On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms

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1 On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms Vladimir Maz ya Tatyana Shaposhnikova Abstract We prove the Gagliardo-Nirenberg type inequality c(n) ( p ) θ ( 1 s ) θ/p u θ p 1 1 θ W s,p u 1 θ L, where < θ < 1, < s < 1, 1 < p <, and u W s,p is the seminorm in the fractional Sobolev space W s,p ( ). The dependence of the constant factor in the right-hand side on each of the parameters s, θ, and p is precise in a sense. Let s (, 1) and let 1 < p. We introduce the space W s,p ( ) of functions in with the finite seminorm ( ) Rn u(x) u(y) p 1/p u W s,p = dxdy. x y n+sp Recently Bourgain, Brezis and Mironescu [1] found the relation lim s 1 (1 s) 1/p u W s,p = c u W 1,p, (1) which subsequently motivated Brezis and Mironescu to conjecture the Gagliardo- Nirenberg type inequality u W s/2,2p c(n, p)(1 s) 1/2p u 1/2 W s,p u 1/2 L (2) (see [2], Remark 5). In [2] one can also read: It would be of interest to establish c u θ W s,p u 1 θ L, < θ < 1, (3) with control of the constant c, in particular when s 1. Authors address: Department of Mathematics, University of Linköping, SE Linköping, Sweden. Both authors were supported by the Swedish Research Council 1

2 In the present paper we prove that (3) holds with c = c(n, p, θ)(1 s) θ/p, which, obviously, contains inequality (2) predicted by Brezis and Mironescu. Our proof is straightforward and rather elementary. In concluding Remarks 1 and 2 we show that the dependence of c on each of the parameters s, θ, and p is sharp in a certain sense. Theorem. For all u W s,p L there holds the inequality ( p ) θ ( 1 s ) θ/p u θ c(n) W p 1 1 θ s,p u 1 θ L, (4) where < s < 1, 1 < p <, and < θ < 1. Proof. Clearly, max{2 θ/p, 2 1 θ } u 1 θ L u θ W s,p. (5) Hence it suffices to prove (4) only for s 1/2. Let B r (x) = {ξ : ξ x < r} and B r () = B r. We introduce the mean value u x,y of u over the ball B x,y := B x y /2 ((x + y)/2). Since u(x) u(y) p/θ 2 1+p/θ ( u(x) u x,y p/θ + u x,y u(y) p/θ ), it follows that where We note that x y >δ ( ) θ/p, 2 D(x) p/θ dx (6) D(x) = ( Rn u(x) u x,y p/θ ) θ/p. dy x y n+ps u(x) u x,y p/θ dy 2p/θ B 1 u p/θ x y n+ps L ps δ ps, (7) where B 1 is the area of the unit sphere. Let U be an arbitrary extension of u onto +1 + = {(x, z) : x, z > } such that U L 1 loc (Rn+1 + ). By U x,y (z) we denote the mean value of U(, z) in B x,y. Using the identity we find p 1/θ (1 s x y (1 s)p = p(1 s) x y u(x) u x,y p/θ x y n+ps dy = z 1+p(1 s), ( x y z 1+p(1 s) u(x) u x,y p dy x y 3 1+p/θ p 1/θ (1 s (J 1 + J 2 + J 3 ), (8) 2

3 where J 1 := J 2 := and J 3 := Clearly, J 1 ( x y ( x y ( x y By Hardy s inequality one has z 1+p(1 s) u(x) U(x, z) p dy x y, z 1+p(1 s) U x,y (z) u x,y p dy x y, z 1+p(1 s) U(x, z) U x,y (z) p dy x y. ( x y ( z z 1+p(1 s) U(x, t) ) p dy dt t x y ( x y ( z z 1 ps U(x, t) ) p dy dt t x y. n ps(1 θ)/θ a J 1 s p/θ z z 1 sp ϕ(t)dt p a s p z 1+p(1 s) ϕ(z) p, ( x y θ B 1 s p/θ ps(1 θ) ( z 1+p(1 s) U(x, z) z 1+p(1 s) U(x, z) Duplicating the same argument, we conclude that J 2 s p/θ dy ( x y x y n ps(1 θ)/θ p dy x y n ps(1 θ)/θ p 1/θδ ) ps(1 θ)/θ. (9) z 1+p(1 s) U x,y (z) Let M denote the n-dimensional Hardy-Littlewood maximal operator 1 (Mf)(x) = sup f(ξ) dξ. r> B r Using the obvious inequality we find from (1) J 2 U ( x,y(z) θ B 1 ( s p/θ ps(1 θ) B r(x) M U ) (x, z), p 1/θ. ) (1) ( z 1+p(1 s) M U ) p δ ps(1 θ)/θ. (11) 3

4 In order to estimate J 3 we use the Sobolev type integral representation in the form given in [3], Ch. 1, Sect. 3 U(x, z) U x,y (z) = n k=1 b k (ξ, x) U(ξ, z) dξ, (12) B x,y x ξ n 1 ξ k where b k (ξ, x) are continuous functions for x ξ admitting the estimate b k (ξ, x) x y n n B x,y. Clearly, (12) implies the estimate U(x, z) U x,y (z) 2n n 1/2 B 1 B r(x) ξ U(ξ, z) dξ, x ξ n 1 where r = x y. Integrating by parts we find ξ U(ξ, z) dξ = r1 n B r(x) x ξ n 1 ξ U(ξ, z) dξ+ B r(x) r ds (n 1) s n ξ U(ξ, z) dξ n x y (M U )(x, z). B s(x) Therefore, ( 2 n n 3/2 ) p/θ J 3 B 1 ( x y (2 n n 3/2 ) p/θ θ ( B 1 (p θ)/θ ps(1 θ) z 1+p(1 s) (M U ) p dy x y n ps(1 θ)/θ z 1+p(1 s) (M U ) p δ ps(1 θ)/θ. (13) Here and in the sequel, for the sake of brevity, by M U we mean (M U )(x, z). Putting estimates (9), (11), and (13) into (8), we arrive at u(x) u x,y p/θ ( dy c(n) (1 s)1/θ x y n+ps 1 θ z 1+p(1 s) (M U ) p δ ps(1 θ)/θ. This estimate together with (7) implies that D(x) is majorized by ( c(n) u L δ θs + ( 1 s 1 θ ) 1/p ( Minimizing the right-hand side, we conclude that Hence and by (6) ) 1/pδ z 1+p(1 s) (M U ) p s(1 θ)). ( 1 s ) θ/p u ( 1 θ θ/p. D(x) c(n) L z 1+p(1 s) (M U ) ) p 1 θ ( 1 s ) θ/p u ( 1 θ θ/p. c(n) L z 1+p(1 s) (M U ) dx) p 1 θ 4

5 Since (see [4], Sect. 2.5), we have Mu L p ( p ) θ ( 1 s c(n) p 1 1 θ n8 n p B 1 (p 1) u L p ) θ/p u z 1+p(1 s) U(x, z) p 1 θ dx L. (14) Now we define U by the formula U(x, z) := ψ(h)u(x + zh)dh, (15) where ψ(h) = B 1 n(n + 1)(1 h ) + with plus standing for the nonnegative part of a real valued function. It follows directly from (15) that n(n + 1)(n + 2) U(x, z) u(x + zh) u(x) dh. z B 1 Hence and by Hölder s inequality n B 1 (n + 1)p (n + 2) p We have Thus, z 1 ps h <1 z 1+p(1 s) U(x, z) p dx z 1 ps h <1 h <1 u(x + zh) u(x) p dxdh. (16) u(x + zh) u(x) p dh = z z 1 ps n ρ n 1 dρ u(x + ρθ) u(x) p dθ = B 1 (ps + n) 1 ρ ps 1 dρ u(x + ρθ) u(x) p dθ. B 1 Rn z 1+p(1 s) U(x, z) p dx n(n + 1)p (n + 2) p Combining (17) with (14) we complete the proof. B 1 (ps + n) u p W s,p. (17) Remark 1. Let ( ) 1/p. u W 1,p = u(x) p dx As a particular case of a more general inequality, Brezis and Mironescu [2] obtained (3) for s = 1. They commented on this in the following way: We do not know any elementary (i.e., without the Littlewood-Paley machinery) proof of (3) when s = 1. 5

6 Obviously, the above proof of (4), complemented by the reference to formula (1), provides an elementary proof of the inequality (1 θ) θ/p u W θ,p/θ c(n, p) u θ W 1,p u 1 θ L. The factor (1 θ) θ/p controls the blow up of the norm in W θ,p/θ as θ 1. Remark 2. Note that passing to the limit as p in both sides of (4) one obtains inequality (3) with p = and with a certain finite constant c. Let us consider the case p 1 when the constant factor in (4) tends to infinity. It follows from (4) that the best value of c(n, p, θ) in the inequality admits the upper estimate c(n, p, θ)(1 s) θ/p u θ W s,p u 1 θ L (18) lim sup p 1 Now we obtain the analogous lower estimate (p 1) θ c(n, p, θ) c(n)(1 θ) θ. (19) lim inf p 1 (p 1) θ c(n, p, θ) 1. (2) In fact, the characteristic function χ of the ball B 1 belongs to W s,p and W θs,p/θ if and only if sp < 1, and there holds χ W θs,p/θ = χ θ W s,p. Putting u = χ into (18), where s = p 1 ε with an arbitrarily small ε >, we obtain 1 c(n, p, θ)((p 1)/p) θ/p, which implies (2). Thus, the growth O((p 1) θ ) of the constant in (4) as p 1 is best possible. References [1] Bourgain J., Brezis H., Mironescu P., Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, J.L. Menaldi, E. Rofman, A. Sulem (Eds.), IOS Press, Amsterdam, 21, [2] Brezis H., Mironescu P., Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evolution equations, 1(4) (22) (to appear). [3] Akilov G.P., Kantorovich L.V., Functional Analysis, Pergamon Press, [4] Iwaniec T., Nonlinear Differential Forms, University Printing House, Jyväskylä, Key words: Gagliardo-Nirenberg inequality, fractional Sobolev norms, interpolation inequalities 6