1 Riesz Potential and Enbeddings Theorems

Transcription

1 Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for all u L R Assume for simlicity that u C c R, so that I u is well-defined, and for r > 0 define the rescaled function If holds for u r, we get R R u ry x y u r x := u rx, x R dy u r y = R R x y c u r x R = c u rx R or, euivalently, after the change of variables z := rx, w := ry, R r R r r u w z w dw dz dy, c r u z dz, R that is, R I u z dz cr + R u z dz If + > 0, let r 0+ to conclude that u 0, while if + < 0, let r to conclude again that u 0 Hence, the only ossible case is when = So in order for to be ositive, we need <, in which case, :=

2 The Subcritical Case < Theorem Let 0 < <, <, and let u L R Then :=, i I u x is well-defined and real valued for L -ae x R, ii if =, then for any t > 0, iii if >, then L { x R : I u x > t } C, t u L R, I u L R C,, u L R 2 Part iii has already been roved in Proosition C3, but we reeat the roofs since we will need to kee track of the constants The roof is due to Hedberg [] and uses the maximal function of u see Definition C27 We refer to [3] for an alternative roof and for more information on the Riesz otential We begin with a reliminary lemma, which is due to Tartar Lemma 2 Let u L R,, and let v L R be such that v x g x for L -ae x R, where g L R is a radial function of the form g x = f x, with f : [0, [0, decreasing Then v x y u y dy g L R M u x R for L -ae x R Proof By the hyotheses on v, v x y u y dy g x y u y dy R R Ste : Assume first that f = χ [0,r, so that g = χ B0,r Then g x y u y dy = u y dy L B x, r M u x R Bx,r = g L R M u x 2

3 Ste 2: ext, consider the case in which f = a i χ [ri,r i, i= where 0 =: r 0 < r < < r n and a > a 2 > > a n Set c i := a i a i+ > 0, i =,, n, where a n+ := 0 Then we can write and In turn, and so R g x y u y dy R f t dt = f = c i χ [0,ri i= a i r i r i = i= g = c i χ B0,ri i= c i r i i= c i χ B0,ri x y u y dy R i= i= c i L B x, r i M u x = g L R M u x, where in the second ineuality we have used Ste Ste 3: The general case follows by observing that every increasing function f : [0, [0, can be aroximated from below by an increasing seuence of simle functions of the tye given in Ste 2 We turn to the roof of Theorem Proof of Theorem Fix r > 0 and for x R write u y I u x = dy Bx,r x y u y + dy =: I + II R \Bx,r x y To estimate I, we aly the revious lemma, taking g x = f x, where { t f t := if 0 < t < r, 0 otherewise Then g L R = β r 0 t t dt = β r, 3

4 and so I g L R M u x = β r M u x On the other hand, using Hölder s ineuality for >, II β = r β / t + dt u L R / r u L R For =, we use instead the fact that x y r ineuality II r u L R Hence, we have roved that to obtain the simler I u x β r M u x + C,, r u L R Fix ε > 0 and choose r := u L R M u x + ε Then I u x β + C,, u M u x + L R ε Since <, letting ε 0+ gives I u x β + C,, u M u L R x 3 ote that in view of Theorem C29i, the revious ineuality imlies that I u x is well-defined and real valued for L -ae x R To rove art ii, assume that = Then by 3, if I u x > t, then t < I u x β = + β + u L R M u x, u L R M u x = and so { x R : I u > t } x R : M u x > t β + u L 4

5 It follows from Theorem C29ii that L { x R : I u > t } 3 β + t u L R Finally, if >, then taking the L norm on both sides of 3 gives I u L R β + C,, u M u x L R R Since =, by Theorem C29iii, we have I u L R This concludes the roof C,, u L R u = C,, u L R L R Examle 3 The ineuality 2 does not hold for = To see this, consider a standard mollifier ϕ ε Then ϕ ε y I ϕ ε x = R x y dy = v ϕ ε x, where v x := x Hence, by Theorem C9ii, I ϕ ε x for L -ae x R If 2 x holds, then I ϕ ε C, ϕ L R ε L R By Fatou s lemma and Theorem C9iv, R x lim inf ε 0 + I ϕ ε L R C, lim ϕ ε ε 0 + L R = C,, R x which is a contradiction, since the integral on the left-hand side is infinite and the one on the right-hand side is finite As a corollary of Theorem, we obtain an alternative roof of the Sobolev Gagliardo iremberg theorem in the case < < 5

6 Theorem 4 Sobolev Gagliardo irenberg s embedding theorem Let < Then there exists a constant C = C, > 0 such that for every function u L, R vanishing at infinity, u x R C u x 4 R In articular, W, R is continuously embedded in L R for all Proof Assume that < < and as in the roof of Theorem 2, that u L R C R with u L R ; R By Exercise 620, for x R, u x u y = I β u x 5 R x y β It is enough to aly Theorem with = 2 The Critical Case ext we discuss the critical case = When u L R, then I u x is not finite for L -ae x R Exercise 5 Prove that the function u x := x log x χ R \B0,2, x R, belongs to L R but I u x = for all x R ote that the roblem is the behavior at To overcome this roblem, there are two alternatives: One should either restrict attention to functions u L R with comact suort, or modify the Riesz otential by considering one of the following variants [ Î u x := R x y χ ] B0, y x y u y dy, x R or, for 0 < <, Ĩ u x := R [ ] x y x 0 y u y dy, x R In what follows, we consider functions with comact suort When +, we have that =, however if u L R, then one cannot conclude that I u belongs to L R 6

7 Exercise 6 Prove that for ε > 0 suffi ciently small the function u x := belongs to L R but I u 0 = χ x log +ε B0, 2, x R, x Theorem 7 Let 0 < < and let u L R \ {0} have suort in a ball B x 0, R Then for every γ 0, β there exists a constant C γ = C γ > 0 such that I u x ex γ C γ R 6 Bx 0,R u L First roof Without loss of generality, we may assume that u L = We roceed as in the first art of the roof of Theorem, with the only difference that we take x B x 0, R The estimate for I does not change, while to estimate II, note that B x 0, R B x, 2R, so that, if 0 < r 2R, u y II = dy Bx,2R\Bx,r x y 2R β t dt u L = β log 2R r r where we used Hölder s ineuality and the fact that = hand, if r > 2R, then II = 0 Hence, we have roved that I u x β r M u x + β log 2R r for x B x 0, R and 0 < r 2R, and I u x β r M u x for x B x 0, R and r > 2R Fix ε, δ > 0 and choose { } δ r := min, 2R M u x + ε β On the other 7

8 Then I u x δ + = δ + δ + β log + [ β δ 2R M u x + ε ] [ ] β log+ 2R M u x + ε β δ β log [ + Since γ < β, there exists ρ > 0 so large that If I u x + ρ δ, then and so ] 2R M u x + ε β δ γ < ρ β + ρ I u x δ I u x + ρ I u x = ρ + ρ I u x, γ I u x < β ρ In turn, log [ + ρ + I u x β δ ex γ I u x + 2R M u x + ε I u x δ β ] 2R M u x + ε β δ Letting ε 0 + gives ex γ I u x + 2R M u x β δ On the other hand, if I u x < + ρ δ, then ex γ I u x ex γ + ρ δ Hence, Bx 0,R ex γ I u x C γ R + 2R M u x β δ Bx 0,R 8

9 The result now follows from Theorem C29iii We now resent a second roof, which does not rely on maximal functions, but does not give as shar a constant γ The following two lemmas are taken from a aer of Serrin [2] Lemma 8 Let 0 < <, let, and let u L R \ {0} have suort in a ball B x 0, R Then I u L Bx β 0,R 2R + u L Bx 0,R Proof Fix x B x 0, R Using the fact that B x 0, R B x, 2R, we have dy dy Bx 0,R x y Bx,2R x y Hence, also by Tonelli s theorem, I u x = Bx 0,R = β 2R Bx 0,R β 2R 0 u y r dr = β 2R Bx 0,R Bx 0,R dy x y u y dy β 2R R u L Bx 0,R, where in the last ineuality we have used Hölder s ineuality Lemma 9 Let 0 < <, let >, and let u L R \ {0} have suort in a ball B x 0, R Then for all x B x 0, R I u x 2R u L Bx 0,R Proof Fix 0 < ε < 2R and x B x 0, R and for t [ε, 2R], define φ t := u y dy Bx,t\Bx,ε Using olar coordinates and Fubini s theorem we have that φ t = t ε r S u y r, ω dh ω dr ote that this shows that φ is absolutely continuous in [ε, 2R] Similarly, we have that the function u y t r F t := dy = Bx,t\Bx,ε x y ε r u y r, ω dh ω dr S 9

10 is absolutely continuous in [ε, 2R], with F t = t φ t for L -ae t [ε, 2R] By the fundamental theorem of calculus and integration by arts, we have F 2R = F 2R F ε = 2R = 2R φ 2R ε F r dr = 2R On the other hand, by Hölder s ineuality, φ r = u y dy u L and so F 2R Bx,r\Bx,ε 2 R + ε 2R ε r φ r dr r φ r dr r, 2R u L + u L ε r + dr 2R u L + = 2R u L 2R u L Using the fact that B x 0, R B x, 2R, we have that u y dy = F 2R Bx 0,R\Bx,ε x y 2R u L Letting ε 0 + and using Lebesgue monotone convergence theorem gives the desired result Second roof The roof follows essentially Theorem 2 in [4] Without loss of generality, we may assume that u L = Let = and write = θ + θ 2, 7 where 0 < θ, θ 2 < Then, given >, for f L B x 0, R, we have f x u y f x u y = x y x y θ f x x y θ 2 0

11 By Hölder s ineuality, we have f x u y f x dy dy θ2 Bx 0,R Bx 0,R x y Bx 0,R Bx 0,R x y u y f x Bx 0,R Bx 0,R x y =: I I 2 By Lemma 8 where and there are relaced here by and θ 2, θ dy I β θ 2 2R θ2+ f L On the other hand, if > θ, by Lemma 9 where and there are relaced here by and θ, we have that Hence, Bx 0,R Bx 0,R Bx 0,R f x x y θ where we have used the fact that θ f x u y x y dy 2R θ θ = 2Rθ f L β θ 2 f L θ u L, by 7 Taking the suremum over all f L B x 0, R, we get Taking Bx 0,R I u x 2R β θ 2 θ 2 :=, we have that θ 2 < <, while from 7, Moreover, 8 becomes Bx 0,R > θ = + > I u x 2R β θ u L 8 u L

12 Taking = k, where k, we obtain k! γk I u x k Since Bx 0,R k= [ 2R k k u L γ β k! k= =: 2R u L x k x k+ = + k γ β x k k Bx 0,R k= β eγ ] k it follows that if γ <, eβ then the series converges Hence, ex γ I u x C γ R u L, Using Theorem 7, we can rove Trudinger s embedding theorem We recall that γ := β, 9 Theorem 0 Suose 2 and let u W, R \{0} have suort in a ball B x 0, R Then for every γ 0, γ there exists a constant C = C, γ > 0 such that u x ex γ Bx 0,R u C γ R L Proof By 5, γ u x γ β I u x Hence if γ < γ = β, then γ < β β, and so we are in a osition to aly Theorem 7 with a =, to conclude that Bx 0,R u x ex γ u L Bx 0,R ex γ β I u x u L C γ R Remark I am unable to find a simle roof of Theorem 29, which does not make use of symmetrization Both roofs of Theorem 7 rely strongly on the fact that u has comact suort 2

13 References [] LI Hedberg, On certain convolution ineualities, Proc Amer Math Soc , [2] J Serrin, A remark on the Morrey otential, Control methods in PDEdynamical systems, , Contem Math, 426, Amer Math Soc, Providence, RI, 2007 [3] EM Stein, Singular integrals and diff erentiability roerties of functions, Princeton Mathematical Series, no 30, Princeton University Press, Princeton, J, 970 [4] S Trudinger, On imbeddings into Orlicz saces and some alications, J Math Mech 7 967,