RESULTS ON FOURIER MULTIPLIERS

Transcription

1 RESULTS ON FOURIER MULTIPLIERS ERIC THOMA Abstract. The problem of giving necessary and sufficient conditions for Fourier multipliers to be bounded on L p spaces does not have a satisfactory answer for general p. In this paper, some of the facts that are known about Fourier multipliers are detailed. The starting point is a quick tour of singular integral theory, leading into the Mikhlin multiplier theorem. An important application to Littlewood-Paley theory and a proof C. Fefferman s ball multiplier theorem is presented. Contents. Introduction. Calderón-Zygmund Operators 3. The Mikhlin Multiplier Theorem 6 4. Littlewood-Paley Theory 9 5. The Ball Multiplier Theorem Acknowledgments 6 References 6. Introduction A Fourier multiplier is a function m from R d to C that defines an operator T m through multiplication on a function s frequency spectrum. In particular, the operator T m is defined on the Schwartz space S(R d ) by T m f = (m ˆf), where ˆf(ξ) = (f) (ξ) = f(x)e πiξ x dx R d and f(x) = (f) (x) = f(ξ)e πix ξ dξ R d are the Fourier transform and inverse Fourier transform of the function f, respectively. It is natural to seek conditions on m under which T m can be extended to a bounded operator on L p. For general p, there is no known characterization of such m, but the Mikhlin multiplier theorem provides convenient sufficient conditions. The majority of this paper is dedicated to proving this theorem and an important application of it. The ball multiplier theorem, which says that T m is unbounded for p and d > when m is the characteristic function of the unit ball, is presented at the end to show the difficulty of a general theory. Theorem.8 will use the Date: 7 September 5.

2 ERIC THOMA fundamental Calderón-Zygmund decomposition of a function, which can be found in [, p. 44]. Most of the proofs in this paper are influenced by those in [] and [].. Calderón-Zygmund Operators The goal of this section is to prove the boundedness of a class of operators on L p for < p <. This result will be used to prove an important theorem on Fourier multipliers. Definition.. A Calderón-Zygmund kernel is a function K : R d \ {} C that satisfies (K) K(x) B x d for all x R d (K) [ x > y ] K(x) K(x y) dx B for all y (K3) K(x) dx = for all r, s > [r< x <s] for some constant B. The Calderón-Zygmund operator T associated with the kernel K is defined by (.) T f(x) = lim K(x y)f(y) dy ε [ x y >ε] for all f S(R d ). Note that the limit in (.) exists for all x. Indeed, using (K3) followed by (K) we have K(x y)f(y) dy [> x y >ɛ] = K(y)(f(x y) f(x)) dy [> y >ɛ] ( B sup ) f(y) dy [> y >ɛ] y d, and the part of the integral with x y > is easily controlled using (K) and Schwartz bounds on f. Note how the limit in (.) and the smoothness of f allows us to use the cancellation property in (K3). The following result allows one to check that (K) holds for a kernel given a bound on the derivative. Theorem.3. Let K : R d \ {} C satisfy K(x) B x d for all x and some B >. Then (.4) K(x) K(x y) dx CB, [ x > y ] for all y and C = C(d) a constant depending only on d, and therefore K satisfies (K). Proof. Fix x, y R d with x > y. Define γ : [, ] R d by γ(t) = ( t)(x y)+tx. Then the image of γ is the line segment connecting x y and x and is contained in B(x, x /). We have K(x) K(x y) = K dγ = ( K)(x ( t)y) y dt (( K)(x ( t)y) y dt B d+ x d y.

3 RESULTS ON FOURIER MULTIPLIERS 3 Using this estimate and on (.4) finishes the proof. [ x > y ] dx C(d) d+ x x C(d) y The Hilbert transform is given by a Calderón-Zygmund operator with kernel /x. More generally, any kernel of the form Ω(x/ x ) x d, where Ω is a smooth function on the unit sphere with mean value, defines a Calderón-Zygmund operator. Conditions (K)-(K3) are sufficient to prove that T can be extended to a bounded operator on L. Together with a weak-l bound, Marcinkiewicz interpolation, and a duality argument, this shows that T is in fact bounded on L p for < p <, failing at the endpoints. Since the weak-l bounds do not require (K3), it is often desirable to consider kernels for which (K3) does not necessarily hold, instead taking as hypothesis boundedness on L. Then L p boundedness holds for this larger class of operators. The failure of boundedness at p = can be seen qualitatively by considering the action of T on the Dirac delta distribution δ. In particular, we have T δ / x d L (R d ). Formally, one approximates δ by an approximate identity and uses a limiting process. Theorem.5. The operator T associated with a kernel K as in Definition. can be extended to a bounded operator on L. Specifically T f L CB f L for all f S, where C = C(d) depends only on the dimension d. Proof. Let K be as in Definition.. We write χ [r< y <s] to represent the characteristic function of the annulus with inner and outer radii r and s. Define (T r,s f)(x) = K(y)χ [r< y <s] (y)f(x y) dy R d for < r < s <. Let m r,s be the Fourier transform of the restricted kernel y K(y)χ [r< y <s] (y). By Plancherel s theorem, we know T r,s f L = (Kχ [r< y <s] f) L = m r,s ˆf m L r,s L f L, and so it suffices to prove sup mr,sl <r<s CB; if this holds, then the pointwise identity T f(x) = lim r,s (T r,s f)(x) and Fatou s lemma finish the proof. To prove this L bound on the Fourier transform, we split the integral as follows (.6) mr,s (ξ) e πix ξ K(x) dx + e πix ξ K(x) dx. [r< x < ξ ] [ ξ < x <s] Using (K3) followed by (K) on the first integral, we have e πix ξ K(x) dx [r< x < ξ ] = (e πix ξ )K(x) dx [r< x < ξ ] π x ξ K(x) dx [ x < ξ ] π ξ B x d+ dx CB. [ x < ξ ]

4 4 ERIC THOMA The second integral bound uses condition (K) and cancellation properties of e πix ξ. We have F := K(x)e πix ξ dx = K(x)e πi(x+ξ/( ξ )) ξ dx [ ξ < x <s] [ ξ < x <s] ( ) = K x ξ ξ e πix ξ dx and therefore (.7) F = [ ξ < x <s] K(x) K [ ξ < x ξ/( ξ ) <s] ( x ) ξ ξ e πix ξ dx + R where R is defined appropriately. The first term in (.7) is controlled in absolute value via (K). The R represents the difference in the integration regions [ ξ < x < s] and [ ξ < x ξ/( ξ ) < s]. We can use (K) to prove R < CB. Specifically, the difference in integration regions can be split into four parts: ξ < x < s and x ξ/( ξ ) ξ ξ < x < s and s x ξ/( ξ ) ξ < x ξ/( ξ ) < s and x ξ ξ < x ξ/( ξ ) < s and s x. Noting that the first region is a subset of the region [ ξ / x ξ/( ξ ) ξ ], condition (K) easily shows the integral over the region to be less than CB. A similar argument works for the other three regions. This finishes the proof. We now prove that T has a weak-l bound via the Calderón-Zygmund decomposition. Theorem.8. Let T be a linear operator that is bounded on L (R d ) such that (T f)(x) = K(x y)f(y) dy R d for all f L with compact support and for all x supp(f), and such that K satisfies (K) with constant B. Then for all f S(R d ) we have the weak-l bound { x R d : } T f(x) > λ CB λ f L for all λ >, where C depends only on the dimension d. Proof. By dividing K by B, we can assume without loss of generality that B =. Fix f S(R d ) and λ >. Using the Calderón-Zygmund decomposition of a function with height λ, we can write f = g + b, where g is the good and b is the bad part of f. The decomposition gives g and b with the following properties

5 RESULTS ON FOURIER MULTIPLIERS 5 (see [, p. 44] for proof). There is a collection B of disjoint cubes for which b = B χ f. We have B < λ f L, and for any B we have λ < f d λ. Here is Lebesgue measure. Finally, we have g λ. Define f = g + χ f(x) dx B f = b χ f(x) dx = B B where f = χ ( f f(x) dx Clearly f = f + f. From g < λ, f d λ, and the cubes and supp(g) are disjoint, we conclude f L d λ. From the definition of f it is clear that f L f L, and a similar argument shows that f L f L. Finally it is clear that the f have mean value zero; that is f (x) dx =. Using Chebyshev s inequality, we have { x R d : } { (T f)(x) > λ { (.9) + ). f x R d : (T f )(x) > λ } x R d : (T f )(x) > λ } C { λ T f L + x R d : (T f )(x) } λ >. Since we assumed T bounded on L, the first term on the last line of (.9) is bounded: C λ T f L CB λ f L CB λ f L f L CB λ f L. So it only remains to estimate the second term. Define to be the dilate of by d for any B (that is, and share a center and has edges longer than by a factor of d). Then { x R d : (T f )(x) > λ } { B + x R d \ B : (T f )(x) > λ } C + (T f )(x) dx λ B R d \ C λ f L + (T f )(x) dx, λ R d \ B

6 6 ERIC THOMA where we have used Chebyshev s inequality and the the property < λ f L B of B. For any x R d \ we have (T f )(x) = K(x y)f (y) dy = (K(x y) K(x y ))f (y) dy since f has mean value, where y is the center of the cube. This expression allows us to exploit (K): (T f )(x) dx K(x y) K(x y ) f (y) dy dx R d \ R d \ f (y) dy f(y) dy. Note how the selection of allows us to apply (K). Since the are disjoint, this finishes the proof. Now we use interpolation between the L and weak-l bounds to obtain L p boundedness of singular integral operators. Theorem.. Let T be a singular integral operator bounded on L (R d ) with kernel that satisfies (K) and (K). Then T can be extended to a bounded operator on L p (R d ) for every < p < with an operator norm T L p L C(p, d)b. p Proof. The statement for < p < is a consequence of Marcinkiewicz interpolation, and the statement for p = is assumed. The statement for < p < follows from a duality argument and is as follows. Let f, g S(R d ). Then T f, g = f, T g where T is a singular integral operator associated with kernel K (x) = K( x). Clearly K satisfies (K) and (K), and a simple duality argument shows that T is bounded on L. Applying the same arguments to T as we have to T, we see that T is bounded on L q for < q. Since < p, we have T f L p = sup T f, g = sup f, T g g L p = g L p = f L p sup T g L p CB f g L p = L. p 3. The Mikhlin Multiplier Theorem A Fourier multiplier is a function m : R d C that defines a corresponding operator T m by f (m ˆf) on the space of Schwartz functions. A natural problem is to determine conditions under which such operators can be extended as a bounded operator on L p. For L, the problem is easy using Plancherel s theorem: the operator is bounded if and only if m is in L (R d ). The Mikhlin multiplier theorem gives sufficient conditions for such an operator to be bounded on L p for < p < ; however, it is by no means a complete classification. To start, we must construct a partition of unity on a geometric scale.

7 RESULTS ON FOURIER MULTIPLIERS 7 Theorem 3.. There exists a function ψ C (R d ) such that supp(ψ) R d \ {} is compact and such that ψ( j x) =. for all x. We call such a ψ a dyadic partition of unity, and define ψ j (x) = ψ( j x). We will construct ψ to be radial, nonnegative, and so that only at most two terms in the sum are nonzero. Proof. Let χ C (R d ) be radial with χ(x) = for x, χ(x) < for < x <, and χ(x) = for x. Define ψ(x) = χ(x) χ(x). It is immediately verified that ψ is radial and nonnegative. Moreover for any positive integer N we have ψ( j x) = χ( N x) χ( N+ x). j= N For N sufficiently large and x, we have N x and N+ x, and so the sum is. This finishes the proof. Theorem 3.. (Mikhlin Multiplier Theorem) Let m : R d \ {} C satisfy γ m(ξ) B ξ γ for any multi-index γ with γ d + and for all ξ. Then there exist constants C(d, p) for < p < such that ( ˆf) L m C(d, p)b f L p p for all f S. This theorem implies the L p boundedness of a wide range of multipliers, e.g. the sgn function in one dimension and smooth functions of compact support. Proof. With ψ a dyadic partition of unity, define m j (ξ) = ψ( j ξ)m(ξ). Let K j = m j, and K(x) = K j (x). j= N for a fixed, large positive integer N. We will prove the estimates (3.3) K(x) C(d)B x d, K(x) C(d)B x d. hold uniformly in N. A calculation gives γ m L j CB j γ for all multiindices γ with γ d +. To see this, let R > be such that supp(ψ) B(, R). Then ( ) γ m(ξ)ψ( j ξ) C α m(ξ) γ α ψ( j ξ) α,α<γ CB α,α<γ ξ α ( χ [B(, j R)]) j( γ α ) sup β ψ(x) CB R α j α j( γ α ) CB j γ. α,α<γ β,x

8 8 ERIC THOMA This estimate and the compactness of supp(ψ) gives γ m L j CB j γ jd. This result and another compactness argument yields γ (ξ i m j ) L C ξi γ m L j + C α m L j CB j( γ ) jd. We have x γ m j (x) L α, α = γ = C ( γ m j ) L C γ m j L CB j(d γ ) and similarly x γ D m j (x) L CBj(d+ γ ). Note that these manipulations are justified by the compact support and differentiability of m j. Since x k C(k, d) γ, γ =k xγ, we arrive at (3.4) m j (x) CB j(d k) x k, Dm j (x) CB j(d+ k) x k for any k d + and all j Z, x R d \ {}. We will prove the first inequality in (3.3) by using the first inequality in (3.4) with k = and k = d +, and the same method can be used to prove the second inequality in (3.3) with the second inequality in (3.4). We have K(x) CB m j (x) j x j x jd + CB m j (x) + x < j m j (x) x < j jd ( j x ) (d+) CB x d + CB x x (d+) = CB x d. This shows that K satisfies (K), independent on N. A similar proof yields K(x) < CB x d, which implies K satisfies (K). Note (m ˆf) L m = ˆf m L L ˆf = m L L f L, and since m L B by hypothesis, we know that m is bounded as a Fourier multiplier on L. Sending N to infinity, the resulting operator satisfies the requirements of a Calderón-Zygmund operator with L boundedness in place of (K3). Therefore, the operator is bounded on L p for < p <, as desired. We remark here that while the Mikhlin multiplier theorem provides useful sufficient conditions for L p boundedness of Fourier multipliers, it is not discerning enough for a satisfactory theory; it does not distinguish between p so long as < p <. An interesting consequence of the Mikhlin multiplier theorem is a class of estimates important for elliptic partial differential equations. Corollary 3.5. A Fourier multiplier m that is homogenous of degree and smooth everywhere except at defines an operator bounded on L p. As a consequence, we have the following Schauder estimate: u (3.6) C(p, d) u x i x L p j L p

9 RESULTS ON FOURIER MULTIPLIERS 9 for u S(R d ), for indices i, j with i, j d, for < p <, and for C(p, d) a constant depending only on d and p. Proof. By writing a difference quotient, we see that the directional derivative of m at x is maximized when taken along the direction perpendicular to x. By homogeneity, we see that this derivative is equal to the directional derivative of m at x/ x scaled by / x. By continuing writing difference quotients, we see that an nth order partial derivative of m at x is equal to the nth order partial derivative of m at x/ x scaled by / x n. By smoothness of m and compactness of the unit circle, we can then bound all partial derivatives of order less than or equal to d + globally. Thus m satisfies the hypotheses of theorem 3., and is a Fourier multiplier bounded from L p to L p for < p <. The Schauder estimate (3.6) follows from the formula ( ) u (ξ) = ξ iξ j x i x j ξ ( u) (ξ) and the fact that m(ξ) = ξ i ξ j / ξ is homogenous of degree. 4. Littlewood-Paley Theory In this section, we develop the basic ideas of Littlewood-Paley theory. Define P j f = (ψ j ˆf) for j Z and f S(R d ), where the ψ j are the functions defined in 3.. Define the Littlewood-Paley square function Sf by Sf = Pj f /. Using Plancherel s theorem, we have Sf L = P j f L = ψ j ˆf = L R d ψ j (x) ˆf(x) dx. From this equality, it is easy to see that Sf L f L. By using the fact that at most two terms of j ψj (x) are nonzero, it can be shown that in fact Sf L C f L for an absolute constant C. It is natural to ask: does a similar inequality hold for p other than? The goal of this section is to prove such an inequality for < p <. Define ρ N = {r = (r j ) N : r j = ±}. For any finite set A let #A be the number of elements in A. The following two lemmas come from probability theory, but here will be presented as analytical facts in the interest of being self-contained. Lemma 4.. For any (a j ) N C with N aj =, we have (4.) # r ρ N : r j a j > λ / 4e λ #ρ N for all λ >.

10 ERIC THOMA Proof. Set S N (r) = N r ja j. Make first the assumption that (a j ) N R. The more general case will follow from decomposition into real and imaginary parts. It is easy to check via an expansion that N N #ρ N e λsn (r) = r j=± e λrjaj = cosh(λa j ). Using the bound cosh(x) exp(x /) from calculus, one has e λsn (r) exp N #ρ N λ a j = e λ /. Combining this with the Chebyshev inequality e λsn (r) e λ # {r ρ N : e λs N (r) > e λ} = e λ # { r ρ N : S N (r) > λ } gives # { r ρ N : S N (r) > λ } e λ e λ / #ρ N = e λ / #ρ N. A similar argument gives # { r ρ N : S N (r) < λ } e λ / #ρ N whence the lemma holds for our restricted (a j ). By decomposing the (a j ) into real and imaginary parts (b j ) and (c j ) and noting r j a j = r j b j + r j c j we have reduced the lemma to one of the previous cases, losing a factor of two in the process. The following result is known as Klinchine s inequality. Lemma 4.3. Let < p <. Then, for any r ρ N and (a j ) N C, one has p/ p p/ (4.4) aj C(p) #ρ N r j a j C(p) aj for some constant C(p). Proof. Let S N (r) = N r ja j as before. By dividing each term of (a j ) by N aj, we may assume N aj =. The upper bound will be obtained through the estimate achieved in Lemma 4.. We have SN (r) p = #{r ρ N : SN (r) > λ}pλ p dλ #ρ N #ρ N Note that SN (r) = 4e λ / pλ p dλ = C(p) <. a jr j + n m c n,m r n r m

11 RESULTS ON FOURIER MULTIPLIERS where c n,m depends on the sequence (a j ). But noting that r n r m is equally often + and, we see that SN (r) = #ρ N aj = #ρn. This shows that equality holds in the lemma for p =. We leverage this result and the lower bound to finish the lemma. By applying Holder s inequality, we have = SN (r) #ρ N #ρ N SN (r) /p p C(p ) /p SN (r) p #ρ N #ρ N /p, SN (r) /p p and the proof is finished. Now that we have established the Mikhlin Multiplier theorem and Klinchine s inequality, we are ready to prove the deep Littlewood-Paley square function estimate. Theorem 4.5. For < p <, the Littlewood-Paley square function Sf has an L p norm comparable to the L p norm of f. That is for some constant C(p, d). Proof. Define the Fourier multipliers C(p, d) f L p Sf L p C(p, d) f L p m N,r (ξ) = j= N r j ψ j (ξ) for every r ρ N and every N. We claim that the hypotheses of the Mikhlin multiplier theorem are satisfied uniformly in N and r. We have γ m N,r (ξ) j= N γ ψ j (ξ) = j= N N j γ ( γ ψ)( j ξ). for any multi-index γ and ξ. Note that j γ is within a factor of two of ξ γ whenever ( γ ψ)( j ξ) is nonzero. Moreover, only an absolutely bounded number of terms of the final sum (here exactly two) are nonzero at any one time. This allows us to assert γ m N,r (ξ) C j= N N ξ γ ( γ ψ)( j ξ) C ξ γ. Note that this inequality works trivially for ξ =. We now use Lemma 4.3 with a j = (P j f)(x) to say (Pj f)(x) p/ C #ρ N j= N j= N r j (P j f)(x) p

12 ERIC THOMA for all x. If we now integrate both sides, take the lim sup in N, and apply the Mikhlin multiplier theorem, we have (Sf)(x) p p C dx lim sup R d N #ρ N r j (P j f)(x) dx C f p p R d. j= N Duality is used to prove the lower bound. Let ψ be a function that is on supp(ψ), compactly supported, and has supp( ψ). Define P j with ψ in place of ψ, and similarly define S. Then clearly P j P j = P j. Let f, g S be arbitrary. Applying Plancherel s theorem twice, we have f, g = ˆf(ξ)ĝ(ξ) dξ R = ˆf(ξ)ψ j (ξ)ĝ(ξ) ψ j (ξ) dξ d R d = P j f, P j g. Therefore, by the Cauchy-Schwarz inequality and Holder s inequality, we have f, g (Pj f)(x) / / ( P k g)(x) R d Sf p Sg p C Sf p g p, k= where the upper bound on Sg comes from the same argument we used for the p upper bound of Sf p applied to S. Noting that f p = sup g p = f, g, we see that this proves the lower bound. The failure of this theorem at p = can be seen through the action of S on the Dirac delta distribution; in particular Sδ / x d. As usual, one must argue by approximating δ with an approximate identity and using a limiting process. The failure at p = is a consequence of failure at p = and duality. 5. The Ball Multiplier Theorem It is a remarkable fact that the operator S given by (5.) Sf(x) = ˆf(ξ)e πix ξ dξ, [ ξ <] initially defined on L (R d ) L p (R d ), is not bounded with respect to the L p norm for p and d > (boundedness holds for d = due to the boundedness of the Hilbert Transform and for p = due to Plancherel s theorem). The operator S corresponds to the Fourier multiplier χ B(,), i.e. the ball multiplier. The proof of this fact relies on the existence of so-called Kakeya sets of arbitrarily small measure. These are subsets of the plane that contain a unit line segment in every direction. We do not construct such sets here, and refer the reader to [, p.433] to see a detailed construction. Instead, we state the lemma we will need.

13 RESULTS ON FOURIER MULTIPLIERS 3 Lemma 5.. For any ɛ >, there exists an integer N and N rectangles of side lengths and N, denoted R,..., R N, such that N (5.3) R j < ɛ, and the translates of each R j by units in the positive direction along its longer side, denoted R j, are mutually disjoint. Therefore N (5.4) R j =. Here denotes Lebesgue measure. Note that the rectangles, which constitute a Kakeya set, have a high degree of overlap, yet point in many different directions. We proceed in the proof of the ball multiplier theorem by contradiction. Along with existence of small Kakeya sets, the following lemma plays a key role. Lemma 5.5. Suppose that the following bound holds for all f L (R d ) L p (R d ) for some p with p : (5.6) Sf L p A p f L p. Let u,..., u M be unit vectors in R d for some fixed integer M. Then / M (5.7) H uj (f j ) / M A p f j L p L p where H u (f)(x) = [ξ u>] ˆf(ξ)e πix ξ dξ. Proof. Let f = (f,..., f M ) and let T (f) = (T f,..., T f M ) for any linear operator T. Define the family of operators T ω by M (5.8) T ω (f) = ω j T (f j ) where ω = (ω j ) M Cm satisfies ω =. Similarly define f ω = M ω j f j. Suppose that we have the bound (for p < ) (5.9) T ω f(x) p dx A p p f ω (x) p dx. R d R d Define φ(ω, u) = u/ u, ω = (u/ u ) ω for u and φ(ω, ) =. Clearly we have (5.) Tω f(x) M = T f(x) φ(ω, T f(x)) = T fj (x) φ(ω, / T f(x)).

14 4 ERIC THOMA Define γ(p, f, x) = [ ω =] φ(ω, T f(x)) p dω. If T f(x) then there exists a rotation such that (T f(x)/ T f(x) ) = (,,..., ) = e. Since rotations preserve the Hermitian inner product and the resulting change of variables preserves dω, we have γ(p, f, x) = φ(ω, e ) p dω = φ(ω, e ) p dω, [ ω =] [ ω =] whence γ(p, f, x) = γ p is independent of f and x and is nonzero. The case T f(x) = is immaterial in the proof. We integrate (5.9) over {ω : ω = } before integrating in x. Using the independence of γ p from x and γ p >, we conclude that M M (5.) / T (f j ) L p (R d ) A p / f j Now let p =. If we assume in place of (5.9) the inequality (5.) T ω f L A f ω L, L p (R d ) then taking ess sup ω = of both sides inside of the L norm and noting ess sup φ(ω, u) = for all u gives (5.) for p =. We now show that (5.9) and (5.) and therefore (5.) hold for operators of interest. Since T ω (f) = T (f ω ), any operator satisfying the bound (5.6) is such an operator. For any set E R d define S E f(x) = ˆf(ξ)e πix ξ dξ. E The hypothesis (5.6) implies an identical bound for S B(,R), where B(, R) is the ball of radius R centered at the origin, for any R >. Indeed, letting δ R (g)(x) = g(x/r), we have δ S B(,R) f = R Sδ Rf = R d Sδ R f L p L p R d A p δ R f L p = A p f L p. L p Let u R d be a unit vector. It is easy to verify S B(Ru,R) f(x) = e πiru x S B(,r) (e πiru x f), in view of which / / M (5.3) S B(Ru,R) (f j ) M A p f j. Lp (R d ) Lp (R d ) As R, the set B(Ru, R) increases to the half space {ξ : ξ u > }. By monotone convergence, we have S B(Ru,R) converges to H u in L, and so a subsequence converges almost everywhere, whence (5.3) proves the lemma. We now formally state and prove the ball multiplier theorem. Theorem 5.4. The operator S cannot be extended to a bounded operator on L p.

15 Proof. Consider the operator S R + defined by RESULTS ON FOURIER MULTIPLIERS 5 (S R +f)(x) = for f L (R ). Put f = χ ( /,/). We have We also have (S R +f)(x) = lim ε + ˆf(ξ)e πi(x+iε)ξ dξ = ( = πi ˆf(ξ)e πixξ dξ ˆf(ξ)e πi(x+iε)ξ dξ. ) e πiyξ e πi(x+iε)ξ dξ f(y) dy f(y) (y x iε) dy where the change in integration order is justified by f(y) e πɛξ dy dξ = e πɛξ dξ <. Let x > / and < ε < /. By the intermediate value theorem, we write / / y x iɛ dy = c x iɛ for some c with / c /, from which it follows (5.5) (SR +f)(x) C, for x > /. x We will use this bound along with the Kakeya construction to show a contradiction in (5.7). Set p <. Let f be the indicator function for the ball of radius 3 in R d ; if n = let f be the constant. Let R,..., R N be the rectangles constructed by lemma 5.. Note that if F (x, x ) = f (x )f (x ) and e is the unit vector in the x direction, we have that (H e F )(x, x ) = (S R +f )(x )f (x ). Let u j, and u j, be unit vectors in the direction of the longest and shortest sides of R j, respectively, and let u j,3 stand in for the remaining n coordinates. Define f j (x, x, x 3 ) = χ Rj (x, x )f(x 3 ) for j =,..., N, where x, x, x 3 are coordinates with respect to u j,, u j,, u j,3. We see (H uj, f j )(x, x, x 3 ) = (S R +χ ( /,/) )(x )χ ( N, N )(x )f(x 3 ) (5.6) C χ x ( N, N )(x )f(x 3 ) c χ R j (x, x ) f(x3 ) for some c > (c = C/5 works), where R j is the translate of R j as defined in lemma 5.. Setting M = N and u j = u j,, we compute that the left side of (5.7) is larger than c v(b 3 ), where v(b 3 ) is defined to be the volume of the ball of radius 3 in R d if d > and is defined to be if d =. This is because j R j =

16 6 ERIC THOMA and the R j are disjoint. Applying Holder s inequality with dual exponents /p and q = /( p/), we conclude that the right side of (5.7) is smaller than N (5.7) A p χ Rj (x, x )f(x 3 ) ( ) /q dx. R d jr j B 3 dx Note that, by our choice of p < and since j R j can be made arbitrarily small via lemma 5., the rightmost factor of (5.7) can be made arbitrarily small. And N χ Rj (x, x )f(x 3 ) N dx = v(b 3 ) R j = v(b 3 ), R d whence (5.7) can be made smaller than c v(b 3 ), contradicting (5.7). Since the L p boundedness of the ball multiplier, (5.6), implies (5.7), we have the theorem for p < and all d >. We now extend the result to p > by duality. Suppose that S is bounded as an operator for some p >. Let p < denote the dual exponent of p. Then, noting that Sf, g = f, Sg, we have Sg L p = sup f, Sg = f L p = sup f L p = Sf, g sup f, g = A p g L p, f L p =A p contradicting the preceding result. Therefore S cannot be extended to a bounded operator on L p (R d ) for p and d >. Acknowledgments. It is a pleasure to thank my mentor, Casey Rodriguez, for his extensive help throughout the learning and writing process, and J.P. May for his help with editing this paper and his leadership of the Chicago REU. References [] Camil Muscalu and Wilhelm Schlag. Classical and Multilinear Harmonic Analysis Vol.. Cambridge University Press. 3. [] Elias M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press. 993.