This talk was given during the Jozef Marcinkiewicz Centenary Conference

Stokolos, Alexander M.

English

Georgia Southern University: Digital Commons@Georgia Southern

Georgia Southern University Digital Commons@Georgia Southern Mathematical Sciences Faculty Presentations Mathematical Sciences, Department of 6-28-2010 Lp Estimates for the Maximal Functions Alexander M. Stokolos Georgia Southern University, astokolos@georgiasouthern.edu Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/math-sci-facpres  Part of the Mathematics Commons Recommended Citation Stokolos, Alexander M.. 2010. "Lp Estimates for the Maximal Functions." Mathematical Sciences Faculty Presentations. Presentation 3. source: http://condor.depaul.edu/astokolo/Marcinkiewicz.pdf https://digitalcommons.georgiasouthern.edu/math-sci-facpres/3 This presentation is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Presentations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact digitalcommons@georgiasouthern.edu. Lp estimates for the maximal functionsAlex Stokolos, Georgia Southern UniversityDedicated to Olga Vladimirovna Voskoboynik(July 1, 2010)Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions!M!(x) = supI!x ;!!"I, I # dyadic.! Hardy–Littlewood–Doob–Cox$M!$p < q$!$p, ! %= 0, 1 < p < &.! Nazarov and Treil (Algebra i Analiz (1996), 32-162):B(f ,F , L) = sup0"!#Lploc(R)!!(M!)p"Q: !!"Q= f ;!!p"Q= F ; supR$Q!!"R = L".Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions!M!(x) = supI!x ;!!"I, I # dyadic.! Hardy–Littlewood–Doob–Cox$M!$p < q$!$p, ! %= 0, 1 < p < &.! Nazarov and Treil (Algebra i Analiz (1996), 32-162):B(f ,F , L) = sup0"!#Lploc(R)!!(M!)p"Q: !!"Q= f ;!!p"Q= F ; supR$Q!!"R = L".Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions!M!(x) = supI!x ;!!"I, I # dyadic.! Hardy–Littlewood–Doob–Cox$M!$p < q$!$p, ! %= 0, 1 < p < &.! Nazarov and Treil (Algebra i Analiz (1996), 32-162):B(f ,F , L) = sup0"!#Lploc(R)!!(M!)p"Q: !!"Q= f ;!!p"Q= F ; supR$Q!!"R = L".Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions! Nazarov and Treil (1996): B(f ,F , L) ' qpF # pqfLp%1 + pLp.! Melas (Advances in Mathematics (2005), 310-340) foundB(f ,F , L).! In particular case p = 2:B(f ,F , L) =#((F +(F + L2 # 2Lf )2 for L ' 2fL2 + 4(F # f 2) for L ) 2f! Slavin, Stokolos and Vasyunin (C. R. Math. Acad. Sci.(2008), 585-588) suggested the alternative approach based onapplication of the Monge-Ampére PDE.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions! Nazarov and Treil (1996): B(f ,F , L) ' qpF # pqfLp%1 + pLp.! Melas (Advances in Mathematics (2005), 310-340) foundB(f ,F , L).! In particular case p = 2:B(f ,F , L) =#((F +(F + L2 # 2Lf )2 for L ' 2fL2 + 4(F # f 2) for L ) 2f! Slavin, Stokolos and Vasyunin (C. R. Math. Acad. Sci.(2008), 585-588) suggested the alternative approach based onapplication of the Monge-Ampére PDE.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions! Nazarov and Treil (1996): B(f ,F , L) ' qpF # pqfLp%1 + pLp.! Melas (Advances in Mathematics (2005), 310-340) foundB(f ,F , L).! In particular case p = 2:B(f ,F , L) =#((F +(F + L2 # 2Lf )2 for L ' 2fL2 + 4(F # f 2) for L ) 2f! Slavin, Stokolos and Vasyunin (C. R. Math. Acad. Sci.(2008), 585-588) suggested the alternative approach based onapplication of the Monge-Ampére PDE.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsLp estimates for the maximal functions! Nazarov and Treil (1996): B(f ,F , L) ' qpF # pqfLp%1 + pLp.! Melas (Advances in Mathematics (2005), 310-340) foundB(f ,F , L).! In particular case p = 2:B(f ,F , L) =#((F +(F + L2 # 2Lf )2 for L ' 2fL2 + 4(F # f 2) for L ) 2f! Slavin, Stokolos and Vasyunin (C. R. Math. Acad. Sci.(2008), 585-588) suggested the alternative approach based onapplication of the Monge-Ampére PDE.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!B(x1, x2) = inf!&0{!(!"Q: !!"Q= x1; !!2"Q = x2}.!B(cx1, c2x2) = cB(x1, x2) * B(x1, x2) =(x1B$1,x2x21%.!B(x1, x21 ) =(x1.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!B(x1, x2) = inf!&0{!(!"Q: !!"Q= x1; !!2"Q = x2}.!B(cx1, c2x2) = cB(x1, x2) * B(x1, x2) =(x1B$1,x2x21%.!B(x1, x21 ) =(x1.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!B(x1, x2) = inf!&0{!(!"Q: !!"Q= x1; !!2"Q = x2}.!B(cx1, c2x2) = cB(x1, x2) * B(x1, x2) =(x1B$1,x2x21%.!B(x1, x21 ) =(x1.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!B(x1, x2) =(x1C$x2x21%, C (1) = 1.! Assuming that Bx1x1Bx2x2 = B2x1x2 we get an ODE for thefunction C with the boundary condition C (1) = 1.! The solution C (t) = t%1/2 givesB(x1, x2) = x1&x1x2.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!B(x1, x2) =(x1C$x2x21%, C (1) = 1.! Assuming that Bx1x1Bx2x2 = B2x1x2 we get an ODE for thefunction C with the boundary condition C (1) = 1.! The solution C (t) = t%1/2 givesB(x1, x2) = x1&x1x2.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!B(x1, x2) =(x1C$x2x21%, C (1) = 1.! Assuming that Bx1x1Bx2x2 = B2x1x2 we get an ODE for thefunction C with the boundary condition C (1) = 1.! The solution C (t) = t%1/2 givesB(x1, x2) = x1&x1x2.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!Bx1,x1 > 0 * concavity up!B'n(k=1"kxk)'n(k=1"kB(xk) ifn(k=1"k = 1.!B(x1, x2) = B*+(I'[0,1]|I |(!!"I, !!2"I),- '(I'[0,1]|I |B(!!"I, !!2"I) =!(I'[0,1]|I |!!"3/2I!!2"1/2I+(I'[0,1]!!"1/2I|I |,! (!"Q.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!Bx1,x1 > 0 * concavity up!B'n(k=1"kxk)'n(k=1"kB(xk) ifn(k=1"k = 1.!B(x1, x2) = B*+(I'[0,1]|I |(!!"I, !!2"I),- '(I'[0,1]|I |B(!!"I, !!2"I) =!(I'[0,1]|I |!!"3/2I!!2"1/2I+(I'[0,1]!!"1/2I|I |,! (!"Q.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!Bx1,x1 > 0 * concavity up!B'n(k=1"kxk)'n(k=1"kB(xk) ifn(k=1"k = 1.!B(x1, x2) = B*+(I'[0,1]|I |(!!"I, !!2"I),- '(I'[0,1]|I |B(!!"I, !!2"I) =!(I'[0,1]|I |!!"3/2I!!2"1/2I+(I'[0,1]!!"1/2I|I |,! (!"Q.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function!Bx1,x1 > 0 * concavity up!B'n(k=1"kxk)'n(k=1"kB(xk) ifn(k=1"k = 1.!B(x1, x2) = B*+(I'[0,1]|I |(!!"I, !!2"I),- '(I'[0,1]|I |B(!!"I, !!2"I) =!(I'[0,1]|I |!!"3/2I!!2"1/2I+(I'[0,1]!!"1/2I|I |,! (!"Q.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function! Taking infimum to the right we getB(x1, x2) ' B(x1, x21 )! which implies!!"3/2Q!!2"1/2Q' !(!"Q! or$!$31 ' $!$ 12$!$22.! From Hölder$!$41 ' $!$ 12$!$332' $!$ 12$!$32.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function! Taking infimum to the right we getB(x1, x2) ' B(x1, x21 )! which implies!!"3/2Q!!2"1/2Q' !(!"Q! or$!$31 ' $!$ 12$!$22.! From Hölder$!$41 ' $!$ 12$!$332' $!$ 12$!$32.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function! Taking infimum to the right we getB(x1, x2) ' B(x1, x21 )! which implies!!"3/2Q!!2"1/2Q' !(!"Q! or$!$31 ' $!$ 12$!$22.! From Hölder$!$41 ' $!$ 12$!$332' $!$ 12$!$32.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation via Bellman Function! Taking infimum to the right we getB(x1, x2) ' B(x1, x21 )! which implies!!"3/2Q!!2"1/2Q' !(!"Q! or$!$31 ' $!$ 12$!$22.! From Hölder$!$41 ' $!$ 12$!$332' $!$ 12$!$32.Alex Stokolos, Georgia Southern University Lp estimates for the maximal functionsInterpolation in Orlicz classesTheorem (Slavin, S., Vasyunin).LetB(x1, x2) = inf!&0{!log(1+!)"Q: !!"Q= x1; !(1+!) log(1+!)"Q = x2}.ThenB(x1, x2) =x21 + w,where w is the unique solution of the equation(1 + w) log(1 + w)w=x2x1.in the domain (1,&).Alex Stokolos, Georgia Southern University Lp estimates for the maximal functions

L\u3csup\u3ep\u3c/sup\u3e Estimates for the Maximal Functions

https://digitalcommons.georgiasouthern.edu/cgi/viewcontent.cgi?article=1009&amp;context=math-sci-facpres

L\u3csup\u3ep\u3c/sup\u3e Estimates for the Maximal Functions

Abstract

Similar works

Full text

Available Versions

Georgia Southern University: Digital Commons@Georgia Southern