The Sobolev inequality on the torus revisited

Transcription

1 Publ. Math. Debrecen Manuscript (April 26, 2012) The Sobolev inequality on the torus revisite By Árpá Bényi an Taahiro Oh Abstract. We revisit the Sobolev inequality for perioic functions on the - imensional torus. We provie an elementary Fourier analytic proof of this inequality which highlights both the similarities an ifferences between the perioic setting an the classical -imensional Eucliean one. 1. Introuction: motivation an preliminaries The Sobolev spaces are ubiquitous in harmonic analysis an PDEs, where they appear naturally in problems about regularity of solutions or well-poseness. Tightly connecte to these problems are certain embeing theorems that relate the norms of Lebesgue an Sobolev spaces for appropriate inices. These theorems are known uner the name of Sobolev inequalities; they are state rigorously in Proposition 1.1 an Corollary 1.2; see also Subsection 2.2. In this note, we use tools from classical Fourier analysis an provie an elementary approach to such inequalities for perioic functions on the -imensional torus. The appeal of Sobolev spaces is ue to the simplicity of their efinition which captures both the regularity an size of a istribution. If k is a positive integer an 1 p, let L p k (R ) enote the space of all u L p (R ) such that the weak erivatives D α u L p (R ) for all α k. In the PDE literature, this space is often enote by W k,p (R ). For non-integer values of s > 0, the complex interpolation of the integer orer spaces L p k (R ) yiels the inhomogeneous (fractional) Sobolev spaces, or as they are also commonly referre to, inhomogeneous Bessel potential Mathematics Subject Classification: Primary 42B35; Seconary 42B10, 42B25. Key wors an phrases: Sobolev spaces, Sobolev inequality, Fourier series, Bessel potential, Riesz potential, maximal function.

2 2 Árpá Bényi an Taahiro Oh spaces. We enote them by L p s(r ), s R +. In fact, on the Fourier sie, these spaces can be efine for all s R. As such, they are Banach spaces, enowe with the norm u L p s (R ) = ( ξ s û(ξ) ) Lp (R ). Here, ξ = (1 + 4π 2 ξ 2 ) 1 2, û enotes the Fourier transform efine by û(ξ) = u(x)e 2πix ξ x, R an u enotes the inverse Fourier transform of u efine by u (x) = û( x). We can also efine the fractional inhomogeneous Sobolev spaces W s,p (R ) by applying the real interpolation metho to the integer orer spaces W k,p (R ). It is worth pointing out, however, that, ue to the ifferent methos of interpolation use (real an complex, respectively), we have W s,p (R ) L p s(r ) unless s is an integer or p = 2. The spaces W s,p (R ) can also be characterize by the L p -moulus of continuity, analogous to (1.13). See the books by Bergh an Löfström [3], Stein [12], an Tartar [15] for more etaile iscussions on L p s(r ) an W s,p (R ). The homogeneous Sobolev spaces L p s(r ) are efine in a similar way, by replacing with in the efinition above 1 : u Lp s (R ) = ( ξ s û(ξ) ) Lp (R ). When p = 2, we simply write H s (R ) = L 2 s(r ) or Ḣs (R ) = L 2 s(r ). Let now T = R /Z enote the -imensional torus. In analogy with the efinition of the Sobolev spaces on the -imensional Eucliean space R, the inhomogeneous Sobolev (or Bessel potential) spaces H s (T ) an L p s(t ) on the torus T are efine via the norms ( ) 1 2 u H s (T ) =, (1.1) n Z n 2s û(n) 2 u L p s (T ) = ( n s û(n) ) Lp (T ). (1.2) Here, u enotes a perioic function on T an û(n), n Z, are its Fourier coefficients efine by û(n) = u(x)e 2πin x x. T 1 Strictly speaking, the homogeneous spaces Lp s (R ) are efine only for the equivalence classes moulo polynomials (corresponing to the istributions supporte at the origin on the Fourier sie).

3 The Sobolev inequality on the torus revisite 3 The fact that H s (T ) = L 2 s(t ) is a simple consequence of Plancherel s ientity. Clearly, we can efine the homogeneous Sobolev spaces on the torus in a similar way: u Ḣs (T ) = ( n Z \{0} n 2s û(n) 2 ) 1 2, (1.3) u Lp s (T ) = ( n s û(n) ) L p (T ). (1.4) The perioic Sobolev inequality inequality is part of the mathematical analysis folklore. It is essentially state in Strichartz paper [14], albeit with no proof. Due to the geometric an topological structure of the torus, the Sobolev inequality on T can be viewe as a particular case of a Sobolev inequality on a compact manifol; see, for example, [1] an [8]. However, our goal here is to provie what we believe is a very natural an elementary proof of this inequality via Fourier analysis which emphasizes the perioic nature of the Sobolev spaces involve; to the best of our knowlege, this argument is missing from the literature. It is plausible that one can infer other proofs of the Sobolev inequality on T from corresponing ones on R (such as the ones implie by the funamental solution of the Laplacian or by isoperimetric inequalities). Our hope is that the expository an self-containe nature of this presentation makes it accessible to a large reaership. The Sobolev inequality on the torus can be unerstoo as an embeing of a perioic Sobolev space into a perioic Lebesgue space. More precisely, we have the following. Proposition 1.1. Let u be a function on T with mean zero. Suppose that s > 0 an 1 < p < q < satisfy s = 1 p 1 q. (1.5) Then, we have u L q (T ) u Lp s (T ). (1.6) Here, an throughout this note, we use A B to enote an estimate of the form A cb for some c > 0 inepenent of A an B. Similarly, we use A B to enote A B an B A. An immeiate consequence of Proposition 1.1 is the same inequality for the inhomogeneous Sobolev spaces L p s(t ) with the natural conition on the inices.

4 4 Árpá Bényi an Taahiro Oh Corollary 1.2. Let u be a function on T. Suppose that s > 0 an 1 < p < q < satisfy s 1 p 1 q. (1.7) Then, we have u L q (T ) u L p s (T ). (1.8) Perhaps unsurprisingly, the appearance of Sobolev spaces on the torus is frequent in works that investigate, for example, nonlinear PDEs in perioic setting. Let us briefly iscuss some applications of these spaces an of the perioic Sobolev inequality in the stuy of the Kortweg-e Vries (KV) equation: u t + u xxx + uu x = 0, (x, t) T R. (1.9) By the classical energy metho, Kato [10, 11] prove local-in-time well-poseness of (1.9) in H s (T) for s > 3/2. This 3/2 critical regularity arises from the Sobolev embeing theorem on T (see (2.1) for the continuous version) applie to the u x term in the nonlinearity, since for each fixe t: u x (, t) L (T) u(, t) Hs (T) for s > 3/2. In the seminal paper [2], Bourgain improve Kato s result an prove wellposeness of (1.9) in L 2 (T) by introucing a new weighte space-time Sobolev space X s,b (T R) whose norm is given by u X s,b (T R) = n s τ n 3 b û(n, τ) L 2 τ l 2, s, b R. n Ever since [2], this so-calle Bourgain space X s,b an its variants have playe a central role in the analysis of nonlinear (ispersive) PDEs an le to a significant evelopment of the fiel. Let S(t) = e t 3 x enote the linear semigroup for (1.9). Then, the X s,b -norm of a function u on T R can be written as the usual spacetime Sobolev H b t H s x-norm of its interaction representation S( t)u: u X s,b = S( t)u H b t H s x. (1.10) Now, in view of (1.10), the perioic Sobolev inequality (1.8) leas to the following estimate: u L 2 t (R;L p x(t)) u X s,0 (T R) for 0 s < 1/2 an 2 p 2/(1 2s). Such estimates are wiely use in multilinear estimates appearing in the I-metho evelope by Collianer, Keel, Staffilani, Takaoka, an Tao; see, for example, [4, Section 8].

5 The Sobolev inequality on the torus revisite 5 Lastly, we present a heuristic argument inicating the connection between Bourgain s perioic L 4 -Strichartz inequality: u L 4 x,t (T R) u X 0,1/3 (T R) (1.11) an the Sobolev inequality. On the one han, by the continuous Sobolev inequality (2.6) an (1.8), we have u L 4 x,t (T R) u X 1/4,1/4 (T R). (1.12) On the other han, in view of the linear part of the equation (1.9), u t + u xxx = 0, we can formally view the three spatial erivatives as equivalent to one temporal erivative. Then, by formally moving the spatial erivative s = 1/4 in (1.12) to the temporal sie, we obtain the temporal regularity b = 1/3 in (1.11), since 1/3 = 1/4 + (1/3)(1/4). Of course, this is merely a heuristic argument showing why b = 1/3 is the natural regularity in (1.11) an the actual proof is more complicate, see [2]. For various relations among the L p t L q x spaces an X s,b spaces by the Sobolev inequality, the perioic L 4 -an L 6 -Strichartz inequalities an interpolation, the reaer is referre to [5, Section 3]. Having iscusse the usefulness of the Sobolev inequality in perioic setting, the next natural question that arises is how it iffers from its Eucliean counterpart. We postpone the answer to this question to the following section. However, in anticipation of this answer, we provie the reaer with the following insight: the perioic Sobolev spaces are intrinsically more elicate in nature than the non-perioic ones, an thus the proofs in the perioic case require a more careful analysis. In orer to justify this claim, let us take a closer look at the ifference (an analogy) between the homogeneous Bessel potential spaces Ḣs (R ) an Ḣs (T ). We begin by recalling the following characterization of the Ḣs (R ) norm by the L 2 -moulus of continuity; see Hörmaner s monograph [9]: u 2 Ḣ s (R ) R R = ξ 2s û(ξ) 2 u(x) u(y) 2 ξ = c R x y +2s xy. (1.13) The proof of (1.13) goes as follows. By the change of variables x x + y, the ouble integral in (1.13) is R R u(x + y) u(y) 2 R e 2πix ξ 1 2 x +2s xy = R x +2s x û(ξ) 2 ξ,

6 6 Árpá Bényi an Taahiro Oh where we use the fact that, for fixe x, the Fourier transform of u(x + y) u(y) as a function of y is given by (e 2πix ξ 1)û(ξ). Now, efine A(ξ) by A(ξ) = ξ 2s e 2πix ξ 1 2 R x +2s x = ξ 2s R sin 2 (πx ξ) 4 x +2s x. (1.14) Then, by the change of variables x tx, we have A(tξ) = A(ξ). Hence, A(ξ) = A is inepenent of ξ. Moreover, with ξ = (1, 0,..., 0), we have A = R sin 2 (πx 1 ) x. 4 x +2s Noting that sin2 (πx 1) x +2(1 s) near the origin an sin2 (πx 1) x 2s near x +2s x +2s infinity, we have A <. Hence, (1.13) follows from (1.14) by choosing c = A. Remark 1.1. Given u L p (R ), ω p (t) = u(x + t) u(x) L P x is calle the L p -moulus of continuity. Hence, we can view (1.13) as the characterization of the Ḣs (R )-norm by the L 2 -moulus of continuity. There is an analogous result for the characterization of the Ẇ s,p (R )-norm by the L p -moulus of continuity; see Stein s book [12, p.141]. We note immeiately that the ouble integral expression in (1.13) is not quite meaningful for perioic functions on T even if we only integrate over T. Nonetheless, we have an analogue of (1.13) for Ḣ s (T ), but the etails of the proof are alreay a little more elicate. T Proposition 1.3. Let 0 < s < 1. Then, for u Ḣs (T ), we have u 2 Ḣ s (T ) T Proof. As before, we have [ 12, 12 ) u(x + y) u(y) 2 x +2s xy = It remains to show that B(n) given by B(n) = n 2s e 2πix n 1 2 [ 1 2, 1 x +2s 2 ) [ 12, 12 ) u(x + y) u(y) 2 n Z \{0} x = n 2s x +2s xy. (1.15) e [ 2πix n , 12 x +2s x û(n) 2. ) sin 2 (πx n) [ 1 2, 1 4 x +2s x (1.16) 2 )

7 The Sobolev inequality on the torus revisite 7 is boune both from above an below uniformly in n Z \ {0}. Note that, in this case, we can not use a change of variables to show that B(n) is inepenent of n. Of course, by extening the integration to R, we have B(n) A(n) = A <, where A(n) is, as in (1.14), inepenent of n. Next, we show that B(n) is boune below by a positive constant, inepenent of n = (n 1, n 2,..., n ). Rearrange n j such that n 1,..., n m are non-zero an n m+1 = = n = 0. By symmetry, assume that n 1 is positive an that n 1 = max(n 1, n 2,..., n ). Now, we restrict the integral in (1.16) to D = { x [ 1 2, 1 2 ] : x < 1 2 n, n jx j > 0,j = 1,..., m } { x 1 = max x j } [ 1 2, 1 2 ). We have n 1 x 1 n x 1 2 on D. Since 2y sin πy for y [0, 1 2 ], we have sin 2 (πx n) (n 1 x 1 ) 2 n 2 x 2, where the last inequality follows from n 1 n an x 1 x. Then, by integration in the polar coorinates, we obtain B(n) n 2 2s D n 2 2s 1 2 n 0 x +2 2s x n 2 2s r 1 2s r 1. x < 1 2 n x +2 2s x This completes the proof of (1.15). 2. The Sobolev inequality This section is evote to a iscussion of the Sobolev inequality on the - imensional torus. As alreay pointe out in the previous section, the inequality is wiely use for perioic PDEs Sobolev s embeing theorem. Sobolev s embeing theorem states that, for sp >, u L (R ) u L p s (R ). (2.1) Notice that the conition sp > is equivalent to s > 1 p = 1 p 1 ; compare this also with (1.5). When p 2, (2.1) follows from Höler s inequality an

8 8 Árpá Bényi an Taahiro Oh Hausorff-Young s inequality. Inee, ( ) 1 u(x) û(ξ) ξ R ξ ps ξ R p ξ sû(ξ) L p (R ) ( ξ s û(ξ) ) L p (R ) = u L p s(r ). This argument, in particular, shows that û L 1 (R ). Hence, it follows from Riemann-Lebesgue lemma that u is uniformly continuous on R, vanishing at infinity. The same argument yiels the corresponing result on T. When p > 2, we nee to procee ifferently. We borrow some ieas from the nice exposition in Grafakos books [6, 7]. Define G s by G s (x) = ( ξ s ) (x). (2.2) Note that G s is the convolution kernel of the Bessel potential J s = (I ) s 2 of orer s, i.e. J s (f) = f G s. Then, the following estimates hol for G s (see [7, Proposition 6.1.5]): G s (x) C(s, )e x 2 for x 2, (2.3) while for x 2, we have x s O( x s +2 ), for 0 < s <, G s (x) c(s, ) log 2 x O( x 2 ), for s =, 1 + O( x s ), for s >. (2.4) When s, G s L p (R ), while when s <, we have G s L p (R ) (near the origin) if an only if sp >. Thus, by Young s inequality, we obtain f G s L (R ) f Lp (R ). (2.5) This proves (2.1) since (2.5) is equivalent to it. Note also that Young s inequality implies that u = f G s is uniformly continuous on R. We will briefly escribe an argument for p > 2 on T at the en of the next subsection The Sobolev inequality. Let s > 0 an 1 < p < q < satisfy (1.5). Sobolev s inequality on R states that for all such s, p, q we have u L q (R ) u Lp s (R ). (2.6)

9 The Sobolev inequality on the torus revisite 9 This is equivalent to the following Hary-Littlewoo-Sobolev inequality: I s (f) Lq (R ) f Lp (R ), (2.7) where s > 0 an 1 < p < q < satisfy (1.5), an I s = ( ) s 2 Riesz potential of orer s. Using [6, Theorem 2.4.6], we have enotes the ( ξ z ) = π 2z+ 2 Γ( +z 2 ) Γ( z 2 ) x z, (2.8) where the equality hols in the sense of istributions (inee, when Re z < 0, the expression in (2.8) is in L 1 loc (R ) an can be mae sense as a function). Now, (2.8) allows us to write I s (f)(x) = 2 s π 2 Γ( s 2 ) Γ( s 2 ) f(x y) y +s y. (2.9) R See Subsection in [7]. Then, one can prove (2.7) by an argument on the physical sie, using (2.9); see [7, Theorem 6.1.3], an also the proof of Proposition 1.1 below. We note in passing that, having establishe (2.9), one may view (2.7) as the enpoint case of Young s inequality. Inee, by a simple application of Young s inequality, one woul obtain I s (f) Lq (R ) x +s L s (R ) f L p (R ), where the first factor on the right-han sie is infinite since x +s barely misses to be in L s (R ). We arrive at last to the Sobolev inequality for perioic functions on T state in Proposition 1.1, which we prove next. As before, (1.6) is equivalent to the following Hary-Littlewoo-Sobolev inequality on T : I s (f) L q (T ) f L p (T ), (2.10) where f has mean zero. The proof of (2.10) follows along the same lines as the proof of (2.7) on R (c.f. [7, Theorem 6.1.3]) once we obtain a formula analogous to (2.8) relating n s an x +s for n Z an x T. Inee, our argument has a simple structure. First, we see that the operator I s is realizable as a convolution with

10 10 Árpá Bényi an Taahiro Oh the istribution on T whose Fourier series is given by n Z \{0} n s e 2πin x. Then, a refinement of the Poisson summation formula, base on some smooth cut-off techniques, gives that, moulo an aitive smooth function on T, this istribution behaves like a constant multiple of x +s. While this is a known fact, see Stein an Weiss [13, Theorem 2.17, p. 256], for the convenience of the reaer we err on the sie of a longer an ifferent argument than in [13] by proviing full etails of the careful analysis require here. Finally, we notice that the (L p, L q ) mapping property of the convolution operator I s can be reuce to that of the convolution operator with x +s which roughly follows as in the -imensional Eucliean case once we o a correct splitting of the -imensional torus Proof of Proposition 1.1. We start by recalling the Poisson summation formula. Lemma 2.1 (Theorem in [6]). Suppose that f, f L 1 (R ) satisfy f(x) + f(x) C(1 + x ) δ for some C, δ > 0. Then, f an f are continuous an, for all x R, we have n Z f(n)e 2πin x = n Z f(x + n). (2.11) Let now η be a smooth function on R such that η(ξ) = 1 for ξ 1 2 an η(ξ) = 0 for ξ 1 4. For 0 < Re s <, efine g(x) = ( η(ξ) ξ s) (x). Then, it is known (see [6, Example 2.4.9]) that g ecays faster than the reciprocal of any polynomial at infinity. Let h(x) = g(x) G(x), where G(x) = π s 2 Γ( s 2 ) Γ( s 2 ) x s. (2.12) Then, h C (R ). We woul like to apply now Lemma 2.1 to g an ĝ = η(ξ) ξ s. However, the ecay of ĝ at infinity is not fast enough (since (1.5) implies s < ) an we have ĝ / L 1 (R ). Hence, Lemma 2.1 is not applicable. Fix φ S(R ) supporte on [ 1 2, 1 2 ) such that R φ(x)x = 1, an let φ ε (x) = ε φ(ε 1 x), ε > 0. The family {φ ε } is an approximation of the ientity. If we now let g ε = g φ ε, then ĝ ε (ξ) = ĝ(ξ) φ ε (ξ) = φ ε (ξ)η(ξ) ξ s satisfies the esire ecay ĝ ε (ξ) C(1 + ξ ) δ for some δ > 0. Clearly, g ε (x) C(1 + x ) δ near infinity thanks to the rapi ecay of g at infinity. Also, g ε is

11 The Sobolev inequality on the torus revisite 11 boune near the origin since x s is integrable near the origin (an thus, g ε is a C function.) Let x [ 1 2, 1 2 ). By Lemma 2.1 we have n Z \{0} φ ε (n)e 2πin x n s = φε (n)η(n)e 2πin x n s n Z = max n j 1 g ε (x + n) + n Z max n j 2 g ε (x + n). (2.13) Note that, for x, y [ 1 2, 1 2 ) an n Z, we have x y + n 1, if max n j 2, x y + n 3, if max n j 1. Let r 1. Since g(x) is a smooth rapily ecreasing function on x 1, we have n Z max n j 2 g ε (x + n) = n Z max n j 2 Lr ([ 1 2, 1 2 ) ) g(x y + n)φ ε (y)y [ 1 2, 1 2 ) Lr ([ 1 2, 1 2 ) ) φ ε L 1 ([ 1 2, 1 2 ) ) g(x) Lr ( x 1) = g(x) Lr ( x 1) <. Also, since h in (2.12) is a smooth function, we have max n j 1 h φ ε (x + n) = max n j 1 Lr ([ 1 2, 1 2 ) ) h(x y + n)φ ε (y)y [ 1 2, 1 2 ) Lr ([ 1 2, 1 2 ) ) φ ε L 1 ([ 1 2, 1 2 ) ) h(x) L r ( x 3 ) = h(x) L r ( x 3 ) <. Motivate by these two estimates, we let H ε (x) = n Z max n j 2 g ε (x + n) + max n j 1 h φ ε (x + n).

12 12 Árpá Bényi an Taahiro Oh Then, H ε is smooth on [ 1 2, 1 2 ) an H ε Lr ([ 1 2, 1 2 ) ) C <, (2.14) where the constant C is inepenent of ε > 0. Moreover, from (2.13), we have φ ε (n)e 2πin x n s = G φ ε (x + n) + H ε (x), (2.15) n Z \{0} max n j 1 for x [ 1 2, 1 2 ), where H ε is smooth, satisfying (2.14). We are now reay to prove (2.10). Let ε > 0. We will first prove φ ε I s (f) L q (T ) f L p (T ) (2.16) for smooth f with mean zero on T, where the implicit constant is inepenent of ε > 0. In the following, we often view f as a perioic function efine on R. By (2.15), we have φ ε I s (f)(x) = (2π) s f(n) φ ε (n) n s e 2πin x n Z \{0} = (2π) s T f(y) max n j 1 n Z \{0} φ ε (n) n s e 2πin (x y) y f(y) ( ) G φ ε (x y + n)y + f(y)h ε (x y)y T T =: I (x) + II(x) (2.17) for x [ 1 2, 1 2 ). Here, for fixe x [ 1 2, 1 2 ), y ranges over x + ( 1 2, 1 2 ] such that x y [ 1 2, 1 2 ). By Young s inequality with 1 r = q 1 p, we have II Lq (T ) H ε Lr (T ) f Lp (T ) f Lp (T ), (2.18) where the implicit constant is inepenent of ε > 0 thanks to (2.14). Next, we estimate I. First, note that for x y [ 1 2, 1 2 ), max n j 1, an z > 2, we have x y + n z / [ 1 2, 1 2 ). Then, recalling (2.12) an changing the orer of integration, we have I (x) f(y) max n x+( 1 j 1 2, 1 2 ] z 2 z s φ ε (x y + n z)zy z 2 z s f(y) φ ε (x y z + n)yz max n x+( 1 j 1 2, 1 2 ] z s F ε (x z)z, z 2

13 where F ε is efine by The Sobolev inequality on the torus revisite 13 F ε (z) = max n j 1 R f(y) φ ε (z y + n)y. (2.19) Here, we are viewing f as a perioic function efine on R. Although the omain of integration in (2.19) is R, the actual integration is over a boune omain since φ ε is supporte on [ 1 2, 1 2 ). Making a change of variables in (2.19) an using the perioicity of f, we have F ε (z) = max n j 1 f(y + n) φ ε (z y)y = c R f(y) φ ε (z y)y z y [ 1 2, 1 2 ) = c f(z y) φ ε (y)y. [ 1 2, 1 2 ) As such, the expression efining F ε (z) is now perioic in z. Going now back to the estimate on I (x), we ivie it into two parts: I (x) z s F ε (x z)z + z s F ε (x z)z z R =: J 1 (F ε )(x) + J 2 (F ε )(x), R< z 2 where R = R(x) > 0 is to be chosen later. Note that J 1 is given by a convolution with z s χ z R (z), which is raial, integrable, an symmetrically ecreasing about the origin. Hence, it can be boune from above by the uncentere Hary-Littlewoo maximal function M(F ε ) (see [6, Theorem ]): ( ) J 1 (F ε )(x) M(F ε )(x) z s z = cr s M(F ε )(x). (2.20) z R We recall that M(f)(x) is efine as the supremum of the averages of f over all balls B(y, δ) := {z R : z y < δ} that contain the point x, that is 1 M(f)(x) = c δ f(z) z. sup δ>0 x y <δ z y <δ By Höler s inequality followe by Minkowski s integral inequality, we have ( ) 1 ( z 2 J 2 (F ε )(x) z (s )p p ( z z >R Fε (x z) ) ) 1 p p z R q f Lp (T ) (2.21)

14 14 Árpá Bényi an Taahiro Oh for x [ 1 2, 1 2 ). Note also that here we use the conition (1.5) on the inices s, p, q, in particular the fact that (s )p = p q. From (2.20) an (2.21), we obtain I (x) R s M(F ε )(x) + R q f L p (T ). (2.22) Choose now R = R(x) > 0 that minimizes (2.22). With we have 2 R = c f p L p (T )( M(Fε )(x) ) p, (2.23) I (x) ( M(F ε )(x) ) p q f 1 p q L p (T ). (2.24) By taking the L q -norm of both sies an then using the bouneness of the Hary-Littlewoo maximal operator M(F ɛ ) on L p (T ) (see [6, Theorem 2.1.6]), we obtain I Lq (T ) f Lp (T ). Combining this with (2.18), we get (2.16). Finally, (2.10) follows from (2.16) by taking ε 0. This completes the proof of Proposition Proof of (2.1) on T. In the remainer of this section, we briefly escribe the proof of (2.1) for p > 2 in the perioic setting. As pointe out at the beginning of Subsection 2.1, the proof of (2.1) for p 2 is ientical to the one on R. If s > 2 an p 2, then from (2.1) (for p = 2) an Lp (T ) L 2 (T ), we have u L (T ) u H s (T ) u L p s (T ). Now, consier the case s 2 an p > 2. With G s as in (2.2), it follows from (2.3) an (2.4) that G s (x) x s, for x 2. (2.25) As in the proof of Proposition 1.1, the main point is to transfer the relation Ĝ s (ξ) = ξ s to the perioic omain T. This is one by Poisson s summation formula, Lemma 2.1. However, the ecay of ξ s at infinity is not fast enough, i.e. ξ s / L 1 (R ), an thus, Lemma 2.1 is not irectly applicable, Hence, we nee to go through a similar moification as before. We omit this part of the argument. Once we o that, the main objective is to estimate the expression I in (2.17) with G s in place of G: I (x) G s (z)f ε (x z)z. z 2 2 If R 2 in (2.23), then there is no contribution from J2 (F ε)(x). In this case, (2.22) becomes I (x) R s M(F ε)(x). Since s > 0, setting R = 2 yiels (2.24).

15 The Sobolev inequality on the torus revisite 15 Since sp >, we have (s )p + = p (s p ) > 0. Thus, using (2.25), Höler s inequality an Minkowski s integral inequality, we get ( I (x) z 2 z (s )p z ) 1 p ( z 2 f L p (T ) for x [ 1 2, 1 2 ). ( Fε (x z) ) p z ) 1 p Hence, we obtain I L (T ) f L p (T ). As pointe out in the proof of Proposition 1.1, the estimate II L (T ) f L p (T ) is rather straightforwar. Combining these two estimates, we obtain (2.1). Acknowlegments. The authors woul like to thank the anonymous referee for thoughtful comments. The first name author is also grateful to Diego Malonao, Virginia Naibo, an Pietro Poggi-Corraini for their hospitality uring his Fall 2011 visit at Kansas State University an for useful iscussions an suggestions that improve the exposition of this note. Á. Bényi is partially supporte by Simons Founation Grant No T. Oh acknowleges support from an AMS-Simons Travel Grant. References [1] T. Aubin, Nonlinear analysis on manifols. Monge-Ampère equations, Vol. 252, Grunleren er Mathematischen Wisserchaften, Springer-Verlag, Berlin-New York, [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets an applications to nonlinear evolution equations. II. The KV-equation, Geom. Funct. Anal. 3, no. 3 (1993), [3] J. Bergh, J. Löfström, Interpolation spaces: an introuction, Springer-Verlag, Berlin-New York, [4] J. Collianer, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-poseness for KV an moifie KV on R an T, J. Amer. Math. Soc. 16, no. 3 (2003), [5] J. Collianer, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Multilinear estimates for perioic KV equations, an applications, J. Funct. Anal. 211, no. 1 (2004), [6] L. Grafakos, Classical Fourier analysis, Vol. 249, Secon eition. Grauate Texts in Mathematics, Springer, New York, [7] L. Grafakos, Moern Fourier analysis, Vol. 250, Secon eition. Grauate Texts in Mathematics, Springer, New York, [8] E. Hebey, Nonlinear analysis on manifols: Sobolev spaces an inequalities, American Mathematical Society, [9] L. Hörmaner, The analysis of linear partial ifferential operators. I. Distribution theory an Fourier analysis, Springer-Verlag, Berlin, [10] T. Kato, On the Korteweg-e Vries equation, Manuscripta Math. 28 (1979), [11] T. Kato, Quasi-linear equations of evolution, with application to partial ifferential equations, Vol. 448, Lecture notes in Math., Springer-Verlag, 1975,

16 16 Árpá Bényi an Taahiro Oh [12] E. Stein, Singular integrals an ifferentiability properties of functions, Vol. 30, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., [13] E. Stein, G. Weiss, Introuction to Fourier analysis on Eucliean spaces, Vol. 32, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., [14] R. Strichartz, Improve Sobolev inequalities, Trans. Amer. Math. Soc. 279, no. 1 (1983), [15] L. Tartar, An introuction to Sobolev spaces an interpolation spaces, Vol. 3, Lecture Notes of the Unione Matematica Italiana, Springer Berlin; UMI, Bologna, DEPARTMENT OF MATHEMATICS WESTERN WASHINGTON UNIVERSITY 516 HIGH STREET BELLINGHAM, WA 98226, USA arpa.benyi@wwu.eu SCHOOL OF MATHEMATICS THE UNIVERSITY OF EDINBURGH JAMES CLERK MAXWELL BUILDING THE KINGS BUILDINGS MAYFIELD ROAD EDINBURGH, EH9 3JZ, SCOTLAND, AND DEPARTMENT OF MATHEMATICS PRINCETON UNIVERSITY FINE HALL, WASHINGTON RD PRINCETON, NJ , USA hirooh@math.princeton.eu