers. For further discussion on multipliers, see Section 7.3 of [4], and Chapter 8 of [3]. Acknowledgement I would like to thank my mentor, Daniel Campos Salas, for suggesting the topic of this paper, for guiding me through the reading and writing process, and for reviewing the drafts. I am also grateful to Professor Peter May for offering me the opportunity to participate in the REU program at the University of Chicago this summer, and for giving me much valuable advice.

22 22 LINGXIAN ZHANG Appendix Proof of Proposition.3. Suppose that f L p, with p [, ], and that µ M. We define for each θ a function f θ : C by f θ τ := fθ τ. hen, f θ p = f p for every θ. If p =, then f µ sup θ f µθ sup θ fθ τ µ τ sup f µ τ = f µ. θ If p [,, define for each θ a function f θ : C by f θ τ := fθ τ. hen, f θ p = f p for every θ, whence p /p f µ p f θ τ d µ τ dθ /p f θ τ dθ p d µ τ = f θ p d µ τ = f p µ, where the second inequality holds by Minkowski s integral inequality. herefore, the inequality f µ p f p µ holds in all possible cases. Proof of Proposition.4. o begin with, suppose that f L and g C. Since g is uniformly continuous, given an arbitrary ɛ >, there exists δ > such that gx gx < ɛ whenever x x < δ. hen, for any two x, x with x x < δ, f gx f gx = fy gx y gx y dy ɛ f. Since f <, we conclude that f g is continuous. Now, suppose that f L and g L. Since C is a dense subset of L, for any given ɛ >, there exists h C such that g h < ɛ, and there exsits δ > such that f hx f hx < ɛ whenever x x < δ. hus, for any two x, x with x x < δ, f gx f gx f g hx + f h gx + f hx f hx 2 f g h + f hx f hx < 2 f + ɛ It then follows that f g is continuous. Proof of Proposition.. Firstly, suppose that f C, and let ɛ > be given arbitrarily. Since f is uniformly continuous, there exists δ > such that fx y fx < ɛ for any two x, y with y δ. hen, for every index N and every x, Φ N f x fx = fx y fx Φ N ydy fx y fx Φ N y dy + y >δ y δ ɛ Φ N y dy

23 2 f 2 f which implies that HE L p CONVERGENCE OF FOURIER SERIES 23 y >δ y >δ Φ N f f 2 f aking N, we obtain that Φ N y dy + ɛ Φ N y dy Φ N y dy + ɛ sup Φ K y dy, K Z + y >δ Φ N y dy + ɛ sup K Z + Φ K. lim sup Φ N f f ɛ sup Φ K. K Z + Since ɛ > may be arbitrarily small, we deduce from the inequality above that lim Φ N f f =. 2 Next, suppose that f L p, with p [,. hen, for any g C, Φ N f f p Φ N f Φ N g p + g f p + Φ N g g p = Φ N f g p + f g p + Φ N gx gx p dx Φ N f g p + f g p + /p Φ N g g p dx sup Φ K + K Z + f g p + Φ N g g, for every index N, and whence that lim sup Φ N f f p sup Φ K + K Z + f g p. /p Since C is a dense subset of L p, taking the infimum over all g C, we obtain that lim Φ N f f p =. 3 Finally, suppose that µ M. Fix any f C, and define a function g : C by gx := f x. hen, g is continuous, and lim sup = lim sup = lim sup = lim sup = lim sup = lim sup fxφ N µxdx fx dµx fx Φ N x y dµy dx fx dµx fxφ N x y dx dµy fx dµx fxφ N x y dx dµy fx dµx g xφ N y x dx dµy g x dµx Φ N g y dµy g y dµy

24 24 LINGXIAN ZHANG lim sup Φ N g y g y d µ y lim sup Φ N g µ =, which implies that lim fxφ N µxdx = fx dµx. herefore, Φ N µ µ in the weak* sense as N. Proof of Lemma 3.4. It is clear from the piecewise monotonicity of K that K is of bounded variation. Since K is continuous, there exists a unique κ M such that κ [a, b] = Kb Ka for all 2 a b 2. Set ν := κ + K 2 δ/2, where δ /2 is the Dirac mass at 2. hen, for every x [ 2, 2], Kx = K x = K whence = κ = = = = 2 [ x, 2 /2 /2 /2 /2 K = 2 = 2 K x ] + K 2 + K 2 /2 χ [ ϕ,ϕ] x dκϕ + K χ [ ϕ,ϕ] x dδ /2 ϕ 2 /2 χ [ ϕ,ϕ] x dκϕ + K χ [ ϕ,ϕ] x dδ /2 ϕ 2 χ [ ϕ,ϕ] x d κ + K δ /2 ϕ 2 χ [ ϕ,ϕ] x dνϕ, /2 /2 /2 Kx dx = 2 /2 /2 χ [ ϕ,ϕ] x dx dνϕ = 2 χ [ ϕ,ϕ] x dνϕ dx /2 ϕ dνϕ, Furthermore, since K is decreasing on [, 2], the restriction of the signed measure µ to B [,/2] is a negative measure. Consequently, the restriction of ν = µ + K 2 δ/2 to B [,/2] is a sum of two positive measures, and thus itself a positive measure. herefore, for every µ M and every θ [ 2, 2], /2 /2 K µθ = χ [ ϕ,ϕ] θ τ dνϕ dµτ /2 /2 /2 = χ [ ϕ,ϕ] θ τ dµτ dνϕ = /2 /2 2ϕ [θ ϕ, θ + ϕ] µ[θ ϕ, θ + ϕ] dνϕ

25 HE L p CONVERGENCE OF FOURIER SERIES 25 /2 /2 2ϕ µ [θ ϕ, θ + ϕ] dνϕ [θ ϕ, θ + ϕ] 2ϕ Mµθ dνϕ = K Mµθ. Proof of heorem 6.3. Let m : Z C be any multiplier for which there is a bounded operator m : L L such that 6. holds. We define a family f t } t> of functions from to C by setting f t := e 2π n t e n n= for each t >. hen, by the Fubini-onelli theorem, f t = for every t >, and whence the family m f t } t> L M is bounded by m. Using a compactness argument similar to that used in the proof of Lemma 2.8, there exists a sequence t k } k=, decreasing to, and a complex measure µ M with µ m such that m f tk µ in the weak* sense as k. Since e n x m f t x dx = mne 2π n t for every t > and every n Z, it follows that for every n Z, mn = mn lim e 2π n t = lim t = lim e n x m f tk x dx = k k mne 2π n t k e n x dµx = ˆµn. References [] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2. [2] G. B. Folland, Real Analysis: Modern echniques and their Applications, John Wiley & Sons Inc., Second Edition, 999. [3] C. Muscalu, W. Schlag, Classical and Multilinear Harmonic Analysis, Cambridge University Press, Volume I, 23. [4] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 97.