BEST CONSTANTS FOR TWO NON-CONVOLUTION INEQUALITIES

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1 BEST CONSTANTS FOR TWO NON-CONVOLUTION INEQUALITIES Michael Chris ad Loukas Grafakos Uiversiy of Califoria, Los Ageles ad Washigo Uiversiy Absrac. The orm of he oeraor which averages f i L ( ) over balls of radius δ x ceered a eiher 0 or x is obaied as a fucio of, ad δ. Boh iequaliies roved are -dimesioal aalogues of a classical iequaliy of Hardy i R. Fially, a lower boud for he oeraor orm of he Hardy-Lilewood maximal fucio o L ( ) is give. 0. Iroducio A classical resul of Hardy [HLP] saes ha if f is i L (R ) for >, he (0.) ( 0 ( x ) / f() d) dx ( ) / f() d x 0 0 ad he cosa /( ) is he bes ossible. By cosiderig wo-sided averages of f isead of oe-sided, (0.) ca be equivalely formulaed as: (0.) ( ( x ) / f() d) dx ( / f() d). x x We deoe by D(a, R) he ball of radius R i ceered a a. Le (Tf)(x) be he average of f L ( ) over he ball D(0, x ). The aalogue of (0.) for is he iequaliy: (0.3) Tf L C () f L Research arially suored by he Naioal Sciece Foudaio 99 mahemaics Subjec Classificaio. Primary 4B5 Tyese by AMS-TEX

2 for some cosa C () which deeds a riori o ad. Our firs resul is ha he bes cosa C () which saisfies (0.3) for all f L ( )is = /( ), he same cosa as i dimesio oe. Aoher versio of Hardy s iequaliy i wih he bes ossible cosa ca be foud i [F]. Nex we cosider a similar roblem. A equivale formulaio of (0.) ad (0.) is (0.4) ( ( x+ x ) / ( f() d) dx / f() d), x x x where f is i L (R ). Le (Sf)(x) be he average of f L ( ) over he ball D(x, x ). We comue he oeraor orm c, of S o L ( ) as a fucio of ad. The recise value of he cosa c, is give i Theorem. I secio 3 a lower boud for he oeraor orm of he Hardy-Lilewood maximal fucio o L ( ) is give. Fially, i secio 4 he orm o L ( ) of he oeraor which averages f over he ball of radius δ x ceered a eiher 0or x is give as a fucio of δ, ad, for ay δ>0. Throughou his oe, ω will deoe he area of he ui shere S ad v he volume of he ui ball i.. Hardy s iequaliy o. I his secio we will rove iequaliy (0.3) wih cosa C () = = /( ). We deoe by A he Lebesgue measure of he se A ad by χ A is characerisic fucio. Theorem. Le f L ( ), where <<. The followig iequaliy holds (.) ( ( D(0, x ) D(0, x ) ad he cosa = /( ) is he bes ossible. ) / f(y) dy) dx ( ) / f(y) dy, Proof Fix f L ( ). Wihou loss of geeraliy, assume ha f is oegaive ad coiuous. Le R + deoe he mulilicaive grou of osiive real umbers wih Haar

3 measure d. The fucio / χ [0,] is i L (R +, d ) wih orm /. For a fixed θ i he ui shere S, he fucio f(θ) / is i L (R +, d ). The grou iequaliy g K L g L K L gives: (.) ( 0 f(rθ)(r) ) ( d dr r ( f(rθ)r ) dr r )( ). Noe ha equaliy holds i (.) if ad oly if equaliy holds i g K L g L K L. This haes i he limi by he sequece g ɛ,n = χ [ɛ,n]. Sice g() = f(θ) /, we coclude ha equaliy is aaied i (.) i he limi by he sequece (.3) f ɛ,n (θ) = / χ ɛ N as ɛ 0ad N. Noe ha Tf is a radial fucio. Exressig all iegrals i olar coordiaes, we reduce (.) o a covoluio iequaliy o he mulilicaive grou R +. We have ( Tf r ) L ( ) = ω v r f(θ) dθd r dr =0 θ S = ω ( ) (.4) v f(rθ)(r) d dθ dr r. S We aly Hölder s iequaliy wih exoes + = o he fucios ad θ =0 f(rθ)(r)/ / d ad he Fubii s heorem o ierchage he iegrals i θ ad r. We obai ha (.4) is bouded above by ω ( ) (.5) v ω f(rθ)(r) d dr S =0 r dθ. Noe ha if f is a radial fucio he (.4) ad (.5) are ideical. We ow aly (.) o majorize (.5) by ω ( ) v S =0 f(rθ) r dr ( ) r dθ = f L ( ) usig he fac ha ω = v. We have ow obaied he iequaliy Tf L f L. Equaliy holds whe he family of fucios (.3) is radial. Therefore he exremal family for iequaliy (.) is x / χ ɛ x N,asɛ 0ad N. 3

4 . A varia of Hardy s iequaliy o. The derivaio of he -dimesioal aalogue of (0.4) is more suble. Le B(s, ) deoe he usual bea-fucio 0 x ( x) s dx. Our secod resul is Theorem. Le << ad c, = ω ω B( ( iequaliy holds for all f i L ( ): (.) ( ( D(x, x ) D(x, x ) ad he cosa c, is he bes ossible. f(y) dy) dx) / c, ), 3 ). The followig ( f(y) dy Proof. We use dualiy. Fix f ad g osiive ad coiuous wih f L ( ) ad g L ( ). We will show ha g(x)(sf)(x) dx c,. We exress boh g ad Sf i olar coordiaes by wriig x = rφ ad y = θ. The relaio x y x is equivale o θ φ /r. We obai g(x)(sf)(x) dx = = r v (.) = v v (S ) (S ) =0 v x f(y)g(x)χ D(x, x )(y) dx dy f(θ)g(rφ)χ φ θ /r d g(rφ)r ( [ ( G(φ) (S ) =0 =0 dr dφ dθ r f(rθ)(r) χφ θ f(rθ)(r) χφ θ d ) d dr dφ dθ r ) / ) ] / dr dφ dθ, r ) /. The brackeed exressio i (.) is he L orm where G(φ) = ( g(rφ) r dr r of he grou (R +, d ) covoluio of he fucio f(θ) wih he kerel χ[0,θ φ] () a r. We herefore esimae (.) by ( θ φ ) (.3) G(φ)F (θ) d dφ dθ, v (S ) 0 where F (θ) = ( ) /. f(rθ) r dr θ φ 0 r Le K(φ θ) = / d 0 = [(φ θ)+ ] /, where N + deoes he osiive ar of he umber N. Nex, we eed he followig: 4

5 Lemma. For ay F, G 0 measurable o S ad K 0 measurable o [, ], (.4) Proof. (S ) F (θ) G(φ) K(θ φ) dφ dθ F L (S ) G L (S ) S K(θ φ) dφ. We may assume ha all hree quaiies o he righ had side of (.4) are fiie. Sice K deeds oly o he ier roduc θ φ, he iegral S K(θ φ) dφ is ideede of θ. Hölder s iequaliy alied o he fucios F ad wih resec o he measure K(θ φ) dθ gives (.5) ( ) / ( F (θ)k(θ φ) dθ S F (θ) K(θ φ) dθ S K(θ φ) dθ S ) / We will ow use (.5) o rove (.4). The lef had side of (.4) is rivially esimaed by ( ( F (θ)k(θ φ) dθ S ) ) / G L S dφ (S ). Alyig (.5) ad Fubii s heorem we boud his las exressio by F L (S ) G L (S ) K(θ φ) dφ. The lemma S is ow roved. Observe ha equaliy is aaied i (.4) if ad oly if boh F ad G are cosas. We ow coiue wih he roof of Theorem. Alyig he lemma ad usig he ( fac ha F ad G have orm oe, we esimae (.3) by v S ) (θ φ) + dθ. To comue his iegral, we slice he shere i he direcio rasverse o φ. For coveiece we may ake φ = e =(, 0,, 0). The area of he slice cu by he hyerlae φ = s is ω ( s ) ad he weigh of his slice is ( s ). We ge ( (θ φ) + ) dθ = ω s ( s ) 3 ds = ω B( ( ) ), 3. S s=0 We ow use ha v = ω o ge he fial esimae c, i (.) which comlees he roof of (.). I remais o esablish ha he cosa c, is he bes ossible. For ay y, le A(y) be he sherical ca {θ S : θ y y }. This ca is oemy if ad oly if y /. For such y, he Lebesgue measure A(y) is ω / y ( s ) 3 ds. Le G() =χ [0,] () / ( s ) 3 ds. A easy comuaio shows ha G L (R +, d ) = 5.

6 ( ) 0 ( s ) 3 s ds. Le h = h ɛ,n be a eleme of he family x / χ ɛ x N ormalized o have L orm oe. We have ( Sh L ( ) = φ S v r h(y) dy) r dφ dr D(rφ,r) ( r = φ S v r θ S h(θ) dθd) r dφ dr =0 θ D(rφ,r) ( r = φ S v r A((r/)φ) h() d ) r dφ dr =0 r = ω ( ) (.7) ω h(r)(r) d G() r dr r. v The covoluio iequaliy g L L g L L L i he grou (R +, d ) wrie as ( ) ( (.8) h(r)(r) d dr G() =0 r h(r) r dr ) G r L (R +, d ) becomes a equaliy as ɛ 0ad N. Iserig (.8) i (.7) we obai Sh L ( ) ( ) ( ω v ( s ) 3 s ds) ω h(r) r dr = c, s=0 sice h L =0 =, ad equaliy is aaied as ɛ 0ad N. Theorem is ow roved. 3. A lower boud for he oeraor orm of he Hardy-Lilewood maximal fucio o L ( ). Le M(f)(x) = su r>0 (v r ) f(y) dy be he usual Hardy-Lilewood maximal fucio o. The family of fucios f ɛ,n (x) = x / χ ɛ x N is exremal y x r for Theorems ad. Le A, be he oeraor orm of M o L ( ). By comuig M(f ɛ,n ) L / f ɛ,n L ad leig ɛ 0ad N we obai a lower boud for A,. Proosiio. For <<, le A, be he bes cosa C ha saisfies he iequaliy Mf L ( ) C f L ( ) for all f i L. The (3.) A, ω ω su δ> ( ) s 3( s + s + δ ) ds δ 6

7 ad he suremum above is aaied for some δ = δ, always less ha. Proof. The followig is oly a skech. Sice x / is i L loc (R ), we ca calculae M( x / ) isead. Observe ha M( x / )=c x / where c = M( x / )(e ) ad e =(, 0,...,0). Also oe ha he suremum of he averages of x / over balls of radius r ceered a e is aaied for some r =+γ where γ>0. We herefore fid ha (3.) c = su γ>0 +γ v ( + γ) r dr Ar r, where A r = {θ S : rθ e < +γ}. Calculaio gives ha A r = ω for r γ ad A r = ω (r γ γ)/r ( s ) 3 ds for + γ>r>γ. We lug hese values io (3.) ad we ierchage he iegraio i r ad s: +γ r=γ s= r γ γ r r ( s ) 3 ds dr s+ s +γ +γ r = r ( s ) 3 r=γ dr r ds. We ow le δ = γ + ad obai (3.). Noe ha he cosa o he righ had side of (3.) reduces o he cosa c, of Theorem whe δ =. 4. Fial Remarks. We ed wih a coule of remarks. Le c, be he cosa of Theorem. We observe ha c,. This ca be show direcly or via he followig iequaliy which ca be foud i [HLP]: (4.) f(x)g(x) dx f(x) g(x) dx, where f ad g are iegrable ad f deoes he symmeric decreasig rearrageme of ay fucio f. Le T ad S be he oeraors of Theorems ad. The osymmeric decreasig rearrageme of he kerel of S is he kerel of T. Takig g o be he kerel of S ad f i L L i (4.), we obai he oiwise iequaliy Sf T f. I follows ha c,. 7

8 For ay δ>0, we defie varias T δ of T ad S δ of S as follows: (T δ f)(x) = f(y) dy ad (S δ f)(x) = D(0,δ x ) D(0,δ x ) f(y) dy. D(x, δ x ) D(x,δ x ) Sice (T δ f)(x) =(Tf)(δx) i is immediae ha he oeraor orm of T δ o L ( )is δ /. To comue he oeraor orm of S δ o L ( ), we reea he roof of Theorem verbaim. We obai he followig resul: Theorem. A. For δ>, he oeraor orm of S δ o L ( ) is ω ω δ ( s ( ) 3 s + s + δ ) ds B. For δ<, he oeraor orm of S δ o L ( ) is ω [ (s ω δ s= ( s ) 3 + s + δ ) ( s s + δ ) ] ds. δ (3.) is of course subsumed i coclusio A above. The secod auhor would like o hak Professor Al Baersei for simulaig his ieres i hese roblems ad also for may useful coversaios. Refereces [BT] A. Baersei II ad B.A. Taylor, Sherical rearragemes, subharmoic fucios ad -fucios i -sace, Duke Mah. J. 43 (976), o., [HLP] G. Hardy, J. Lilewood ad G. Pólya, Iequaliies, The Uiversiy Press, Cambridge, 959. [F] W. G. Faris, Weak Lebesgue saces ad Quaum mechaical bidig, Duke Mah. J. 43 (976), o., [PS] G. Pólya ad G. Szegö, Isoerimaric iequaliies i Mahemaical Physics, Priceo Uiv. Press, 95. [S] E. M. Sei, Sigular iegrals ad differeiabiliy roeries of fucios, Priceo Uiv. Press,

9 [SO] S. L. Sobolev, O a heorem of fucioal aalysis, Ma. Sb. (N.S.) 4 (938), o. 46, ; Eglish Traslaio: Amer. Mah. Soc. Trasl. () 34 (963), [SS] E. M. Sei ad J. O. Srömberg, Behavior of maximal fucios i for large, Arkiv för Ma. (983), [SW] E. M. Sei ad G. Weiss, Iroducio o Fourier Aalysis o Euclidea saces, Priceo Uiv. Press, 97. Dearme of Mahemaics, UCLA, Los Ageles, CA Dearme of Mahemaics, Washigo Uiversiy, S Louis, MO Curre address: Loukas Grafakos Dearme of Mahemaics, Uiversiy of Missouri-Columbia, Columbia, Missouri 65 9