INTEGRAL OPERATORS C. DAVID LEVERMORE

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1 INTEGRAL OPERATORS C. DAVID LEVERMORE Abstact. We give bounds on integal oeatos that act eithe on classical Lebesgue saces o on eak Lebesgue saces. These include Hölde-Young bounds fo oeatos ith egula kenels, Hady-Littleood bounds fo oeatos ith eakly singula kenels, and Caldeon- Zygmund bounds fo stongly singula convolution oeatos ove Euclidean sace.. Intoduction Let (X, Σ µ, dµ) and (Y, Σ ν, dν) be ositive σ-finite measue saces. Let M(dµ) and M(dν) be the saces of all comlex-valued dµ-measuable and dν-measuable functions esectively. As usual, functions in these saces ae consideed identical if they ae equal almost eveyhee. We conside linea integal oeatos K of the fom (.) Ku(y) = k(x, y) u(x) dµ(x), hee the kenel k is a comlex-valued measuable function ith esect to the σ-algeba Σ µ ν. We seek conditions on k that imly the oeato K is bounded o even comact fom X to Y hee (X, X ) and (Y, Y ) ae Banach saces of functions that ae contained ithin M(dµ) and M(dν) esectively. We ill fist obtain such esults fo classical Lebesgue saces namely, fo cases hee X = L (dµ) and Y = L q (dν) fo some, [, ]. We ill then extend these esults to eak Lebesgue saces namely, to cases hee X = L (dµ) o Y = Lq (dν) fo some, q (, ). These notes ill assume that you have some familiaity ith the classical Lebesgue saces, but ill be self-contained egading the eak Lebesgue saces... Bounded Linea Oeatos. Recall that a linea oeato K that mas a nomed sace (X, X ) into a nomed sace (Y, Y ) is said to be bounded if it mas bounded subsets of X into bounded bounded subsets of Y. It is easy to sho that this is equivalent to the oety that thee exists a constant C < such that (.2) Ku Y C u X fo evey u X. It easy to see that evey bounded linea oeato is continuous. It is not had to sho that the convese is also tue. The notions of bounded and continuous theeby coincide fo linea oeatos acting beteen nomed saces. It is customay to efe the teminology bounded linea oeato ove that of continuous linea oeato. The eason fo this efeence is the fact that the had at of shoing a linea oeato is continuous is usually establishing the bound (.2). Date: July 5, 29.

2 2 C. DAVID LEVERMORE A Banach sace is a comlete nomed sace. When Y is a Banach sace and K is defined ove a dense linea subsace of X, it suffices to establish (.2) fo evey u in that subsace. Because (.2) imlies that K is unifomly continuous ove bounded subsets of X, thee is a unique extension of K to evey u X so that (.2) holds. The sace of all bounded linea oeatos fom a nomed sace (X, X ) into a nomed sace (Y, Y ) is denoted B(X, Y). Fo each K B(X, Y) e define K B(X,Y) to be the infimum of all constants C such that (.2) holds. It is easy to check that B(X, Y) is a linea sace and that B(X,Y) is a nom on B(X, Y). Moeove, if S is any dense linea subsace of X then one can sho that (.3) K B(X,Y) = su u S Ku Y : u X = Ku Y = su : u. u S u X Finally, B(X, Y) equied ith this nom is a Banach sace heneve Y is a Banach sace. In aticula, the dual sace of X, defined by X = B(X, C), is alays a Banach sace..2. Comact Linea Oeatos. Recall that a linea oeato K that mas a nomed sace (X, X ) into a nomed sace (Y, Y ) is said to be comact if it mas bounded subsets of X into totally bounded subsets of Y. Because evey totally bounded subset of a metic sace (hence, of a nomed sace) is also bounded, it is theefoe clea that evey comact linea oeato is also a bounded linea oeato. It is easy to sho that K being comact is equivalent to the oety that K mas the unit ball of X into a totally bounded subset of Y. The folloing theoem is at the heat of many aguments that a given linea oeato is comact. It states that any bounded linea oeato hich can be aoximated by comact linea oeatos is also comact. Theoem.. Let (X, X ) and (Y, Y ) be nomed saces. Let K B(X, Y). If thee exists a sequence K n n N B(X, Y) such that () K n K in B(X, Y) as n, (2) each K n is comact, then K is comact. Poof. Let B X be the unit ball in X. We ill sho that the set KB X Y is totally bounded. Let ǫ >. Because K n K in B(X, Y) as n, thee exists m N such that K m K B(X,Y) < 3 ǫ. Because K m is comact, the set K m B X is totally bounded. This imlies thee exists a finite set S = u j k j= B X such that fo evey u B X thee exists a u j S such that Fo this u and u j e see that K m u K m u j Y < 3 ǫ. Ku Ku j Y Ku K m u Y + K m u K m u j Y + K m u j Ku j Y K m K B(X,Y) u X + K m u K m u j Y + K m K B(X,Y) u j X < 3 ǫ + 3 ǫ + 3 ǫ = ǫ. Hence, fo evey u B X thee exists a u j S such that Ku Ku j Y < ǫ. Theefoe the set KB X is totally bounded, heeby the oeato K is comact.

3 INTEGRAL OPERATORS 3 The evious theoem ould not be of much hel unless thee exists a sufficiently lage class of comact linea oeatos ith hich to build the aoximating sequences that it equies. This class is often ovided by a class of finite ank oeatos. Definition.. Let (X, X ) and (Y, Y ) be nomed saces. We say that K B(X, Y) has finite ank if the ange of K is a finite dimensional subsace of Y, in hich case the dimension of the ange is called the ank of K. The fact that finite ank oeatos ae comact is a consequence of the Bolzano-Weiestass Theoem, hich imlies that bounded subsets of Euclidean sace ae totally bounded. 2. Hölde-Young Bounds None of the bounds esented belo ee actually deived by eithe Hölde o Young in the geneal setting given hee. Rathe, they genealize cetain bounds fist deived by them [2]. 2.. Lebesgue Saces. Fo any ositive σ-finite measue sace (X, Σ µ, dµ) and any (, ) e define the Lebesgue sace L (dµ) by (2.) L (dµ) = u M(dµ) : u(x) dµ(x) <. It is easy to check that L (dµ) is a linea sace []. Fo evey (, ) e define the magnitude of u L (dµ) by ( ) (2.2) [u] L = u(x) dµ(x). It is clea fom (2.) that fo evey u M(dµ) e have u L (dµ) if and only if [u] L <. It is also clea that [λ u] L = λ [u] L fo evey u L (dµ) and evey λ C. Fo evey [, ) the Minkoski inequality imlies that [ ] L satisfies the tiangle inequality, and is theeby a nom []. In that case L (dµ) is a Banach sace equied ith the nom ( ) (2.3) u L = [u] L = u(x) dµ(x) Moeove, fo evey (, ) one can sho that [ ] L fails to satisfy the tiangle inequality, and is theeby not a nom. Hoeve, in that case L (dµ) is a Fechét sace equied ith the metic (2.4) d(u, v) L = [u v] L = u(x) v(x) dµ(x). Finally, e define the Lebesgue sace L (dµ) by (2.5) L (dµ) = u M(dµ) : ess su u(x) <. x X You can sho that L (dµ) is a Banach sace equied ith the nom ( (2.6) u L = ess su u(x) = inf α > : µ Eu (α) ) =, x X hee E u (α) = x X : u(x) > α. Hee e adot the usual convention that inf =..

4 4 C. DAVID LEVERMORE 2... Hölde Inequalities. Fo any ositive σ-finite measue sace (X, Σ µ, dµ) the basic Hölde inequality goes as follos [, 3]. Let, [, ] satisfy the duality elation (2.7) + =. Then fo evey u L (dµ) and v L (dµ) e have uv L (dµ) ith (2.8) u(x) v(x) dµ(x) u L v L. The basic Hölde inequality has the folloing genealization. Let, 2,, n, [, ] satisfy the elation (2.9) = 2 n. Then fo evey u L (dµ), u 2 L 2 (dµ),, u n L n (dµ) e have u u 2 u n L (dµ) ith (2.) u u 2 u n L u L u 2 L 2 u n L n L Riesz Reesentation Theoem. Fo evey, [, ] that satisfy the duality elation (2.7), the basic Hölde inequality (2.8) shos that evey v L defines a bounded linea functional l v (L ) = B(L, C) by l v (u) = v(x) u(x) dµ(x), and that l v (L ) v L, hee by (.3) e have (2.) l v (L ) = su lv (u) u L u L : u. We claim that l v (L ) = v L. This is clealy tue hen v =, so suose that v L >. Fo (, ] the agument is easy. One sees that u = v sgn(v) L ith u L = v, and l L v(u) = v. L Because u, e infe fom (2.) that l v (L ) v L. Fo = e have =. Because v L >, e see fom (2.6) that fo evey α such that < α < v L one has µ(x : v(x) > α) >. Let E α Σ µ such that E α x : v(x) > α and < µ(e α ) <. One sees that u α = Eα sgn(v) L ith u α L = µ(e α ), and l v (u) = v(x) dµ(x) α µ(e α ). E α Because u α, e infe fom (2.) that l v (L ) α. Because this holds fo evey α such that < α < v L, it follos that l v (L ) v L. Hence, fo evey [, ] e have l v (L ) = v L. The maing v l v is theefoe an isomety fom L (dµ) into (L (dµ)) fo evey [, ]. The L Riesz Reesentation Theoem assets that this isomety is onto fo evey [, ). It is a consequence of the Radon-Nikodym Theoem [] o the L -Pojection Theoem [3]. One of the most useful consequences of the L Riesz Reesentation Theoem is the folloing chaacteization of functions in L (dµ).

5 INTEGRAL OPERATORS 5 Lemma 2.. Let u M(dµ), [, ], and C [, ). Then u L (dµ) ith u L C if and only if (2.2) u(x) v(x) dµ(x) C v L fo evey v L (dµ). Poof. The foad imlication ( = ) follos diectly fom the basic Hölde inequality (2.8). Fo = the othe diection simly follos by taking v = sgn(u) in (2.2). Fo (, ] define the linea functional l u by l u (v) = u(x) v(x) dµ(x) fo evey v L (dµ). It follos fom (2.2) that l u (L (dµ)). Because [, ) the L Riesz Reesentation Theoem then imlies that thee exists L (dµ) such that l u (v) = (x) v(x) dµ(x) fo evey v L (dµ). Hence, u M(dµ) satisfies (u(x) (x) ) v(x) dµ(x) = fo evey v L (dµ). Fo evey E Σ µ such that µ(e) < e have that v = E sgn(u ) is in L (dµ) fo evey [, ]. The above condition theefoe imlies that u M(dµ) satisfies u(x) (x) dµ(x) = fo evey E Σµ such that µ(e) <. E We theeby conclude that u = L (dµ). It follos that v = u sgn(u) L (dµ) ith v L = u L. By then setting v = u sgn(u) into (2.2) e infe that u L C Paied Bounds. Let K denote fomal adjoint of K, hich is given by (2.3) K v(x) = k(x, y)v(y) dν(y). The oeato K is bounded fom L (dµ) to L q (dν) if and only if K is bounded fom L q (dν) to L (dµ) hee, q [, ] ae detemined by the duality elations (2.4) + =, and q + =. Moeove, K B(L q (dν),l (dµ)) = K B(L (dµ),l q (dν)). We ill use the folloing citeion to establish the boundedness of both K and K. Lemma 2.2. Let k M(dµ dν) and C [, ) such that fo evey u L (dµ) and v L q (dν) e have (2.5) k(x, y) u(x) v(y) dµ(x) dν(y) C u L v L q. Then K B(L (dµ), L q (dν)) and K B(L q (dν), L (dµ)) ith (2.6) K B(L,L q ) = K B(L q,l ) C.

6 6 C. DAVID LEVERMORE Remak. The measues dµ and dν ill be doed fom the notation fo noms as e did in (2.5) hen thee is no confusion about hat measues ae involved. Poof. By (2.5) and the Fubini-Tonelli Theoem e see that fo evey u L (dµ) and v L q (dν) e have Ku M(dν), K v M(dµ), and v(y) Ku(y) dν(y) = k(x, y) u(x) v(y)dµ(x) dν(y) = u(x)k v(x) dµ(x). Hence, fo evey u L (dµ) e have Ku M(dν) and v(y) Ku(y) dν(y) C u L v L q fo evey v Lq (dν), By Lemma 2. e infe that Ku L q (dν) and that Ku L C u L. Because this holds fo evey u L (dµ), e conclude that K B(L (dµ), L q (dν)) and that K B(L,L q ) C. Similaly, fo evey v L q (dν) e have K v M(dµ) and u(x) K v(x) dµ(x) C u L v L q fo evey u L (dµ), By Lemma 2. e infe that K v L (dµ) and that K v L C v L q. Because this holds fo evey v L q (dν), e conclude that K B(L q (dν), L (dµ)) and that K B(L q,l ) C Iteated Nom Bounds. We no give to basic bounds of the tye (2.5). Lemma 2.3. Let, q [, ]. Let the kenel k satisfy the bound (( ) (2.7) k L (dν;l (dµ)) = k(x, y) dµ(x) dν(y) Then fo evey u L (dµ) and v L q (dν) e have ) <. (2.8) k(x, y) u(x) v(y) dµ(x) dν(y) k Lq (L ) u L v L q. Similaly, let the kenel k satisfy the bound (( (2.9) k L (dµ;l q (dν)) = k(x, y) q dν(y) Then fo evey u L (dµ) and v L q (dν) e have ) dµ(x) ) <. (2.2) k(x, y) u(x) v(y) dµ(x) dν(y) k L (L q ) u L v L q. Remak. The saces L q (dν; L (dµ)) and L (dµ; L q (dν)) ae called iteated saces. They ae equied ith the so-called iteated noms L (dν;l (dµ)) and L (dµ;l q (dν)) defined above by (2.7) and (2.9). The bounds (2.8) and (2.2) ae called iteated nom bounds. Poof. Let I(k, u, v) denote the quantity on the left-hand side of (2.8) and (2.2) namely, let k(x, (2.2) I(k, u, v) = y) u(x) v(y) dµ(x) dν(y). In ode to ensue that this quantity makes sense, e assume fo the moment that u, v, and k ae simle functions ith esect to the measues dµ, dν, and dµ dν esectively.

7 INTEGRAL OPERATORS 7 The fist iteated nom bound (2.8) is deived as follos. By the basic Hölde inequality (2.8) one has ( ) k(x, y) u(x) dµ(x) k(x, y) dµ(x) u L (dµ). Uon fist using this bound and then alying the Hölde inequality again, e deive the bound ( ) k(x, I(k, u, v) = y) u(x) dµ(x) v(y) dν(y) ( ) k(x, y) dµ(x) v(y) dν(y) u L (dµ) k L (dν;l (dµ)) u L (dµ) v L q (dν). The fist iteated nom bound (2.8) then follos by a density agument. The second iteated nom bound (2.2) is deived by simly evesing the oles of x, u,, and dµ ith those of y, v, q, and dν. By the Hölde inequality one has ( ) k(x, q y) v(y) dν(y) k(x, y) q dν(y) v L q (dν). Uon fist using this bound and then alying the Hölde inequality again, e obtain the bound ( ) k(x, I(k, u, v) = y) v(y) dν(y) u(x) dµ(x) ( ) k(x, y) q dν(y) u(x) dµ(x) v L q (dν) k L (dµ;l q (dν)) u L (dµ) v L q (dν). The second iteated nom bound (2.2) then follos by a density agument. Remak. The Minkoski inequality fo integals [] imlies that (2.22) k L (dν;l (dµ)) k L (dµ;l q (dν)) heneve, k L (dµ;l q (dν)) k L (dν;l (dµ)) heneve. In the fist case e can conclude that the fist iteated nom bound (2.8) is the shae one, heeby e conclude by Lemma 2.2 that K B(L, L q ) and K B(L q, L ) ith (2.23) K B(L,L q ) = K B(L q,l ) k L (L ) fo evey k L q (dν; L (dµ)). In the second case e can conclude that the second iteated nom bound (2.2) is the shae one, heeby e conclude by Lemma 2.2 that K B(L, L q ) and K B(L q, L ) ith (2.24) K B(L,L q ) = K B(L q,l ) k L (L q ) fo evey k L (dµ; L q (dν)). Remak: When eithe < and k L q (dν; L (dµ)) o < and k L (dµ; L q (dν)) then e can conclude that the bounded oeatos K and K fom (2.23) and (2.24) ae moeove comact. This is because one can sho that the finite-ank kenels ae dense in the saces L q (dν; L (dµ)) and L (dµ; L q (dν)). The classical Hilbet-Schmidt comactness citeion is the secial case = q = 2.

8 8 C. DAVID LEVERMORE Remak: When = q in the iteated saces L (dν; L (dµ)) and L (dµ; L (dν)) coincide ith L (dν; L (dµ)) = L (dµ; L (dν)) = L (dµ dν). Moeove, the iteated noms given by (2.7) and (2.9) also coincide ith k L (dν;l (dµ)) = k L (dµ;l (dν)) = k L (dµ dν). If these ae finite then K is bounded fom L (dµ) to L (dν) and K is bounded fom L (dν) to L (dµ). If moeove < then K and K ae also comact by the evious emak. Remak: In some cases the iteated nom bounds (2.8) and (2.2) ae sha. Secifically, it can be shon that hen [, ] and q = one has K B(L,L ) = K B(L,L ) = k L (L ) hile hen = and q [, ] one has fo evey k L (dν; L (dµ)), K B(L,L q ) = K B(L q,l ) = k L (L q ) fo evey k L (dµ; L q (dν)) Young Integal Oeato Bounds. The esults of the evious section include the folloing. If k L (dµ dν) then fo evey u L (dµ) and v L (dν) e have (2.25) k(x, y) u(x) v(y) dµ(x) dν(y) k L (dµ dν) u L v L. If k L (dµ; L (dν)) fo some [, ) then fo evey u L (dµ) and v L (dν) e have (2.26) k(x, y) u(x) v(y) dµ(x) dν(y) k L (dµ;l(dν)) u L v L. If k L (dν; L (dµ)) fo some [, ) then fo evey u L (dµ) and v L (dν) e have (2.27) k(x, y) u(x) v(y) dµ(x) dν(y) k L (dν;l (dµ)) u L v L. In this section e sho that if k L (L )(dµ, dν) = L (dν; L (dµ)) L (dµ; L (dν)) fo some [, ) then, in addition to the bounds (2.26) and (2.27), e have an entie family of Young integal oeato bounds. Theoem 2.. Let k L (L )(dµ, dν) fo some [, ). Let, q [, ] satisfy the elation (2.28) + q + = 2. Then fo evey u L (dµ) and v L q (dν) e have (2.29) k(x, y) u(x) v(y) dµ(x) dν(y) k L (dµ;l (dν)) k L (dν;l (dµ)) u L v L q. Moeove, e have K B(L (dµ), L q (dν)) and K B(L q (dν), L (dµ)) ith (2.3) K B(L,L q ) = K B(L q,l ) k L (dµ;l (dν)) k L (dν;l (dµ)).

9 INTEGRAL OPERATORS 9 Remak. The case = is aleady coveed by (2.25) because in that case elation (2.28) ould equie that = q =. The case [, ) and = is aleady coveed by (2.26) because in that case elation (2.28) ould equie that q =. The case [, ) and q = is aleady coveed by (2.27) because in that case elation (2.28) ould equie that =. Poof. The case q = is coveed by (2.26) ith = because in that case elation (2.28) ould equie that = =. The case = is coveed by (2.27) ith = because in that case elation (2.28) ould equie that q = =. Theefoe e only have to establish the case hen, q, [, ) satisfy elation (2.28). We ill aly the geneal Hölde inequality (2.) to the thee functions U(x, y) = k(x, y) v(y) q, V (x, y) = k(x, y) u(x), W(x, y) = u(x) v(y) q. Relation (2.28) imlies that One theeby sees that + =, q + q =, + =. k(x, y) u(x) v(y) = U(x, y) V (x, y) W(x, y). Because elation (2.28) also imlies that + + =, the geneal Hölde inequality (2.) yields I(u, v, ) = k(x, y) u(x) v(y) dµ(x) dν(y) = U(x, y) V (x, y) W(x, y) dµ(x) dν(y) U L (dµ dν) V L (dµdν) W L (dµ dν) = k q L (dµ;l (dν)) v L q k L (dν;l (dµ)) u L u L v q L q = k L (dµ;l (dν)) k L (dν;l (dµ)) u L v L q, heeby the Young integal oeato bound (2.29) holds. Assetion (2.3) then follos fom Lemma Inteolation Bounds. The family of Young integal bounds (2.29) belongs to the lage class of inteolation bounds. We ill develo inteolation bounds in the folloing setting. Suose that fo some, q,, q [, ] the kenel k satisfies the bounds k(x, y) u(x) v(y) dµ(x) dν(y) C u L v L q (2.3) fo evey u L (dµ) and v L q (dν), k(x, y) u(x) v(y) dµ(x) dν(y) C u L v L q fo evey u L (dµ) and v L q (dν), These bounds imly the oeato K belongs to B(L (dµ), L q (dν)) and to B(L (dµ), L q (dν)), hee the usual duality elations q + = and q q + = hold. Inteolation ill allo us q to extend all of these esults to othe saces. Rathe than develo the full Riesz-Thoin inteolation theoy fo L saces [], hee e ill simly emloy the folloing elementay inteolation lemma.

10 C. DAVID LEVERMORE Lemma 2.4. If the kenel k satisfies the bounds (2.3) fo some, q,, q [, ] then fo evey t [, ] it satisfies the inteolation bound k(x, y) u(x) v(y) dµ(x) dν(y) C t C t u L v L q (2.32) fo evey u L (dµ) and v L q (dν), hee t [, ] satisfies t + t (2.33) =, and, q [, ] satisfy the inteolation elations = +, t t q = +. t q tq Moeove, e have K B(L (dµ), L q (dν)) and K B(L q (dν), L (dµ)) ith (2.34) K B(L,L q ) = K B(L q,l ) C t C t. Hee the usual duality elations + =, and q + = hold. Poof. Suose that eithe o q q, because otheise thee is nothing to ove. When t = o t = then (2.32) educes to the fist o second bound of (2.3) esectively. We theeby only need to establish (2.32) hen t (, ). Because (2.32) clealy holds hen eithe u = o v =, e only need to conside cases hen both u and v. We fist conside the case hen neithe = = no q = q =. Because t (, ) e see fom (2.33) that, q [, ). Hence, heneve u L (dµ) and v L q (dν) e obseve that u L (dµ), u L (dµ), v q q L q (dν), and v q q L q (dν) ith u L = u L (2.35), q q v q L q q = v L, q q q u L = u v q L q q = v Given (2.33), fo evey λ (, ) the classical Young s inequality gives L, t u v = u v q t q u t v q tq λt t u v q q + tλ u t v q q. Uon multilying this inequality by k, integating ith esect to dµ dν ove X Y, and using the assumed bounds (2.3) along ith the obsevations (2.35), e find that fo evey u L (dµ), evey v L q (dν), and evey λ (, ) e have the bound (2.36) k(x, y) u(x) v(y) dµ(x) dν(y) λt q t C u q L v L + q tλ C t u L v When u and v the ight-hand side above attains its minimum ove λ (, ) at q λ = C u L v t t L q q. C u L v q L q The inteolation bound (2.32) is then obtained by setting this value of λ into (2.36). The cases hen eithe = = o q = q = can be teated in the same fameok. When = = e can set = and = = in the above agument and it goes though as itten. Similaly, hen q = q = e can set q = and q q = q q = in the above agument and it goes though as itten. We have theefoe established the inteolation bound (2.32) fo all cases. Assetion (2.34) then follos fom Lemma 2.2. L q. q q L. q

11 INTEGRAL OPERATORS We no aly the Inteolation Lemma 2.4 to a kenel k hich fo some, s [, ] satisfies the bounds ( ( ) s ) k L s (dν;l (dµ)) = k(x, y) s dµ(x) dν(y) <, (2.37) ( ( ) s ) k L s (dµ;l (dν)) = k(x, y) s dν(y) dµ(x) <. Without loss of geneality e can assume s because in that case k L s (L ) k L (L s ) fo each of the above noms. We can assume moeove that < s because hen = s the bounds in (2.37) coincide, so the Inteolation Lemma cannot yield futhe boundedness esults. Remak: In the symmetic setting in hich (X, Σ µ, dµ) = (Y, Σ ν, dν) and k(y, x) = k(x, y), the to bounds in (2.37) educe to the single bound ( ( ) s ) (2.38) k L s (dµ;l (dµ)) = k(x, y) s dµ(x) dµ(y) <. This setting is common in alications. The iteated nom bounds of Section 2.2 sho that the bounds (2.37) on k imly the folloing. Because k L s (dν; L (dµ)), then fo evey u L (dµ) and v L s (dν) e have (2.39) k(x, y) u(x) v(y) dµ(x) dν(y) k L s(dν;(l (dµ)) u L v Ls. Because k L s (dµ; L (dν)), then fo evey u L s (dµ) and v L (dν) e have (2.4) k(x, y) u(x) v(y) dµ(x) dν(y) k L s(dµ;(l (dν)) u Ls v L. In this section e sho that because k L s (L )(dµ, dν) = L s (dν; L (dµ)) L s (dµ; L (dν)) then, in addition to the bounds (2.39) and (2.4), e have a family of inteolation bounds. Theoem 2.2. Let k L s (L )(dµ dν) fo some, s [, ] such that < s. Let, q [s, ] satisfy the elation (2.4) + q + + s = 2. Then fo evey u L (dµ) and v L q (dν) e have the inteolation bound (2.42) k(x, y) u(x) v(y) dµ(x) dν(y) k t L s (dν;l (dµ)) k t L s (dµ;l (dν)) u L v L q, hee t and t ae given by (2.43) t = s s = q s, t = s = s q s. Moeove, e have K B(L (dµ), L q (dν)) and K B(L q (dν), L (dµ)) ith (2.44) K B(L,L q ) = K B(L q,l ) k t L s (dν;l (dµ)) k t L s (dµ;l (dν)). When s < the oeatos K and K ae also comact.

12 2 C. DAVID LEVERMORE Remak: When s = this educes to the Young Integal Oeato Theoem 2.. Poof. Uon alying the Inteolation Lemma 2.4 to the bounds (2.39) and (2.4) ith =, q = s, = s, q =, C = k L s (dν;l (dµ)), and C = k L s (dµ;l (dν)), fo evey t [, ] e obtain the inteolation bound (2.45) hee (2.46) k(x, y) u(x) v(y) dµ(x) dν(y) k t L s (dν;l (dµ)) k t L s (dµ;l (dν)) u L v L q fo evey u L (dµ) and v L q (dν), = t + ts, q = t s + t. It is clea fom (2.46) that, q [s, ] and that (2.47) + q = + s. The elation is equivalent to elation (2.4). Convesely, if, q [s, ] and elation (2.47) holds then thee exists a unique t [, ] such that and q ae given by (2.46) namely, the unique t given by (2.43). Hence, bound (2.45) is exactly bound (2.42). Assetion (2.44) then follos fom Lemma 2.2. Finally, the fact that the oeatos K and K ae comact hen s < follos because in that case one can sho that the finite-ank kenels ae dense in the saces L s (dν; L (dµ)) and L s (dµ; L (dν)). Remak: When [, 2] (so that ) and s = then fo evey [, ] one sees that K B(L (dµ), L (dν)) and K B(L (dν), L (dµ)) ith K B(L,L ) = K B(L,L ) k t L (dν;l (dµ)) k t L (dµ;l (dν)), hee t is given by (2.43). In this case q =. Remak: Let satisfy 2 = + s. Notice that is the hamonic mean of and s, so that [s, ]. One sees that K B(L (dµ), L (dν)) and K B(L (dν), L (dµ)) ith In this case q =. K B(L,L ) = K B(L,L ) k 2 L s (dν;l (dµ)) k 2 L s (dµ;l (dν)). 3. Hady-Littleood Bounds The inteolation bound (2.42) cannot be alied to kenels ove R D R D of the fom k(x, y) = x y D fo some (, ) hen dµ and dν ae each Lebesgue measue because in that case both k L s (dν;l (dµ)) and k L s (dµ;l (dν)) ae not finite. This oblem as ovecome by bounds that ge out of the ioneeing ok of Hady and Littleood [2]. Thei ok led to a class of saces that allo the teatment of such kenels namely, the eak Lebesgue saces.

13 INTEGRAL OPERATORS Weak Lebesgue Saces. Fo any ositive σ-finite measue sace (X, Σ µ, dµ) and any (, ) e define the eak Lebesgue sace L (dµ) by (3.) L u (dµ) = M(dµ) : su α µ(e u (α)) <, α> hee E u (α) = x X : u(x) > α. It is easy to check that L (dµ) is a linea sace. Fo evey (, ) it is clea that L (dµ) L (dµ). Indeed, fo evey u L (dµ) and evey α > the Chebyshev inequality yields µ(e u (α)) = dµ(x) u(x) dµ(x) u(x) dµ(x) = α α α [u] L (dµ). E u(α) It theeby follos that E u(α) su α µ(e u (α)) [u] L (dµ) <, α> heeby u L (dµ). In geneal L (dµ) is lage than L (dµ). Fo examle, hen X = R D and dµ is the unsual Lebesgue measue on R D then it can be shon that the function u(x) = x D is in L (dµ) but it is clealy not in L (dµ). Fo evey (, ) e define the magnitude of evey u L (dµ) by ( (3.2) [u] L = su α> α µ(e u (α)) ). It is clea fom (3.) that u L (dµ) if and only if [u] L <. Hoeve, [ ] L is not a nom. While it satisfies [λu] L = λ [u] L fo evey u L (dµ) and λ C, it fails to satisfy the tiangle inequality. Hoeve, the next esult shos thee is an equivalent nom fo (, ). Theoem 3.. Fo evey (, ) and evey u M(dµ) e define (3.3) u L = su u(x) dµ(x) : µ(e) (, ). E Σ µ µ(e) E Fo evey u M(dµ) e can sho that u L (dµ) if and only if u L <. Moeove, (3.4) [u] L u L [u] L fo evey u L (dµ). Remak. It is easily checked fom definition (3.3) that L is a nom. The esult stated above and oved belo shos that the sace L (dµ) is chaacteized by the finiteness of this nom fo evey (, ). Finally, if u L (dµ) fo some (, ) then by alying the Hölde inequality inside the suemum of (3.3) and using the fact that E L = µ(e) shos that u L u L. Hee E denotes the indicato function of the set E. Poof. Fist assume that u L <. We claim this imlies µ(e u (α)) < fo evey α >. Indeed, suose otheise. Then µ(e u (α)) = fo some α >. One can then constuct a sequence E n n N Σ µ such that E n E u (α) = x X : u(x) > α and µ(e n ) (n, ) fo evey n N. It follos that µ(e n ) E n u(x) dµ(x) µ(e n ) µ(e n ) α = µ(e n ) α as n.

14 4 C. DAVID LEVERMORE But by (3.3) this contadicts u L <. Hence, µ(e u (α)) < fo evey α >. Moeove, heneve µ(e u (α)) > e have µ(e u (α)) (, ) and by (3.3) µ(e u (α)) = dµ(x) u(x) dµ(x) E u(α) α E u(α) α u L µ(e u(α)). This imlies that µ(e u (α)) u L /α fo evey α >. It theeby follos fom (3.2) that ( [u] L = su α> α µ(e u (α)) ) u L <, heeby u L (dµ) and the fist inequality in (3.4) holds. No assume that u L (dµ), heeby [u] L <. We see fom (3.2) that (3.5) µ(e u (α)) [u] L α fo evey α >. The key ne tool e ill use is the so-called laye-cake decomosition of u(x), u(x) = u(x) dα = u(x) >α dα. Let E Σ µ such that µ(e) (, ). The Fubini-Tonelli theoem then yields (3.6) u(x) dµ(x) = u(x) >α dα dµ(x) = Eu(α) dµ(x) dα. E E We see fom (3.5) that the above inne integal can be bounded as Eu(α) dµ(x) min µ(e), µ(e u (α)) min µ(e), [u] L. α E When this bound is laced into (3.6), and the vaiable of integation is escaled aoiately, e obtain u(x) dµ(x) Eu(α) dµ(x) dα min µ(e), [u] L dα E E α ( = µ(e) ) [u] L min, dα = µ(e) α [u] L dα + α dα ( = µ(e) [u] L + ) = µ(e) [u] L. It then follos fom definition (3.3) that u L = su u(x) dµ(x) : µ(e) (, ) [u] E Σ µ µ(e) L <, E E heeby u L < and the second inequality in (3.4) holds.

15 INTEGRAL OPERATORS Fist Hady-Littleood Bound. The deivation of this bound is staightfoad. Theoem 3.2. Let (, ). Fo evey kenel k that satisfies (3.7) k L (dµ;l (dν)) = ess su su k(x, y) dν(y) : ν(e) (, ) <, x X E Σ ν ν(e) E the integal oeato K defined by (.) satisfies the bound (3.8) Ku L (dν) k L (dµ;l (dν)) u L (dµ) fo evey u L (dµ). Remak. Bound (3.8) shos that the oeato K is bounded fom L (dµ) into L (dν) ith (3.9) K B(L,L ) k L (L ) fo evey k L (dµ; L (dν)). This esult should be comaed to (2.24) ith = and q =. Poof. Fo evey E Σ ν such that ν(e) (, ) e see by Fubini-Tonelli and (3.7) that Ku(y) dν(y) k(x, y) u(x) dµ(x) dν(y) ν(e) E ν(e) E [ ] = ν(e) E k(x, y) dν(y) k L (dµ;l (dν)) u L (dµ). u(x) dµ(x) By taking the suemum ove all such E and using (3.3) e obtain (3.8) Second Hady-Littleood Bound. The deivation of this bound again uses laye-cake decomositions, hich ee intoduced in the oof of Theoem 3.. Theoem 3.3. Let, q, (, ) satisfy the elation (3.) + q + = 2. Thee exists a ositive constant C,q, such that fo evey kenel k that satisfies C µ = ess su suγ µ(e k (γ))(y) <, (3.) y Y C ν = ess su x X γ> suγ ν(e k (γ))(x) γ> <, hee E k (γ) = (x, y) X Y : k(x, y) > γ, the integal oeato K defined by (.) satisfies the bound (3.2) Ku L Hee e ill establish (3.2) ith C,q, C µ C (3.3) C,q, ν [u] L fo evey u L (dµ). = = +.

16 6 C. DAVID LEVERMORE Remak. Notice that C,q, given by (3.3) is univesal in the sense that it is indeendent of the undelying measue saces (X, Σ µ, dµ) and (Y, Σ ν, dν). Remak. Conditions (3.) on k ae equivalent to the nom conditions k L (dν;l (dµ)) = ess su su k(x, y) dµ(x) : µ(e) (, ) <, y Y E Σ µ µ(e) E k L (dµ;l (dν)) = ess su su k(x, y) dν(y) : ν(e) (, ) <. x X E Σ ν ν(e) E Indeed, by folloing the agument that led to (3.4) e can sho that (3.4) C µ k L (dν;l (dµ)) C µ, C ν k L (dµ;l (dν)) C ν. By elacing C µ and C ν in the second Hady-Littleood bound (3.2) accodingly, e obtain Ku L C,q, k L (dν;l (dµ)) k L (dµ;l (dν)) [u] L fo evey k L (L )(dµ, dν) and u L (dµ), hee C,q, is given by (3.3). This is a eak Lebesque sace analog of bound (2.3) that e obtained fom the Young integal oeato bound (2.29). Poof. Because the bound (3.2) clealy holds hen eithe u = o k =, e only need to conside the case hen u and k. We can theeby nomalize u so that [u] L = and assume that C µ and C ν ae stictly ositive. Let E Σ ν such that ν(e) (, ). Define I E (k, u) = Ku(y) dν(y). By the laye-cake decomositions k(x, y) = E k(x,y) >γ dγ, u(x) = u(x) >α dα, and the Fubini-Tonelli Theoem e have I E (k, u) k(x, y) u(x) dµ(x) dν(y) E (3.5) k(x,y) >γ u(x) >α E (y) dµ(x) dν(y) dγ dα. We can obtain thee ue bounds of the double integal ove X Y in (3.5) by successively elacing each of the thee indicato functions by. This ocedue yields k(x,y) >γ u(x) >α E (y) dµ(x) dν(y) U(α) ν(e), k(x,y) >γ u(x) >α E (y) dµ(x) dν(y) K µ (γ) ν(e), k(x,y) >γ u(x) >α E (y) dµ(x) dν(y) K ν (γ) U(α),

17 hee (3.6) U(α) = INTEGRAL OPERATORS 7 u(x) >α dµ(x), K µ (γ) = ess su K ν (γ) = ess su x X y Y k(x,y) >γ dν(y). k(x,y) >γ dµ(x), By then using the minimum of these thee ue bounds in (3.5) e obtain (3.7) I E (k, u) min U(α) ν(e), K µ (γ) ν(e), K ν (γ) U(α) dγ dα. The nomalization [u] L = and hyothesis (3.) on k imlies that fo evey α, γ (, ) e have U(α) α, K µ(γ) C µ γ, K ν(γ) C ν γ. When these bounds ae laced into (3.7) and the vaiables of integation ae aoiately escaled e obtain ν(e) I E (k, u) min, ν(e) C µ C ν, dγ dα α γ α γ = ν(e) q C µ C q ν min α, γ, dγ dα. α γ We then see fom definition (3.3) that Ku L su I (dν) E Σ ν ν(e) E (k, u) : ν(e) (, ) C,q, q ith C,q, given by C,q, = = = = α min α, γ, γ dγ dα + dγ dα α γ α dα + + γ + α dα dγ + γ dγ + = C µ C ν, α γ dα dγ ( + + ) =. Theefoe the second Hady-Littleood bound (3.2) holds ith C,q, given by (3.3). Remak. We can deive othe bounds by these methods. Fo examle, if k L (L )(dµ dν) fo some (, ) then fo evey u L (dµ) e can sho that Ku L k L (dµ;l (dν)) k L (dν;l (dµ)) [u] L.

18 8 C. DAVID LEVERMORE 3.4. Thid Hady-Littleood Bound. The deivation of this bound again uses laye-cake decomositions much as they ee used in the oofs of Theoems 3. and 3.3. Hoeve hee the agument ill be fa moe technical because e ill be oking ith classical Lebesgue saces athe than just eak Lebesgue saces. Theoem 3.4. Let, q, (, ) satisfy the elation (3.8) + q + = 2. Thee exists a ositive constant C,q, such that fo evey k that satisfies C µ = ess su suγ µ(e k (γ))(y) <, (3.9) y Y C ν = ess su x X γ> suγ ν(e k (γ))(x) γ> <, hee E k (γ) = (x, y) X Y : k(x, y) > γ, one has the bound (3.2) k(x, y) u(x) v(y) dµ(x) dν(y) C,q, C Hee e ill establish (3.2) ith µ C ν fo evey u L (dµ) and v L q (dν), u L v L q (3.2) C,q, = q ( ) + ( ) + q 2. Remak. Notice that C,q, given by (3.2) is univesal in the sense that it is indeendent of the undelying measue saces (X, Σ µ, dµ) and (Y, Σ ν, dν). Remak. By elacing C µ and C ν in the thid Hady-Littleood bound (3.2) ith th -oes of the noms k L (dν;l (dµ)) and k L (dµ;l (dν)) in accod ith the bounds (3.4), e obtain k(x, y) u(x) v(y) dµ(x) dν(y) C,q, k L (dν;l hee C,q, is given by (3.2). (dµ)) k L (dµ;l (dν)) u L v L q fo evey k L (L )(dµ, dν), u L (dµ), and v L q (dν), Remak. Bound (3.2) and the above emak sho that the oeato K defined by (.) is bounded fom L (dµ) into L q (dν) ith (3.22) K B(L,L q ) C,q, C µ C ν C,q, k fo evey k L (L )(dµ, dν). L (dν;l (dµ)) k L (dµ;l (dν)) This should be comaed ith bound (2.3) that e obtained fom the Young integal oeato bound (2.29). Fo each (, ) that bound equies the kenel k to be in the moe estictive class L (L )(dµ, dν), but includes the cases = o q =. Fom (3.8) and (3.2) e see that C,q, as eithe (, ) (, ) o (, ) (, ), heeby bound (3.22) beaks don in these limits. The beakdon at (, ) should be contasted ith bound (3.9), in hich the ange of K is L (dν) athe than L (dν).

19 INTEGRAL OPERATORS 9 Poof. Because the bound (3.2) clealy holds hen eithe u =, v =, o k =, e only need to conside the case hen u, v, and k. We can then nomalize u and v so that ( ) ( ) (3.23) u L = u(x) dµ(x) =, v L q = v(y) q q dν(y) =, and assume that C µ and C ν ae stictly ositive. Define k(x, I(k, u, v) = y) u(x) v(y) dµ(x) dν(y). Fo any set E, let E denote its indicato function. By the laye-cake decomositions k(x, y) = k(x,y) >γ dγ, u(x) = u(x) >α dα, v(y) = v(y) >β dβ, and the Fubini-Tonelli Theoem e have (3.24) I(k, u, v) = k(x,y) >γ u(x) >α v(y) >β dµ(x) dν(y) dγ dα dβ. We can obtain thee ue bounds of the double integal ove X Y in (3.24) by successively elacing each of the thee indicato functions by. This ocedue yields k(x,y) >γ u(x) >α v(y) >β dµ(x) dν(y) U(α)V (β), k(x,y) >γ u(x) >α v(y) >β dµ(x) dν(y) K µ (γ)v (β), k(x,y) >γ u(x) >α v(y) >β dµ(x) dν(y) K ν (γ)u(α), hee (3.25) U(α) = V (β) = u(x) >α dµ(x), K µ (γ) = ess su y Y v(y) >β dν(y), K ν (γ) = ess su x X k(x,y) >γ dµ(x), k(x,y) >γ dν(y). By then using the minimum of these thee ue bounds in (3.24) e obtain (3.26) I(k, u, v) min U(α)V (β), K µ (γ)v (β), K ν (γ)u(α) dγ dα dβ. Hyothesis (3.9) on k imlies that fo evey γ (, ) e have K µ (γ) C µ γ, K ν(γ) C ν γ. When these bounds ae laced into (3.26) e obtain (3.27) I(k, u, v) min U(α)V (β), C µv (β) γ, C νu(α) dγ dα dβ. γ

20 2 C. DAVID LEVERMORE The integal ove γ in (3.27) can be evaluated exactly. When U(α)V (β) = it vanishes. When < C ν U(α) C µ V (β) e obtain min U(α)V (β), C µv (β) γ Similaly, hen < C µ V (β) C ν U(α) e obtain min U(α)V (β), C µv (β), C νu(α) γ γ, C νu(α) ( Cν V (β)) dγ = U(α)V (β) dγ + U(α) γ ) = C ν U(α)V (β) ( + = C ν U(α)V (β). dγ = C µ U(α) V (β). ( Cν V (β)) C ν γ dγ Because heneve U(α)V (β) > e have that C ν U(α)V (β) < C µ U(α) V (β) if and only if C ν U(α) < C µ V (β), e see that in all cases min U(α)V (β), C µv (β) γ, C νu(α) γ dγ = min C ν U(α)V (β), C µ U(α) V (β). When this evaluation is laced into (3.27), e obtain (3.28) I(k, u, v) min C ν U(α)V (β), C µ U(α) V (β) dβ dα. At this oint e ould like to bound the ight-hand side of (3.28) using only the fact that u and v satisfy the nomalizations (3.23). Definitions (3.25) of U(α) and V (β), the Fubini-Tonelli Theoem, and ou nomalizations (3.23) fo u and v imly that α U(α) dα = α u(x) >α dµ(x) dα = α u(x) >α dα dµ(x) (3.29) u(x) = α dα dµ(x) = u(x) dµ(x) =, β q V (β) dβ = β q v(y) >β dν(y) dβ = β q v(y) >β dβ dν(y) v(y) = β q dβ dµ(y) = q v(y) dν(y) = q. We theefoe e ould like to bound the ight-hand side of (3.28) using only these identities. It is clea fom (3.28) that fo evey λ (, ) e have the bound [ λ q α q ] I(k, u, v) C ν U(α)V (β) dβ dα + C µ U(α) V (β) dβ dα (3.3) = [ λ q α q C ν U(α)V (β) dβ dα + λ q α q λ β q C µ U(α) V (β) dαdβ ].

21 INTEGRAL OPERATORS 2 Next e aly the Hölde inequality to bound the inne integals above as λ q α q ( λ q α q ) ( V (β) q ) dβ β dβ β q V (β) dβ, (3.3) λ β q ( λ β q ) ( U(α) ) dα α dα α U(α) dα Because elation (3.8) imlies that q (3.32) λ q α q β α q λ β q dβ = q dα = q = q ) (λ q q α q ) (λ q q β and. =, e see that ) = (λ α, q ) = (λ q β q. Hence, by using (3.29) and (3.32) to evaluate the ight-hand sides of (3.3), e obtain λ q α q ( ) ( ) V (β) dβ λ α, q q (3.33) λ β q ( ) ( ) U(α) q dα λ q β q. Uon lacing these esults into (3.3) and again using (3.29) to evaluate the emaining integals, e see that fo evey λ (, ) e have the bound [ ( ] (3.34) I(k, u, v) ) q C ν λ + (C ) µ λ. q The ight-hand side of (3.34) attains its minimum ove λ (, ) hen ( ) ( ) C ν λ C µ λ =. We can use the fact that + =, hich follos fom elation (3.8), to solve the above equation and find that this minimum is attained at λ = (C µ /C ν ) ( / ). By lacing this value of λ into (3.34) and again using elation (3.8), e obtain [ ( ) ( ) I(k, u, v) ( ) ( q + q = q = q = q [ ( ( ( ) + ) + ) + ( ) + ( ) ]C ) ( ) + [ q + ] ( ) + C µ C ν. ( C µ C ν µ C ν ) + ] C µ C ν Theefoe the thid Hady-Littleood bound (3.2) holds ith C,q, given by (3.2).

22 22 C. DAVID LEVERMORE 4. Convolution Oeatos Let (G, +) be an Abelian gou ith Haa measue dm defined ove the σ-algeba Σ m. (Recall that the Haa measue is a ositive measue that is tanslation invaiant; it is unique u to a ositive constant facto.) Given to functions and u defined ove G, e define thei convolution to be the function u that is fomally given by (4.) u(y) = (y x) u(x) dm(x). This can be vieed as an integal oeato of the fom (.) hee X = Y = G, dµ = dν = dm, Σ µ = Σ ν = Σ m and k(x, y) = (y x). Such oeatos ae called convolution oeatos. In this setting, is called the convolution kenel. In this section e deive bounds that ensue the convolution (4.) mas beteen eithe classical Lebesgue saces o eak Lebesgue saces. We ill stat by secializing the Young integal oeato bound (2.29), and the Hady-Littleood bounds (3.8), (3.2) and (3.2) to this setting. We ill then give a Caldeon-Zygmund bound in the setting hee G = R D and dm is Lebesque measue ove R D. 4.. Young Convolution Inequality. The Young convolution inequality follos diectly fom Young integal oeato bound (2.29). Coollay 4.. Let, q, [, ] satisfy the elation (4.2) + q + = 2. Fo evey u L (dm), v L q (dm), and L (dm) e have (4.3) (y x) u(x) v(y) dm(x) dm(y) u L v L q L. Poof. Because k(x, y) = (y x) and L (dm), e see that k L (L )(dm, dm) ith k L (dm;l (dm)) = L (dm) <. Because elation (4.2) imlies both that, q [, ] and that elation (2.28) holds, it theefoe follos fom (2.29) that the Young convolution inequality (4.3) holds Hady-Littleood-Sobolev Inequalities. These inequalities ae secializations of the Hady-Littleood inequalities to convolution kenels k(x, y) = (y x). In the Young convolution inequality (4.3) the function sits in L (dm). When (, ) the Hady-Littleood- Sobolev inequalities allos this class to be extended to L (dm). The fist Hady-Littleood-Sobolev inequality is an immediate coollay of the fist Hady- Littleood bound (3.8). Coollay 4.2. Let (, ). Fo evey u L (dm) and L (dm) e have (4.4) u L u L L. Poof. Because k(x, y) = (y x) and L (dm), e see that k L (dm; L (dm)) ith k L (dm;l (dm)) = L (dm) <. It theefoe follos fom (3.8) that the fist Hady- Littleood-Sobolev inequality (4.4) holds. The second Hady-Littleood-Sobolev inequality is an immediate coollay of the second Hady-Littleood bound (3.2).

23 INTEGRAL OPERATORS 23 Coollay 4.3. Let, q, (, ) that satisfy elation (4.2). Then thee exists a ositive constant C,q, G, such that fo evey u L (dm) and L (dm) e have (4.5) u L Hee e ill establish (4.5) ith C,q, G, [u] L [] L. (4.6) C,q, G, =. q Remak. Notice that C,q, G, given by (4.6) is univesal in the sense that it is indeendent of G. Poof. Because k(x, y) = (y x) and L (dm), e see that k L (dm; L (dm)) ith C m = [] L (dm) <. Because elations (4.2) and (3.) ae the same, it follos fom (3.2) that the second Hady-Littleood-Sobolev inequality (4.5) holds. Fomula (4.6) fo C,q, G, follos fom fomula (3.3). The thid Hady-Littleood-Sobolev inequality is an immediate coollay of the thid Hady- Littleood bound (3.2). Coollay 4.4. Let, q, (, ) that satisfy elation (4.2). Then thee exists a ositive constant C,q, G such that fo evey u L (dm), v L q (dm), and L (dm) e have (y (4.7) x) u(x) v(y) dm(x) dm(y) C,q, G u L v L q [] L. Hee e ill establish (4.7) ith (4.8) C,q, G = q ( ) + ( ) + q 2. Remak. Notice that C,q, G given by (4.8) is univesal in the sense that it is indeendent of G. Fo a discussion of sha values fo C,q, G hen G = R D see [3]. Poof. Because k(x, y) = (y x) and L (dm), e see that k L (dm; L (dm)) ith C m = [] L (dm) <. Because elations (4.2) and (3.8) ae the same, it follos fom (3.2) that the thid Hady-Littleood-Sobolev inequality (4.7) holds. Fomula (4.8) fo C,q, G follos fom fomula (3.2) Caldeon-Zygmund Inequality. The eceding theoy cannot be alied to integal oeatos ith moe singula kenels such as the classical Hilbet tansfom H, hich is fomally defined fo functions ove R by (4.9) Hu(y) = PV π u(y x) x Hee the PV indicates that the integal is undestood in the sense of incile value, namely as the limit [ (x) ǫ (x) ] (4.) PV dx = lim x ǫ + x dx + (x) ǫ x dx. Such a limit ill exist hen sufficiently egula nea x = because /x takes diffeent signs on eithe side of x =, hich leads to cancellation. Caldeon-Zygmund theoy bounds integal oeatos ith singula convolution kenels that ae on the bodeline of being locally integable ovided thee is cancellation nea the singulaity. We ill not give a vey geneal esult hee. Rathe, e ill give ithout oof a secial esult that has ide alicability [4]. dx.

24 24 C. DAVID LEVERMORE We secialize to the case in hich G = R D and dm is the usual Lebesgue measue on R D. Caldeon-Zygmund theoy imlies the folloing. Theoem 4.. Let be a comlex-valued function ove R D that has the factoed fom ( ) z (4.) (z) = h( z ) j, z hee h is Lischitz continuous aay fom z = and satisfies su z D h( z ) : z > <, hile j is Lischitz continuous ove S D and satisfies the cancellation condition (4.2) j(o) ds(o) =. S D Hee ds denotes the usual Lebesgue suface measue on S D. Fo evey ǫ > define the function ǫ by ǫ (z) = z >ǫ (z), and the oeato K ǫ by (4.3) K ǫ u(y) = ǫ (y x) u(x) dm(x). Then fo evey (, ) thee exists a ositive constant C that is indeendent of ǫ such that fo evey ǫ > the oeato K ǫ satisfies the bound (4.4) K ǫ u L C u L fo evey u L (dm), Moeove, fo evey u L (dm) the limit (4.5) Ku = lim ǫ K ǫ u exists in L (dm), and the oeato K so defined satisfies the bound (4.6) Ku L C u L fo evey u L (dm) Summay. Ou esults egading the convolution of to functions ae summaized in the folloing table. L L q L fo, q, [, ] such that + q = +. L L L fo (, ). L L q L fo, q, (, ) such that + = +. q L Lq L fo, q, (, ) such that + q = +. CZ L L fo (, ). The fist item follos fom the Young convolution inequality, the second fom the fist Hady- Littleood-Sobolev inequality, the thid fom the second Hady-Littleood-Sobolev inequality, the fouth fom the thid Hady-Littleood-Sobolev inequality, and the last fom the Caldeon- Zygmund inequality, hee CZ denotes all functions of the Caldeon-Zygmund fom (4.). Remak. The ange given fo the second item in the table cannot be educed to L so as to be consistent ith the fouth item ith q =. Indeed, let u(x) = x D fo some (, ) and v(x) be a ositive, smooth, aidly deceasing function. One can sho that lim x D x y D v(y) dm(y) = v(y) dm(y) >. x Hence, u L, v L, but u v L.

25 INTEGRAL OPERATORS 25 Remak. The ange given fo the thid item in the table cannot be educed to L. Indeed, let, q, (, ) such that + = +. Set u(x) = D q x and v(x) = x D q. One can sho that fo some C > u v(x) = x y D y D q dm(y) = C x D. Hence, u L, v L q, but u v L. 5. Examles fom Patial Diffeential Equations Hee e aly the foegoing theoy to examles of integal oeatos that aise in the study of atial diffeential equations. Examle. Fo D > 2 the Geen function g of x ove R D is given by g(x) = S D x D+2. If u is the solution of the Poisson equation x u = f fo some f L (dm) then fomally hee Because e see that u = g f, x u = ( x g) f, 2 x u = ( 2 x g) f, x g(x) = D 2 S D x D+ x x, 2 x g(x) = D 2 ( S D x D D x x ) I. x 2 x g(x) = D 2 S D x D+, x 2 (D 2)(D ) g(x) = x D, S D g L D D 2 (dm), x g L D D (dm), x 2 g CZ(dm), hee CZ(dm) denotes the set of all functions that have the Caldeon-Zygmund fom (4.). Hence, if f L (dm) then u L D D 2 (dm) x u L D D (dm) 2 x u L (dm) hen (, D 2 ), hen (, D), hen (, ). The last esult shos that solutions of the Poisson equation gain to deivatives. Examle. The Geen function g of x + κ 2 ove R 3 is given by g(x) = 4π e κ x x If u is the solution of x u + κ 2 u = f fo some f L (dm) then fomally hee u = g f, x u = ( x g) f, 2 x u = ( 2 x g) f, x g(x) = 4π 2 x g(x) = 4π e κ x x x ( + κ x ) 2 x, e κ x ( + κ x ) x 3 ( 3 x x I x 2. ) + κ2 4π e κ x x x x x 2.

26 26 C. DAVID LEVERMORE Because e see that x g(x) = 4π e κ x ( + κ x ), x 2 g L q (dm) fo evey q [, 3) and g L 3 (dm), Hence, if f L (dm) then x g L q (dm) fo evey q [, 3 2 ) and xg L 3 2 (dm). u L (dm) x u L (dm) fo evey [, ] hen ( 3, ), 2 fo evey [, ) hen = 3, 2 3 fo evey [, 3 2 ) hen (, 3), 2 fo evey [, 3) hen =, fo evey [, ] hen (3, ), fo evey [, ) hen = 3, 3 fo evey [, ) hen (, 3), 3 fo evey [, 3) hen =, 2 In aticula, e see that u L (dm) and x u L (dm). Finally, notice that x 2 g = H (x) + H 2 (x) hee H and H 2 ae the matix-valued functions H (x) = ( e κ x ( + κ x ) 3 x x ) I, H 4π x 3 x 2 2 (x) = κ2 e κ x x x. 4π x x 2 Because H (x) = e κ x ( + κ x ), H 2π x 3 2 (x) = κ2 4π e see that H CZ(dm) hile e κ x H 2 L q (dm) fo evey q [, 3), and H 2 L 3 (dm). In aticula, e see that if f L (dm) then x u = H f + H 2 f L (dm), hen (, ). Theefoe, as ith the Poisson equation, solutions of x u + κ 2 u = f gain to deivatives. 2 Refeences [] G.B. Folland, Real Analysis: Moden Techniques and Thei Alications, Second Edition, John Wiley & Sons, Ne Yok, Ne Yok, USA, 999. [2] G.H. Hady, J.E. Littleood, and G. Pólya, Inequalities, Cambidge Univesity Pess, Cambidge, UK, 952. [3] E.H. Lieb and M. Loss, Analysis, Gaduate Studies in Mathematics 4, Ameican Mathematical Society, Povidence, Rhode Island, USA, 997. [4] E.M. Stein, Singula Integals and Diffeentiability Poeties of Functions, Pinceton Univesity Pess, Pinceton, Ne Jesey, USA, 97. (C.D. Levemoe) Deatment of Mathematics and Institute fo Physical Sciences and Technology, Univesity of Mayland, College Pak, MD , USA addess: lvm@math.umd.edu x,