On pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa)

Transcription

1 On pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa) Abstract. We prove two pointwise estimates relating some classical maximal and singular integral operators. In particular, these estimates imply well-known rearrangements inequalities, L p ω and BLO-norm inequalities. Introduction. For a locally integrable function f on R n, define the Hardy-Littlewood and Fefferman-Stein maximal functions by 1 Mf(x) = sup f(y) dy, x f # (x) = sup x 1 f(y) f dy, where f = 1 f, the supremum is taken over all cubes containing x, and denotes the Lebesgue measure of. We also define the Calderón-Zygmund maximal singular integral operator by T f(x) = sup ε>0 x y >ε f(y)k(x y) dy, where the kernel k(x) satisfies the standard conditions: (1) k(x) c x n, R 1 < x <R 2 k(x) dx = 0 (0 < R 1 < R 2 < ), k(x) k(x y) c y α x n+α ( y x /2, α > 0). Let ω be a non-negative, locally integrable function. Given a measurable set E, let ω(e) = E ω(x)dx. We say that ω satisfies Muckenhoupt s condition A if there exist c, δ > 0 so that for any and E, ω(e) c( E / ) δ ω() Mathematics subject classification: Primary 42B20, 42B25. 1

2 For ω A, it is well known (see [7, 9, 10]) that (2) T f p,ω c Mf p,ω, (3) Mf p,ω c f # p,ω for all p > 0, where f p,ω ( R n f(x) p ω(x)dx) 1/p. BMO estimates for T f go back to [15, 16]: (4) T f c f. For the Hardy-Littlewood maximal function a BMO estimate was established later in [4]: (5) Mf c f. These estimates were strengthened in [13] and [3] respectively: (6) T f BLO c f, (7) Mf BLO c f. The space BLO [8] consists of all functions f L 1 loc(r n ) such that f BLO = sup(f inf f) <. It is easy to see that BLO BMO, moreover f 2 f BLO. Note that the estimates (2), (3) were proved in [7, 9, 10] with the help of so-called good λ inequalities. Afterwards, rearrangements inequalities for Mf, f #, T f were obtained (see [1, 2, 5]), which also imply (2), (3). The non-increasing rearrangement of f with respect to ω [6, p.32] is defined by fω(t) = sup inf f(x) (0 < t < ). x E ω(e)=t If ω 1 we use the notation f (t). A key role in our work is played by the maximal function (see [11, 19]) m λ f(x) = sup(fχ ) (λ ) (0 < λ < 1). x In terms of this function we establish pointwise estimates for the operators Mf, f #, T f. In particular, these estimates imply all the above mentioned results, namely rearrangements inequalities, L p ω and BLO-norm estimates (2) (7). Our main results are the following. 2

3 Theorem 1. For any function f L p (R n ) (1 p < ) and for all x R n, m λ (T f)(x) c λ,n Mf(x) + T f(x) (0 < λ < 1). Theorem 2. For any function f L 1 loc(r n ) and for all x R n, m λ (Mf)(x) c λ,n f # (x) + Mf(x) (0 < λ < 1). Inequalities (2) (7) follow from these theorems in view of the next main lemma. Lemma 1. Let f and g be non-negative functions on R n. Suppose that for any λ, 0 < λ 1/2, there exists a constant c λ > 0 so that for all x R n, and let ω A. Then m λ f(x) c λ g(x) + f(x) (i) there exists a constant c > 0 so that for all t > 0; (ii) if f ω(+ ) = 0, then f ω(t) c g ω(2t) + f ω(2t) f L p ω c p g L p ω (0 < p < ); (iii) if g L, then f BLO c g. The proof of (iii) is essentially based on the inequality (8) f c n sup inf ((f c)χ c ) (λ ) which was proved by F. John [12] and J.-O. Strömberg [19] in the cases 0 < λ < 1/2 and λ = 1/2 respectively. For λ > 1/2 this inequality fails. First, we prove Theorems 1,2, and then Lemma 1. Proof of Theorem 1. Here it is convenient to use the maximal function m λ f(x) = sup(fχ B ) (λ B ), B x 3

4 where the supremum is taken over all balls B centered at x. It is easy to see that for any cube containing x there is a ball B centered at x which contains such that B = c n. From this property, for any x R n we have (9) m λ f(x) m λ/cn f(x). By (9), it suffices to get the required estimate for m λ. Let B be an arbitrary ball with center at x. From the definition T it follows that (10) T (fχ R n \2B )(x) T f(x). Further, by (1), the standard arguments (see, for example, [18, p.59]) show that for all y B, (11) T (fχ R n \2B )(y) cmf(x) + T (fχ R n \2B )(x). On the other hand, by weak type (1, 1) of T [17, p. 42] we have (T (fχ 2B )) (λ B ) c f(y) dy cmf(x). B 2B From this and (10), (11) we get ((T f)χ B ) (λ B ) cmf(x) + T f(x). Taking the upper bound over all balls B centered at x proves the theorem. Proof of Theorem 2. We shall use the following elementary property of cubes: if cubes 1 and 2 intersect then either or (as usual, k denotes the cube concentric with and having edge lenght k times as large). Let be an arbitrary cube containing the point x. Take an arbitrary point y and suppose a cube contains y. If 3, then f f f 3 + f 3 M((f f 3 )χ 3 )(y) + Mf(x). Assume now that 3. Then 3 and in this case Thus, for all y, f f f 3 + f 3 3 n f # (x) + Mf(x). Mf(y) = max sup y 3 4 f, sup y 3 f

5 M((f f 3 )χ )(y) + 3 n f # (x) + Mf(x). Using the weak type (1, 1) of the operator M, we get ((Mf)χ ) (λ ) (M((f f 3 )χ 3 )) (λ ) + 3 n f # (x) + Mf(x) c f f n f # (x) + Mf(x) cf # (x) + Mf(x). 3 Taking the upper bound over all x yields the theorem. Proof of Lemma 1. Choose λ so that c(2 n λ) δ = 1/4, where c, δ are the constants from the definition of A, and put c = c λ. Let E be an arbitrary set with ω(e) = t. Applying the Calderón- Zygmund decomposition to the function χ E and number λ, we get pairwise disjoint cubes i such that (12) λ i < E i 2 n λ i. From the definition A it follows that ( ) δ E i ω( i ) c(2 n λ) δ ω( i i ). ω(e) = i ω(e i ) c i i So, we have ω( i i ) 4t. From this and the left-hand inequality of (12) we obtain inf f(x) inf inf f(x) inf (fχ x E i x E i i i ) (λ i ) inf i inf m λ f(x) = x i inf m λ f(x) (m λ f) x i i ω(4t). Taking the supremum over all sets E with ω(e) = t, we get f ω(t) (m λ f) ω(4t). From this and simple properties of rearrangement it follows that f ω(t) (c g + f) ω(4t) c g ω(2t) + f ω(2t). So, we get (i). Iterating this inequality we obtain (ii) in a standard way (see, for example, [14]). It remains to prove part (iii). This follows immediately from the following BLO criterion. 5

6 Lemma 2. Let λ 1/2. Then a non-negative function f belongs to BLO iff m λ f f L. Moreover, f BLO m λ f f. Proof. Define A = m λ f f. It is clear that (13) (fχ ) (λ ) A + inf f for any cube. From this it follows that inf c ((f c)χ ) (λ ) ((f inf f)χ ) (λ ) = (fχ ) (λ ) inf f A. Since λ 1/2, by John and Strömberg s theorem (see (8)) it follows that f BMO and f ca. Further, note that for any cube, f inf x ( f(x) f + f(x) ) ( ( f f + f ) χ ) ( ) ( (f f )χ ) ( /2) + (fχ ) ( /2) 2 f + (fχ ) ( /2). From this and (13) we get Thus, f BLO = sup(f inf f) sup(2 f + (fχ ) ( /2) inf f) sup(2ca + (fχ ) (λ ) inf f) (2c + 1)A. Conversely, let f BLO. Then (fχ ) (λ ) ((f f )χ ) (λ ) + f 1 λ f + f BLO + inf f (2/λ + 1) f BLO + inf f. The lemma is proved. m λ f(x) (2/λ + 1) f BLO + f(x). Acknowledgement. This work was done during my stay at the Institute of Mathematics of Wroc law University in the Spring Semester of I would like to thank Professor A. Hulanicki for his hospitality. I am grateful to Professor V.I. Kolyada for useful discussions about the subject of this paper. 6

7 References [1] R.J. Bagby and D.S. Kurtz Covering lemmas and the sharp function, Proc. Amer. Math. Soc. 93 (1985), [2] R.J. Bagby and D.S. Kurtz, A rearranged good λ inequality, Trans. Amer. Math. Soc. 293 (1986), [3] C. Bennett, Another characterization of BLO, Proc. Amer. Math. Soc. 85 (1982), [4] C. Bennett, R. DeVore and R. Sharpley, Weak-L and BMO, Ann. of Math. 113 (1981), [5] C. Bennett and R. Sharpley, Weak-type inequalities for H p and BMO, Proc. Symp. Pure Math. 35 (1979), [6] K.M. Chong and N.M. Rice, Equimeasurable rearrangements of functions, ueen s Papers in Pure and Appl. Math. 28, ueen s University, Kingston, Ont., [7] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 15 (1974), [8] R.R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), [9] A. Córdoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math 57 (1976), [10] C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math. 129 (1972), [11] B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), [12] F. John, uasi-isometric mappings, Seminari di Analisi, Algebra, Geometria e Topologia, Rome, [13] M.A. Leckband, Structure results on the maximal Hilbert transform and two-weight norm inequalities, Indiana Univ. Math. J. 34 (1985),

8 [14] A.K. Lerner, On weighted estimates of non-increasing rearrangements East J. Approx. 4 (1998), [15] S. Spanne, Sur l interpolation entre les espaces L pφ k, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 20 (1966), [16] E.M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Proc. Symp. Pure. Math. 10 (1967), [17] E.M. Stein, Singular Integrals and differentiability properties of functions, Princeton Univ. Press, [18] E.M. Stein and G.Weiss Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, [19] J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), lerner@paco.net 8