A PROOF OF CARLESON S THEOREM BASED ON A NEW CHARACTERIZATION OF THE LORENTZ SPACES L(p, 1) FOR 1 < p < AND OTHER APPLICATIONS

Transcription

1 PROOF OF CRLESON S THEOREM BSED ON NEW CHRCTERIZTION OF THE LORENTZ SPCES L(p, ) FOR < p < ND OTHER PPLICTIONS GERLDO SORES DE SOUZ bstract. In his 95 nnals of Mathematics paper entitled Some New Functional Spaces, G. G. Lorentz [] introduced the function spaces denoted by Λ(α), < α <, defined as the set of real measurable functions for < x < for which f Λ(α) α x α f (x)dx <, where f is the decreasing rearrangement of f. In this paper we give two simple characterizations for Λ(/p) for < p < based on a generalization of the special atom space introduced by G. De Souza in earlier works [3], [6], [7], and []. The space Λ(/p) is nowadays denoted by L(p, ). s an application, we give a proof of Carleson s Theorem on the convergence of Fourier series on L(p, ) and, more generally, on L(p, r) for p, r >. lso we have a simple proof of a theorem of Stein and Weiss on operators in L(p, ).. Preliminaries In this section, we state several definitions that will be used throughout this paper with references to the original source. Definition.. real-valued function f defined on [, π] belongs to the space L(p, ) for < p < if f L(p,) 2π f (t)t p dt <, where f is the decreasing rearrangement of f. This space was originally introduced by G. G. Lorentz [] in 95 where it was denoted by Λ(/p). Definition.2. generalized special atom is a function b : [, π] R, b(t) 2π or for any α (, ] and µ-measurable subsets X,, B of [, π], b(t) µ(x) α [ χ (t) χ B (t) where X B, B, µ() µ(b), µ is a measure on subsets of [, π], and χ E denotes the characteristic function of the set E. ] Date: October 5, Mathematics Subject Classification Key words and phrases. Lorentz Spaces, Special tom Spaces, Generalized Lipschitz Spaces, Duality, Equivalence of Banach Spaces, Besov-Bergman Spaces.

2 2 GERLDO SORES DE SOUZ Definition.3. For < α, let (b n ) n be a sequence of generalized special atoms, (C n ) n a sequence of real numbers, and µ a measure on subsets of [, π]. We define the generalized special atom spaces by { } (µ, α) f : [, π] R; f(t) C n b n (t); C n <. We endow (µ, α) with the norm f (µ,α) inf C n, where the infimum is taken over all possible representations of f. The notion of special atoms and the spaces formed by special atoms as well as certain generalized spaces were introduced originally by G. De Souza, see [3], [6],[7], [9]. In those works, intervals and lengths were used. Definition.4. For < α and µ a measure on sets of [, π], we define the space B(µ, α) as { } B(µ, α) f : [, π] R; f(t) a n d n (t); a n <, where d n (t) µ α ( n ) χ n (t), n are µ-measurable sets in [, π], and a n s are real numbers. We endow B(µ, α) with the norm f B(µ,α) inf a n, where the infimum is taken over all possible representations of f. This space also was introduced by G. De Souza in his early work, see [3], [4], [6], [7]. Definition.5. For < α and µ a measure on sets of [, π], we define the space Λ(µ, α) as { } Λ(µ, α) f : [, π] R; µ α f(x)dµ(x) f(x)dµ(x) (X) B < M for µ-measurable subsets X,, B of [, π] such that X B, B. We endow Λ(µ, α) with the norm [ ] f Λ(µ,α) sup X B, B µ α f(x)dµ(x) f(x)dµ(x) (X) B Note that this space is a natural generalization of the Lipschitz spaces. In fact if we take µ as the Lebesgue measure, X [x h, x+h], [x h, x), B [x, x+h], and µ α (X) (2h) α, then for a differentiable f, we get µ α (X) f (x)dµ(x) B f (x)dµ(x) f(x + h) + f(x h) 2f(x) (2h) α. The space Λ(µ, α) in this form has been introduced by G. De Souza in his ealier work, see [3], [6], [7], [23].

3 CHRCTERIZTION OF L(p, ) 3 Definition.6. For < α and µ a measure on sets of [, π], we define the space Lip(µ, α) as { } Lip(µ, α) f : [, π] R; µ α f(x)dµ(x) (X) < M, where is a µ-measurable set in [, π]. norm is defined on Lip(µ, α) as f Lip(µ,α) sup µ α f(x)dµ(x) (). This space was originally introduced by G. G. Lorentz in 95, see [], [2]. 2. Known Results In this section, we state some known results and sketch very briefly the proof of some of them or make some comments. For more details, see [], [2], [23], [24]. Theorem 2.. The spaces ((µ, α), (µ,α) ), (B(µ, α), B(µ,α) ), (Λ(µ, α), Λ(µ,α) ), and (Lip(µ, α), Lip(µ,α) ) for < α are Banach spaces. The proof follows using direct application of standard techniques for Banach spaces. Theorem 2.2. The spaces (µ, α) and B(µ, α) are the same as Banach spaces and the norms are equivalent, that is (µ, α) B(µ, α) with M f B(µ,α) f (µ,α) N f B(µ,α), where M and N are absolute constants. Clearly (µ, α) is continuously contained in B(µ, α). In fact if f (µ, α), then f(t) [ ] C n µ α χ n (t) χ Bn (t) (X n ) C n µ α (X n ) χ C n n (t) µ α (X n ) χ n (t) ( ) α µ(n ) C n µ(x n ) µ α ( n ) χ n (t) ( ) α µ(bn ) C n µ(x n ) µ α (B n ) χ B n (t) Since X n n B n, µ( n) µ(x n ), µ(b n) µ(x n ), we have f B(µ,α) 2 Therefore, f B(µ,α) f (µ,α). For the other inequality, please refer to De Souza and Pozo [24]. C n. Theorem 2.3. The spaces Λ(µ, α) and Lip(µ, α) for < α < are equivalent as Banach spaces that is Λ(µ, α) Lip(µ, α) with M f B(µ,α) f Λ(µ,α) N f B(µ,α), where M and N are absolute constants. gain, one of the inequalities is easily seen, that is Lip(µ, α) Λ(µ, α) and f Λ(µ,α) 2 f Lip(µ,α). For the other inequality, just note that f(t) dµ(t) µ /p () µ /p ( B) f(t)dµ(t) f(t)dµ(t), sup µ( B) where B ( B) (B ). B

4 4 GERLDO SORES DE SOUZ Theorem 2.4 (Duality). φ is a bounded linear functional on (µ, α), < α < if and only if there is a unique g Λ(µ, α) so that φ(f) π f(x)g(x)dµ(x) with φ g Λ(µ,α). That is, (µ, α) Λ(µ, α), where (µ, α) is the dual space of (µ, α). Theorem 2.5 (Duality). φ is a bounded linear functional on B(µ, α), < α < if and only if there is a unique g Lip(µ, α) so that φ(f) π f(x)g(x)dµ(x) with φ g Lip(µ,α). That is, B (µ, α) Lip(µ, α), where B (µ, α) is the dual space of B(µ, α). The proofs of these two duality Theorems follow easily after a pair of Holder type inequalities. That is π f(x)g(x)dµ(x) f (µ,α) g Λ(µ,α), f (µ, α), g Λ(µ, α) and π f(x)g(x)dµ(x) f B(µ,α) g Lip(µ,α), f B(µ, α), g Lip(µ, α). For a complete proof see De Souza and Pozo [24]. Theorem 2.6 (Duality-G.G. Lorentz). φ is a bounded linear functional on L( α, ), < α <, if and only if there is a unique g Lip(µ, α) so that φ(f) with φ g Lip(µ,α). That is, L ( α, ) Lip(µ, α). π f(x)g(x)dµ(x) gain this duality Theorem is due to G.G. Lorentz []. It also follows from the Holder type inequality π f(x)g(x)dµ(x) f L( α,) g Lip(µ,α), f L(, ), g Lip(µ, α). α 3. Main result In this section, we state and prove the main result which is the characterization of L(p, ), < p < as B(µ, /p) and (µ, /p). Theorem 3.. f L(p, ) if and only if f B(µ, /p) for < p <. Moreover N f B(µ,/p) f L(p,) M f B(µ,/p), where N and M are absolute constants. Proof. Let us show that B(µ, /p) L(p, ), < p <. To that end, all we need is to estimate f L(p,) where f(t) χ (t), is a µ-measurable set in [, π]. In fact χ (t) L(p,) µ() p(µ()) p χ (t)t p dt χ [,µ()] (t)t p dt t p dt

5 CHRCTERIZTION OF L(p, ) 5 That is χ (µ()) p. p Now if f B(µ, α), then f(t) C n d n (t) with C n <, where d n (t) χ (µ( n )) n (t), n a µ-measurable set in [, π]. p Then f L(p,) C n d n L(p,) p C n so that taking the infimum, we get We have the following situations: f L(p,) p f B(µ,/p), < p <. () B(µ, /p) L(p, ) for < p < and f L(p,) p f B(µ,/P ) (2) B (µ, /p) Lip(µ, /p) by Theorem 2.5 (3) L (p, ) Lip(µ, /p) by Theorem 2.6 (4) B(µ, /p) is dense in L(p, ). Easily shown with standard technique. s a consequence of these facts, the embedding operator I : B(µ, /p) L(p, ) defined by I(f) f is a Banach space isomorphism. That is B(µ, /p) L(p, ) with equivalent norms. Note that (µ, /p) B(µ, /p), < p < by Theorem 2.2. Therefore we have the following result. Theorem 3.2. The spaces (µ, /p), B(µ, /p) and L(p, ) for < p < are equivalent as Banach spaces and the norms are equivalent. 4. pplication In this section, we give a simple proof of a well-known theorem due to Guido Weiss and Elias Stein given in [25] and [26] concerning linear operators acting on the Lorentz space L(p, ). Theorem 4. (Stein and Weiss). If T is a linear operator on the space of measurable functions and T χ X M(µ()) p, < p < where X is a Banach space, then T can be extended to all L(p, ); that is T : L(p, ) X and T f X M f L(p,). Proof. fter this new characterization of L(p, ) as the space B(µ, /p), < p <, given in Theorem 3., this result is an immediate consequence of the representation of f as f L(p, ) f B(µ, /p) f(t) C n d n (t) with C n < and d n (t) χ (µ( n )) n (t), n s µ-measurable sets in [, π] so that p T f(t) C n T (d n ((t)).

6 6 GERLDO SORES DE SOUZ Consequently, T f X Therefore T f X C n T d n X and, by hypothesis, T d n X M(µ( n )) p. C n and so T f X M f L(p,) 5. Comments. Prof. Richard O Neil from SUNY at lbany: If f(t) 2 3 n and C j >, then he added that if f(t) n j n C j q j (t) j where q j (t) χ j (t) χ Bj (t), j B j, C j >, then n j C j (µ( j B j )) p p f L(p,). n C j χ j (t) where j C j (µ( j )) p p f L(p,). lso Note L(p, ) L p for < p < and because of this new characterization, L(p, ) is a much easier space to work with than L p. The space which we denoted in this paper by Lip(µ, α) is denoted in the literature by L( α, ) for < α < and is called the weak-l p space with norm given by f Sup{t α f (t)}, where f is t> the decreasing rearrangement of f. One can show that the norms f and f Lip(µ,α) are equivalent. Finally, if we define m(f, y) µ({x : f(x) > y }), then by a change of variable y f (t), t m(f, y) and integration by parts, we get f L(p,) f (t)t p dt p (m(f, y)) p dy. Prof. Richard O Neil: This last integral is sort of infinitesimal version of your atomic decomposition. Indeed it was this formula that led to the remark that an operator of restricted weak type (p, q) was the same as a strong operator from L(p, ) to L(p, ). 2. One of the most interesting observations that we made in the process to obtain the new characterization of L(p, ) for < p < is that T f(x) sup S n (f, x), n where S n (f, x) is the n th partial sum of the Fourier Series of f is Theorem 5.. If T f(x) sup S n (f, x), then T χ L(p,) Mµ() p for p > n and so T f L(p,) M f L(p,). is a µ-measurable set. Proof. If we take the definition of the norm g L(p,) as g L(p,) p µ{x : g(x) > λ} /p dλ

7 CHRCTERIZTION OF L(p, ) 7 which is equivalent to the one in Definition., and use Hunt s inequality given in [27], which is T χ L(p,) p µ{x : T χ (x) λ} /p dλ [ µ() λ ( + log λ ) ] /p dλ + C ( µ()e Cλ ) /p dλ where C is a positive constant, we get T χ L(p,) Cµ() /p. Consequently, by using the new characterization of L(p, ), we get T f L(p,) C f L(p,). Note: This direct proof using Hunt s inequality was mentioned to the author by Loukas Grafakos during the 23 th Mini-Conference on Harmonic nalysis and Related reas held at uburn University on December 4-5, 29 after the talk given by the author on the subject. s a Corollary of Theorem 5., we have that Corollary 5.2. If f L(p, ), p > and S n (f, x) is the n-th partial sum of the Fourier series of f, then S n (f, x) f(x) almost everywhere. lso we note that L(p, ) L(p, ) with f L(p, ) C f L(p,). It follows by Theorem 5. that for p p, p, p > a) T χ L(p, ) M (µ()) /p b) T χ L(p, ) M (µ()) /p Therefore using the interpolation Theorem.4.9 in [25], we get (5.) T f L(p,r) M f L(p,r), for The inequality (5.) leads to the following corollary. p θ p + θ p, < θ <, r >. Corollary 5.3. If f L(p, r), p, r >, then S n (f, x) f(x) almost everywhere. Corollary 5.4 (Carleson s Theorem on Convergence of Fourier Series). If f L p, then S n (f, x) f(x) almost everywhere. Proof. Set p r in Corollary 5.3 since L(p, p) L p. cknowledgement: I would like to thank Eddy Kwessi and sh bebe for useful discussion and comments.

8 8 GERLDO SORES DE SOUZ References [] G. G. Lorentz, Some New Functional Spaces, nnals of Mathematics, vol5 N5(95), [2] G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math. (95),4-429 [3] G. S. De Souza, Spaces formed by special atoms, PhD dissertation, 98 SUNY at lbany. [4] G. S. De Souza, The atomic Decomposition od Besov-Bergman-Liptschitz Spaces, PMS, vol. 94, N 4 (985), [5] G. S. De Souza, The Bloch decomposition of weighted Besov Spaces, Colloquium Mathematicum, vol. LX/LXI(989), [6] S. Bloom, G. De Souza Weighted Lipschitz Spaces and analytic characterizations, constructive approximations, 994, #, [7] S. Bloom, G. De Souza(989) tomic deconmposition of generalized Lipschitz Spaces, Illinois Journal of Mathematics, Vol 33, #2, (989), [8] G. S. De Souza, Fourier Series and the Maximal Operator on the Weighted special tom Spaces, The international Journal of Mathematics and mathematical Sciences, Vol 2, #3, (989), [9] G. S. De Souza, R. O Neil and G. Sampson Several Characterizations for the special tom Spaces with pplication, Revista Mathematica iberoamericana, Vol 2, #3, (986), [] G. S. De Souza,G. Sampson Function in nthe Dirichlet Space such that its Fourier Series Diverges lmost Everywhere PMS, Vol 2, #3, (984). [] G. S. De Souza, Fourier Series and the Maximal Operator on the Weighted special tom Spaces, The international Journal of Mathematics and mathematical Sciences, Vol 2, #3, (989), [2] G. S. De Souza, Two More Characterizations of Besov-Bergman-Lipschitz Spaces, Real nalysis Exchange, Vol, #, ( ), [3] G. S. De Souza, Spaces Formed by Special toms I, The Rocky Mountain Journal of Mathematics, Vol 4, #2, (984), [4] G. S. De Souza, On The Boundary Value of Besov-Bergman-Lipschitz Spaces, Real nalysis Exchange, Vol, #, ( ), [5] G. S. De Souza, On the Convergence of Fourier Series,The international Journal of Mathematics and mathematical Sciences, Vol7, #4, (984), [6] G. S. De Souza, G. Sampson Real Characterization of the Predual of Bloch Functions, The Journal of London Mathematical Society, Vol 2, #27, (983), [7] G. S. De Souza, Two Theorems on Integration of Operators, Journal of Functional nalysis, Vol 46, (982), [8] G. S. De Souza, The Dyadic Special tom Spaces, Proceedings of Conference on Harmonic nalysis, Lectures Notes in Mathematics, #99, Springer-Verlag,(982), [9] O. Blasco, G. S. De SouzaSpaces of nalytic Functions on the Disc Where the Growth of M p(f, t) depends on a Weigth, Journal of Mathematical nalysis and pplications, Lectures Notes in Mathematics,vol. 47 #2,(99), [2] G. S. De Souza, Functions Spaces and Duality, Proceedingsof International Symposium on pproximation Theory, Benemarita Universidad utonoma de Puebla, Pueblo-Mexico, March 26-pril, 2. [2] G. S. De Souza, Spaces formed by Special toms, Origin, Motivation, Generalizations and pplications, Paper Presented at The Internatinal Conference on Extremal problems in Complex and Real nalysis In People Friendship University of Russia, Moscow-Russia, May 22-26, 27. [22] G. S. De Souza, Maximal Operator and Fourier Series, Proceedings of the 7 th Brazilian Colloqium of Mathematics, (99), IMP-Brazil, [23] G. S. De Souza, The Spaces formes by weighted special atoms and their characterization, Paper presented at the Internatinal Cinference on Extremal problems in Complex and Real nalysis, People Friendship University of Russia-Moscow, Russia, May 22-24, 27. [24] G. S. De Souza, Miguel J. Pozo Immersions of L p spaces in Lipschitz subspaces of continuous functions and duality Theorem, Paper presented at the 22 nd uburn Mini-conference on Harmonic nalysis and related area, November 2-22, 28. [25] L. Grafakos, Classical and Modern Fourier nalysis, Pearson Prentice Hall, US 24. [26] E. M. Stein, G. Weiss, n extension of Theorem of Marcinkiewicz and some of its applications, J. Math. Mech. (959)

9 CHRCTERIZTION OF L(p, ) 9 [27] R.. Hunt On the convergence of Fourier Series, Proc. Conference Southern Illinois University, Edwarville, Ill. 967 uburn University, Department of Mathematics and Statistics, uburn L US address: desougs@auburn.edu