TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS. 1. Introduction We denote the Fourier transform of a complex-valued function f(t) on R d by

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1 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS LOUKAS GRAFAKOS Abstract. In these lectures we discuss examples that are relevant to two questions in the theory of singular integrals. The first question is the L p boundedness of the maximal operator formed by dilates of Mihlin-Hörmander multipliers, while the second concerns the L p boundedness of a well-nown object, the classical L 2 -bounded Calderón-Zygmund homogeneous singular integral associated with an integrable function on the sphere that is very rough.. Introduction We denote the Fourier transform of a complex-valued function f(t) on R d by f(τ) = f(t)e 2πiτ t dt R d and its inverse Fourier transform by f (τ) = f( τ). Many linear operators can be expressed in terms of their action on the Fourier transform of the input function. In particular, convolution operators are identified by operators given by multiplication on the Fourier transform, i.e. operators of the form T m (f) = ( fm). Here we will always be interested in L 2 -bounded convolution operators for which the corresponding multiplying functions m (called the Fourier multipliers) must be essentially bounded functions. The Fourier multiplier associated in this way with an operator bounded on L p (R d ) is called an L p Fourier multiplier. The space of all L p Fourier multipliers on R d will be denoted by M p (R d ). This is a Banach space (in fact algebra) with norm m Mp = T m L p L p. The classical Mihlin multiplier theorem [3] states that if a function m(ξ) on R d satisfies (.) α ξ m(ξ) C α ξ α for all multiindices α with α [ d 2 ] +, then it must be an Lp Fourier multiplier for all < p <. This theorem was extended by Hörmander [2] to functions m satisfying the weaer condition (.2) sup ϕ(ξ)m(2 ξ) L 2 Z β (dξ) < for some β > d/2. Here ϕ is a smooth nonzero bump supported in the annulus < ξ < 2 not vanishing on a smaller annulus and L 2 β is the Sobolev space of functions with β Date: April 27, Mathematics Subject Classification. Primary 42B2, 42B25. Secondary 46B7, 47B38. Key words and phrases. Calderón-Zygmund singular integrals, Mihlin-Hörmander multiplier theorem, maximal multipliers. Wor supported by the NSF under grant DMS This article is an expansion of the lectures the author gave at the conference entitled Harmonic Analysis and its Applications at Sapporo during the period August 22-24, 25.

2 2 LOUKAS GRAFAKOS derivatives in L 2. The space L 2 β is one of the Sobolev spaces Lp γ with norm f L p γ = ( f(ξ)( + ξ 2 ) γ/2) L p (dξ), where p < and γ R. We remar that in Hörmader s version of this multiplier theorem the Sobolev space L 2 β in (.2) can be replaced by Lr γ, where γ > d/r and r 2; however the least restrictive condition is when r = 2. By duality an L p Fourier multiplier must always be an L p Fourier multiplier (where p = p/(p )) and hence by interpolation it must be an L q Fourier multiplier for all q between p and p. Finding examples of functions that are L q Fourier multipliers for some q > 2 but not L s Fourier multipliers for some s > q may not be an easy tas. A question of this sort will be addressed in section 5. In the next section we will discuss a problem concerning the L p boundedness of the supremum of a family of Mihlin-Hörmander Fourier multipliers. 2. Maximal Mihlin-Hörmander Fourier multipliers Suppose that we are given a bounded function on R d that satisfies condition (.) (or even (.2)). The question that we would lie to address is whether the maximal operator M m (f)(x) = sup ( f(ξ)m(tξ)) (x) t> is bounded from L p (R d ) into itself. This question is motivated by the almost everywhere convergence questions ( f(ξ)m(tξ)) (x) m() f(x) for almost all x as t ( f(ξ)m(tξ)) (x) m( ) f(x) for almost all x as t, provided, of course, the quantities m() and m( ) exist. We recall that in the usual proof of the Mihlin-Hörmander multiplier theorem one obtains a wea type (, ) estimate using the trivial L 2 estimate and a smoothing condition on the ernel. Then the boundedness for the remaining p s follows by interpolation and duality. By changing our point of view, we may consider M m as a linear map from (2.3) L p (R d ) L p (R d, L (R + ) and we may as whether the classical scalar argument argument based on the wea type (, ) estimate holds in this setting. In the context of the vector-valued setting described in (2.3) the corresponding multiplier satisfies Mihlin s condition but for the wea type (, ) argument to go through one needs to now an initial estimate at a single exponent. In the scalar case, one uses Plancherel s theorem to obtain the L 2 estimate for free but in the vector-valued case the L 2 (R d, L (R + )) estimate cannot be obtained using Plancherel s theorem, in fact as we will shortly see, it may fail. The underlying problem here is that the Banach space L is not a UMD space and for this reason many analogues of some of the scalar results in the theory of singular integrals do not hold in the Banach-valued setting. Theorem. (M. Christ, L. Grafaos, P. Honzí, and A. Seeger [4]) There exists a bounded function m such that for all multiindices α there are constants C α such that sup ξ sup ξ α (ϕ(ξ)m(2 ξ)) C α,

3 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS 3 hence m is an L p Fourier multiplier for all < p <, but M m is unbounded on L p (R d ). We discuss some of the ideas of the proof of this result. Proof. Let S = {,, i, i}. Enumerate the set of all sequences of length N formed by elements of S as follows: S N = {s, s 2,..., s 4 N }. Let Φ be a smooth function supported in 6 8 ξ 8 satisfying Φ = on 7 8 ξ 9 8. Define for N 4 N m N (ξ) = s l (ν)φ(2 Nl 2 ν ξ) and also m(ξ) = l= ν= m N (2 N8N ξ). It is straightforward that for all multiindices α there are constants C α so that N= α m(ξ) C α ξ α and it is also easy to chec that for any Z we have α (m(2 ξ)ϕ(ξ)) C α. Pic ψ with Fourier transform supported in B(, /8) with ψ L p =. Let g N (x) = e 2πi2j x ψ(x) and note that ĝ N (ξ) = j= ψ(ξ 2 j (,,,..., )). j= Also let f N,p (x) = N 2 (2 N8 N ) d p g N (2 N8N x) and notice that in view of the Littlewood-Paley Theorem (see [8]) we have that g N L p N /2 while f N,p L p C p. The main ingredient we need is the following lower estimate whose proof we postpone momentarily: (2.4) sup (m N (2 ξ)ĝ N (ξ)) L N4 N p c N. This implies that M m (f N,p ) L p c N = c N f N,p L p. The reason for this is that m(ξ) = n= m n(2 n8n ξ) and m n (2 n8n 2 ξ) f N,p (ξ) = m n (2 n8n 2 ξ)ĝ N (2 N8N ξ) = for all N4 N unless n N. It remains to prove (2.4). We observe that sup Re(c z) z / 2. c {,,,i, i}

4 4 LOUKAS GRAFAKOS Thus for all x R d and all j {, 2,..., N} there is a c j (x) {,,, i, i} such that Re [ c j (x) e 2πi2j x ψ(x) ] ψ(x) / 2. Therefore there is a κ x {, 2,..., 4 N } such that We then have R d l= ν= s κx = (c (x), c 2 (x),..., c N (x)). sup (m N (2 ξ)ĝ N (ξ)) (x) N4 N [ 4 N Re s l (ν)φ(2 Nl ν 2 Nκx ξ) j= ] ψ(ξ 2 j e )e 2πix ξ dξ, as easily follows by taing = Nκ x. Our choice of exponents maes the previous expression inside the the square bracets zero unless l = κ x and j = ν. Also Φ = on support( ψ) and hence this expression is at least Re [ s x(j)( ψ(ξ 2 j )) (x) ] N ψ(x) / 2 which proves (2.4). j= 3. A positive result related to the previous counterexample We recall the main observation in the previous section which can be rephrased as follows: (3.5) sup (m N (2 ξ)ĝ N (ξ)) L N4 N p c N g N L p. Replacing N4 N by N we see that the supremum of a family of N Mihlin-Hörmander multipliers has operator norm on L p at least as big as a constant multiple of (log N ) 2. The question we would lie to address is whether this lower estimate is sharp. We precisely formulate our question. Question: Suppose that m j, j N, are Mihlin multipliers satisfying α m j (ξ) C α ξ α, uniformly in j for all α [ d 2 ] +. What is the growth as N of the smallest constant A(N) such that sup (m j f ) L A(N) f j N p (R d L ) p (R d ) holds for all f? The counterexample in the previous section shows that for N we have A(N) c log N and we would lie to now if the converse inequality also holds for some other constant c. The following theorem answers this question.

5 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS 5 Theorem 2. (L. Grafaos, P. Honzi, and A. Seeger [9]) Let < r < 2 and suppose sup α m j (ξ) ξ α B j N for all α [ d 2 ] +. Then for any < p < there is a constant C d,p such that for all N we have sup (m j f ) L C d,pb log N f L j N p (R d ) p (R d ). Therefore for N we have that A(N) c log N and this shows that A(N) grows indeed lie the square root of the logarithm of N as N. We will outline a proof of this theorem in the next section, but before we do so, it will be illuminating to discuss a model case that contains the core idea and forms the basic outline of the proof in the general case. The model case comes from the theory of Rademacher multipliers. Let us recall the Rademacher functions defined on the interval [, ] as follows: r (t) = r (t) = χ [,/2] χ [/2,] r 2 (t) = χ [,/4] χ [/4,/2] + χ [/2,3/4] χ [3/4,] etc. The inspiration comes by studying the growth in N of the L p norms of simple-looing maximal functions of the form sup a j r j, N j where a j is a fixed matrix and r j is the j-th Rademacher function. Let us denote the sequence (a j ) j by a. It turns out that (3.6) sup i N j a j r j L 2 ([,]) C(N) sup N a l 2 ([,]) where C(N) grows lie log N as N. To see this we set F = j a j r j. One has the following exponential decay of sums of Rademacher functions (see [5], [8]) (3.7) {s [, ] : F (s) > λ} 2 e λ 2 /4 a 2 l 2. Then for N we have sup F 2 = λ {s [, ] : sup F (s) > λ} dλ L 2 ([,]) N = un un... dλ + u N λ dλ + u N... dλ λ {s [, ] : F (s) > λ} dλ. =

6 6 LOUKAS GRAFAKOS We now use (3.7) and calculate the integrals in question. We obtain sup N using the optimal choice of This proves (3.6). F 2 L 2 ([,]) 2 u2 N + = u N 2 λ e 2 u2 N + 2N e u2 N /4 c log N u N = 4 log N λ2 /4 a 2 l 2 dλ sup N sup a 2 l, 2 N sup a l 2. N 4. the general case a 2 l 2 We now adapt the idea of the previous section to prove Theorem 2. Proof. Let D be the dyadic cubes in R d of sidelength 2. We recall the dyadic averaging operator E, the martingale difference operator E, and the martingale square function S(f) associated with the family of dyadic cubes: E (f) = χ Q f(t) dt, Q Q D Q D (f) = E + (f) E (f), ( S(f) = D (f) 2) /2, The ey element in the proof is the Chang-Wilson-Wolff inequality [3] : {x R d : sup E (g) E (g) > 2λ, S(g) < ελ} C d e c d ε 2 {x R d : sup E (g) > λ} which is valid for all functions g, all λ >, ε (, ), and for some fixed constants C d, c d (both depending on d). Recall that we denote by T m the operator f ( fm). Start with sup N ( T m (f) = p 4 p L p ) λ p p {sup T m (f) > 4λ} dλ and control the measure of the set that appears in the previous line by the sum of three terms: {sup T m (f) > 4λ} I λ + II λ + III λ, where I λ = {sup T m (f) E (T m (f)) > 2λ, G p (f) < ε N λ/(ab)}, II λ = {G p (f) > ε N λ/(ab)}, III λ = {sup E (T m (f)) > 2λ}.

7 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS 7 Here A is a constant and f G p (f) is an L p bounded maximal operator which controls the square function applied to each T m, precisely it satisfies: ( ) S(T m (f)) A sup sup ξ α ξ α m (ξ) G p (f) A B G p (f) ξ α [ d 2 ]+ for all Schwartz functions f. (For a precise definition of G p, see [9].) To estimate ) (4.8) (p 4 p λ p p I λ dλ we use the Chang Wilson Wolff theorem to write I λ {sup T m (f) E (T m (f)) > 2λ, G p (f) < ε N λ/(ab)} { T m (f) E (T m (f)) > 2λ, G p (f) < ε N λ/(ab)} = { T m (f) E (T m (f)) > 2λ, S(T m (f)) < ε N λ} = = Insert this estimate in (4.8) to obtain (p 4 p ) λ p p I λ dλ C d e c d/ε 2 N {sup E l (T m (f)) > λ}. l C d ( N = ) e c d/ε 2 N sup E l (T m (f)) p p L c B [ Ne c ] d/ε 2 p N p l provided we choose ε N = c / log N. Here we used that the maximal operator g sup E l (g) l is controlled by the Hardy-Littlewood maximal function and is therefore L p bounded for all < p <, while all T m are L p bounded with norm at most a multiple of B. Next we turn our attention to the corresponding integral for term II λ ) (p 4 p λ p p II λ dλ. Using that II λ = {G p (f) > (ε N /AB)λ}, where G p is L p bounded, we deduce that ) (p 4 p λ p p II λ dλ C B A r B G p (f) L p C p f L p. ε N ε N This last expression is equal to C rb log N f L p since ε N was chosen to be c / log N. Finally we need to control (4.9) (p 4 p ) λ p p III λ dλ. B

8 8 LOUKAS GRAFAKOS It turns out that for any < r < p one has an estimate (see [9]) (4.) E (T m (f)) C r B 2 N r (MMM( f r )) r whenever m (ξ) = on ξ 2 N (M is the Hardy-Littlewood maximal operator). This assumption can be made on each multiplier m, =,..., N as follows: woring with f such that f is compactly supported, we may assume that the multipliers m are supported in a finite union of dyadic annuli, which, by changing scales, may assume that do not intersect the ball ξ 2 N. Insert estimate (4.) in (4.9) to obtain (p 4 p ) λ p p III λ dλ ( N E (T m (f)) p ) p L p = which is trivially controlled by B log N f L p. B N p 2 N r f L p 5. A problem involving homogeneous singular integrals Mihlin-Hörmander multipliers correspond to ernels K(y) on R d that are singular at the origin, satisfy an estimate K(y) C y d for some C < and all y, and possess a certain amount of smoothness. This smoothness suffices to guarantee the boundedness on all L p (R d ) ( < p < ) for the corresponding Fourier multiplier operator (given by convolution with K) as well as its wea type (, ) property. In this section, we study a problem concerning Fourier multipliers given by convolution with ernels that are homogeneous of degree d on R d. Such ernels are determined by their restriction on the unit sphere S d. Let K be such a ernel and let Ω be its restriction on S d. One may chec that the function Ω(y/ y ) y d, y coincides with a principal value distribution on R d if and only if Ω has mean value zero on the sphere. Only in this case one can mae sense of convolution with K. Let therefore Ω be an integrable function on S d with mean value zero. We will be considering Calderón-Zygmund singular integrals of the form (5.) T Ω (f)(x) = f p.v. Ω(x/ x ) x d = lim ε y >ε f(x y) Ω(y/ y ) y d dy, where f is a Schwartz function on R d. This type of singular integrals were introduced by Calderón and Zygmund in []. If Ω is odd then the method of rotations (see [2]) gives (5.2) T Ω (f)(x) = π H θ (f)(x) Ω(θ) dθ, 2 S d where H θ is the directional Hilbert transform H θ (f)(x) = lim ε π t >ε f(x tθ) dt t. A simple argument using change of variables yields that the operator H θ is bounded on L p exactly when H (,,...,) is; the latter is the Hilbert transform in the first variable and the identity operator in the remaining variables and hence it is trivially bounded on L p (R d ) (and is of wea type (, ).)

9 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS 9 Thus the boundedness of T Ω on L p (R d ) for Ω odd is an easy consequence of (5.2) and of the boundedness of H θ on L p (R d ) (which is uniform in θ). We point out that, as of this writing, the wea type (, ) boundedness of T Ω for Ω odd, remains an open question. The problem of the L p boundedness of T Ω is therefore interesting for Ω even. We begin our discussion by recalling the results of Calderón and Zygmund [2] who showed that if Ω lies in the space L log L(S d ), then T Ω is bounded on L p (R d ) for all < p <. The more delicate issue of the wea type (, ) property of T Ω was shown much later by Christ and Rubio de Francia [5] (for d 7, published only the case d = 2) and Seeger [4] for all d. We note that M. Weiss and A. Zygmund [6] have constructed examples of even functions Ω in L (S d ) such that T Ω is unbounded on L 2 (even when restricted to continuous functions) and therefore on all other L p. For an operator to be bounded on L 2 (R d ) a certain condition on Ω is required. A calculation using the Fourier transform gives that the multiplier corresponding to the ernel Ω(y/ y ) y d is the function ξ Ω(θ) log S d ξ θ dθ. Therefore we have the equivalence Ω(θ) log S d ξ θ dθ < + T Ω : L 2 L 2 essup ξ = and hence condition (5.3) essup ξ = S d Ω(θ) log dθ < + ξ θ implies the L 2 boundedness of T Ω. Since condition (5.3) arises naturally, it is reasonable to as whether it implies the boundedness of T Ω on L p for some (or all) p 2, The underlying question here is whether the p-independence boundedness property in Calderón-Zygmund theory holds for rough ernels. This question was answered in the negative by P. Honzí and D. Ryabogin, in collaboration with the author, who constructed an example of an even function Ω on S d such that the corresponding operator T Ω is bounded on L p exactly when p = 2. In fact these authors have obtained the following sharper result: Theorem 3. (L. Grafaos, P. Honzí, D. Ryabogin []): Let α < 2. Then there exists Ω L (S d ) with mean value zero such that essup ξ = S d Ω(θ) log +α ξ θ but T Ω is unbounded on L p (R d ) for all p > α. 2 Taing α = yields the previous case. dθ < + 6. The second counterexample In this section we discuss the counterexample of Theorem 3. In the proof we restrict our attention to the case d = 2 and we note that higher dimensional examples can be constructed using the two-dimensional example.

10 LOUKAS GRAFAKOS Let M p (Z) be the space of multipliers on L p ([, ]), i.e. the space of all bounded sequences (b m ) m Z such that the linear operator (6.4) h(x) ( ) b m h(t)e 2πimt dt e 2πimx m Z maps -periodic functions h in L p ([, ]) to functions in L p ([, ]). The M p (Z) norm of the sequence (b m ) m is then the norm of the operator in (6.4) on L p ([, ]) x xn I I n length of arcs - = (n).. A... A n length of arcs A j = L n Figure. The points x,..., x n lie on a straight line perpendicular to the vertical coordinate axis. The arcs A j lie in the first quadrant of the unit circle and have length L n, a quantity to be determined. The cones I j lie in the second quadrant and they meet the unit circle on arcs of length (n) centered at the points x j / x j. The centers of the arcs A j and the points x j / x j form an angle of size π/2. The basis {e 2πix } = of Lp ([, ]), p 2, is not unconditional. This means that for all n =, 2,... there exist complex sequences a n and εn such that n (6.5) ε n L an e2πix c p p n 2 n p a n L [,] e2πix, p [,] = for some constant c p. To see this we consider the sequence of a n = for all for which the L p norm if calculated explicitly and gives n /p and a random sequence of ± s, which by the Khintchine s inequality gives the constant n. Rephrased in the language of multipliers, estimate (6.5) is saying that for some constant c p we have = (...,,...,, ε n, ε n 2,..., ε n n,,...) Mp(Z) c pn 2 p.

11 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS We may choose the sequence {ε n }n = to be maximal in the sense that its M p norm is the supremum of the M p norms of all other sequences of l norm that satisfy (6.5) for the choice of a n. We now define m(ω)(ξ) = ω(θ) log ξ θ dθ S d and we also define a similar quantity m α (ω)(ξ) = ω(θ) log +α S d ξ θ dθ, while for each integer n we define a even function n 3 Ω n = ε n C(n) ( ) j χ A rotated by jπ 2, = } j= {{ } ω n where A are the arcs of Figure. Here C(n) is a constant chosen so that m α (ω n )(x / x ) = /2 for all. Finally we denote by D(n) the constant It is not difficult to chec that D(n) = m(ω n )(x / x ). C(n) L n log L n α D(n) log L n α. while for all x / 3 j= (I rotated by jπ 2 ) S we have m(ω n )(x) (log n) log L n α m α (ω n )(x) (log n)+α log L n α. It follows from these estimates that (6.6) Ω L n (S ) n (log n) log L n. On the other hand we have where o n provided m(ω n )(x ) = D(n) ε n + is an error term which satisfies i n o n D(n)/4 n 4n L n. ε n i m(ω ɛn i )(x ) = D(n)ε n + on, We now tae a loo at certain multiplier norms. Using a deleeuw type argument [6] we can restrict the M p (R 2 ) norm of a multiplier to its values at the points (x ) which lie on a line parallel to the horizontal coordinate axis and we can thus estimate from below the

12 2 LOUKAS GRAFAKOS M p (R 2 ) norm of the multiplier by the M p (Z) norm of the sequence of the first coordinates of its values at the points (x ). This way we obtain m(ω n ) Mp(R 2 ) c p (..,, m(ω n )(x ),..., m(ω n )(x n ),,...) Mp(Z) [ c p D(n) (...,, ε n,..., ε n n,,...) Mp(Z) ] D(n) (...,, on,..., o n n,,...) Mp(Z) 2 c p D(n) (...,, ε n,..., ε n n,,...) M p (Z), since the inequality (...,, o n,..., o n D(n) n,,...) > (...,, ε n,..., ε n Mp(Z) 2 n,,...) Mp(Z) would contradict the maximal choice of (ε n )n =. We now recall that (...,, ε n,..., ε n n,,...) Mp(Z) c p n 2 p, which implies that m(ω n ) Mp(R 2 ) c D(n) n 2 p log L n α n 2 p. We finally choose the L n s. We had the restriction n 4n L n while the need to mae the expression on the right in equation (6.6) equal to a constant forces us to choose log L n n log n. With this choice of L n and all the facts we have accumulated so far we have T Ωn L p L p = m(ω n) Mp(R 2 ) c log L n α n 2 p (log n) α n 2 p α. We have now constructed a sequence of even integrable functions Ω n with L norm at most a constant such that T Ωn L p L p when 2 p > α. To complete the proof we need some functional analysis. Let B α the Banach space of all even integrable functions Ω on S with mean value zero with norm Consider the family of linear maps Ω Bα Ω L (S ) + m α (Ω) L (S ) <. Ω T Ω (f) : B α L p (R 2 ) indexed by the set U = {f L p (R 2 ) : f L p = }. If no claimed Ω existed, then for all Ω B α we would have sup T Ω (f) L p C(Ω) <. f U

13 TWO COUNTEREXAMPLES IN THE THEORY OF SINGULAR INTEGRALS 3 The uniform boundedness principle implies the existence of a constant K < such that T Ω L p L p = sup T Ω (f) L p K Ω Bα f U for all Ω B α. But this contradicts the construction of Ω n s for 2 p > α. 7. conditions that distinguish between p s Theorem 3 suggests that one should loo for conditions on Ω that distinguish boundedness on L p (R d ) for different values of p s. A natural condition that one should introduce in the study of this problem is the following: CL(α) essup Ω(θ) log +α dθ < + ξ = S d ξ θ A result of Stefanov and the author [] says that CL(α) implies the L p boundedness of T Ω whenever p 2 < α 2(2 + α). This restriction was weaened by Fan, Guo, Pan [7] to p 2 < α 2( + α). Since α > α/(2(α + )), it remains an open question to find out if L p boundedness holds for values of p in between. We pose therefore the following question: (a) Assume that CL(α) holds for some α < /2. Does it follow that whenever T Ω : L p L p α p 2 α 2(α + ) For α /2 the counterexample does not wor and one may guess that in this case T Ω is bounded on L p for the whole range (b) Assume that CL(α) holds for some α /2. Does it follow that T Ω : L p L p for all < p <? The author would lie to than the organizers of the meeting in Sapporo for their hospitality and for the inspiring mathematical atmosphere they created during the conference. References [] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (952), [2] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (956), [3] S. Y. A. Chang, M. Wilson, and T. Wolff. Some weighted norm inequalities concerning the Schrödinger operator, Comment. Math. Helv. 6 (985), [4] M. Christ, L. Grafaos, P. Honzí, and A. Seeger, Maximal functions associated with multipliers of Mihlin-Hörmander type, Math. Zeit. 249 (25), [5] M. Christ and J.-L. Rubio de Francia, Wea type (, ) bounds for rough operators II, Invent. Math. 93 (988), [6] K. deleeuw, On L p multipliers, Ann. Math, 8 (964), ?

14 4 LOUKAS GRAFAKOS [7] D. Fan, K. Guo, and Y. Pan, A note of a rough singular integral operator, Math. Ineq. and Appl., 2 (999), [8] L. Grafaos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ 24. [9] L. Grafaos, P. Honzi, and A. Seeger, On Maximal functions for Mihlin-Hörmander multipliers, Advances in Math., to appear. [] L. Grafaos, P. Honzi, and D. Ryabogin, On the p-independence boundedness property of Calderón- Zygmund theory, J. für die Reine und Angew. Math., to appear. [] L. Grafaos and A. Stefanov, L p bounds for singular integrals and maximal singular integrals, Indiana Univ. Math. J., 47 (998), [2] L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 4 (96), [3] S. G. Mihlin, On the multipliers of Fourier integrals, (Russian) Dol. Aad. Nau SSSR (N.S.) 9 (956), [4] A. Seeger, Singular integral operators with rough convolution ernels, Jour. Amer. Math. Soc., 9 (996), [5] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 97. [6] M. Weiss and A. Zygmund, An example in the theory of singular integrals, Studia Math. 26 (965),. Department of Mathematics, University of Missouri, Columbia, MO 652, USA address: louas@math.missouri.edu