ON FOURNIER GAGLIARDO MIXED NORM SPACES

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 36, 2, ON FOURNIER GAGLIARDO MIXED NORM SPACES Robert Algervi and Vitor I. Kolyada Karlstad University, Department of Mathematics Universitetsgatan, Karlstad, Sweden; Karlstad University, Department of Mathematics Universitetsgatan, Karlstad, Sweden; Abstract. We study mixed norm spaces V (R n ) that arise in connection with embeddings of Sobolev spaces W (R n ). We prove embeddings of V (R n ) into Lorentz type spaces defined in terms of iterative rearrangements. Basing on these results, we introduce the scale of mixed norm spaces V p (R n ). We prove that V V p and we discuss some questions related to this embedding.. Introduction Let W p (R n ) ( p < ) be the Sobolev space of all functions f L p (R n ) for which all first-order wea derivatives f/ x D f exist and belong to L p (R n ). The classical Sobolev theorem asserts that for any function f in W p (R n ) ( p < n) (.) f q c D f p, q = np n p. Sobolev proved this inequality in 938 for p >. For p = inequality (.) was proved independently by Gagliardo (958) and Nirenberg (959). The core of Gagliardo s approach [7] is the following: Lemma.. Let n 2. Assume that g L (R n ) ( =,..., n) are nonnegative functions on R n. Then ( n ) /(n ). (.2) g (ˆx ) /(n ) dx g (ˆx ) dˆx R n R n As usual, for any vector x R n and any =,..., n we denote by ˆx the (n )-dimensional vector obtained from x by removal of its th coordinate. We write x = (x, ˆx ). Assume now that f W (R n ). Then for any =,..., n and almost all x R n, f(x) = x D f(u, ˆx ) du = x D f(u, ˆx ) du. It follows that for almost all x R n (.3) f(x) D f(u, ˆx ) du g (ˆx ), =,..., n. 2 R doi:.586/aasfm Mathematics Subject Classification: Primary 46E3, 46E35; Secondary 26B35. Key words: Mixed norms; rearrangements, embeddings, Sobolev spaces.

2 494 Robert Algervi and Vitor I. Kolyada Thus, ( f(x) n /n g (ˆx )). 2 Using this estimate and applying Lemma., we obtain (.4) f n/(n ) ( n ) /n. D f 2 This implies inequality (.) for p =. However, a more general statement can be derived from (.2). Let V (R n ) L ˆx (R n )[L x (R)], n, be the space of measurable functions on R n with the finite mixed norm (.5) f V ψ L (R n ), where ψ (ˆx ) = ess sup x R f(x). We observe that f V has a clear geometric interpretation: it is the n-dimensional measure of the essential projection of the set {(x, y) R n [; ): y f(x) } into the hyperplane x = (see Theorem 3. below). Throughout this paper, we denote also n (.6) V (R n ) = V (R n ), f V = Note that by (.3), for any function f W (R n ) f V. (.7) f V 2 D f ( n). Gagliardo s lemma immediately implies the following theorem: Theorem.2. Let n 2. If f V (R n ), then f L n (R n ) and ( n ) /n. f n f V As usual, for any < p < we denote p = p/(p ). By (.7), Theorem.2 implies (.4). It is well nown that the left-hand side in (.) can be replaced by a stronger Lorentz norm f q,p. Recall that the Lorentz space L q,p (R n ) ( < q, p < ) is defined as the class of all measurable functions f on R n such that ( [ f q,p t /q f (t) ] p dt ) /p <, t where f denotes the nonincreasing rearrangement of f. Note that the quasi-norm q,p is a norm if and only if p q < (see [2]). For a fixed q, the Lorentz spaces L q,p increase as the secondary index p increases (see [2, p. 27]). The following strengthening of (.) holds: (.8) f q,p c D f p, p < n, q = np n p

3 On Fournier Gagliardo mixed norm spaces 495 (see [5], [6] for p >, [7] for p = ). In this paper we consider (.8) only in the case p =. There are numerous proofs of (.8) in this limiting case; most of them are related to rearrangements, properties of level sets, and geometric inequalities. A very interesting approach given by Fournier [6] was based on the following refinement of Theorem.2. Theorem.3. Let n 2. If f V (R n ), then f L n, (R n ) and ( n ) /n. (.9) f n, n f V By virtue of (.7), inequality (.9) immediately implies (.8) for p =. Thus, embedding (.) W (R n ) L n, (R n ) can be split into two successive steps (.) W (R n ) V (R n ) and V (R n ) L n, (R n ). Note that similar splitting for embedding Wp (R n ) L q,p (R n ) in the whole range p < n was obtained in [9]. Different extensions of Theorem.3 and their applications have been studied in the wors [3], [9], [4]. The motivation for this paper was twofold. On the one hand, it was motivated by Theorems.2 and.3. These theorems show that the integrability properties of functions of several variables can be controlled by the behaviour of the L -norms of their linear sections. Following this idea, we obtain stronger versions of inequality (.9) expressed in terms of iterative rearrangements (see Sections 2 and 4 below). We observe that these results were also inspired by embeddings of Sobolev spaces into modified Lorentz spaces proved in [8]. On the other hand, smoothness or integrability properties of functions reflect on the behaviour of their linear sections. In particular, inequality (.7) shows that for any f W (R n ) the L -norms of x -sections fˆx (x ) = f(x, ˆx ) are integrable functions of ˆx in R n. It is also natural to study other norms of linear sections. For example, let us consider L -norms. Assume that f W (R n ) and set ϕ (ˆx ) = f(x) dx ( =,..., n). R If n = 2, then ϕ L (R) ( =, 2) (it follows from (.7)). Let n 3. It is easily seen that ϕ W (R n ) ( =,..., n). Applying inequality (.8), we obtain that ϕ L (n ), (R n ) ). Thus, (.2) W (R n ) n L (n ), ˆx (R n )[L x (R)]. These observations (together with embeddings proved in Section 4) led us to the definition of the scale of mixed norm spaces n (.3) V p (R n ) = L p, ˆx (R n )[L r p, (R)] x

4 496 Robert Algervi and Vitor I. Kolyada ( p (n ), r p = p /(n )). For p = we have V (R n ) = V (R n ) (see Section 5) and thus the space V is included to the scale. We prove that (.4) V (R n ) V p (R n ), p (n ), n 2. By virtue of the first embedding in (.), for n 3 and p = (n ) (.4) implies (.2). We obtain also some results concerning endpoints in the estimates of V p - norms (see Remar 5.4 and Theorem 5.7 below). In this paper we do not study relations between spaces V p with different values of p. We notice only that this problem is not immediate. In particular, the spaces V p ([, ] 2 ) do not form a monotone scale. We observe also that the scales of spaces of the type (.3) provide a flexible control of the growth of linear sections of functions. As in the wors [3], [6], [9], [4], embeddings of these spaces can be applied to obtain optimal results in various problems. Acnowledgements. remars. The authors are grateful to the referee for his/her useful 2. Iterative rearrangements For a measurable set E R, we denote by mes E the Lebesgue measure of E in R. Let f be a measurable function on R n. Recall that nonincreasing rearrangement of f is a nonnegative and nonincreasing function f on R + (, + ) which is equimeasurable with f (see [2, p. 37]). We assume in addition that the rearrangement is left continuous on R + (then it is defined uniquely). By S (R n ) we denote the class of all measurable and almost everywhere finite functions f on R n, for which mes n {x R n : f(x) > y} < for all y >. It is easy to see that f S (R n ) if and only if f (t) as t +. Further, we consider rearrangements with respect to specific variables. Let f S (R n ) and let n. Fix ˆx R n, and consider the function fˆx (x ) = f(x, ˆx ). By Fubini s theorem, fˆx S (R) for almost all ˆx R n. We denote the rearrangement of f with respect to x by R f. That is, we set R f(t, ˆx ) = (fˆx ) (t), t >. This function is defined almost everywhere on R + R n. Moreover, R f is a measurable function equimeasurable with f (see [8]). Let P n denote the set of all permutations σ = (,..., n ) of the numbers, 2,..., n. Let f S (R n ) and σ P n. The R σ -rearrangement of f is defined as the function R σ f(t) = R n R f(t), t R n +. That is, we obtain R σ f from f by rearranging f succesively with respect to the variables x,..., x n, starting with x. In so doing, we replace successively the arguments x,..., x n by the arguments t,..., t n. It is easy to see that R σ f decreases monotonically with respect to each variable. In view of the above observation, R σ f is equimeasurable with f. In what follows we set π(t) = t, t R n +.

5 On Fournier Gagliardo mixed norm spaces 497 For < p, s < and σ P n, the space Lσ p,s (R n ) is defined as the class of all functions f S (R n ) such that f L p,s σ ( R n + [ ] s π(t) /p dt ) /s R σ f(t) < π(t) (see [4]). Set also It was proved in [8] that L p,s (R n ) = σ P n L p,s σ (R n ). (2.) f p,s 2 /s /p f L p,s σ for < s p < and σ P n. Thus, for any σ P n (2.2) L p,s σ L p,s if < s p < (for p < s the converse embedding holds). The ey point of the proof is the following: if a function F defined on R n + is nonnegative and nonincreasing with respect to each variable, then for any t R n + (2.3) mes n {s R n + : F (s) F (t)} π(t). Basing on this observation, we give an alternative proof of (2.) with a better constant. Theorem 2.. Let f S (R n ) and σ P n. For all < s p <, (2.4) f L p,s f L p,s σ. Proof. Set F (t) = R σ f(t). We may suppose that (2.5) mes n {t R n + : F (t) = y} = for all y. Fix a > and set A ν = {t R n + : f (a ν+ ) F (t) < f (a ν )}, ν Z. Let t A ν. Then by (2.3) and (2.5), Thus, we have f s L p,s σ π(t) mes n {s R n + : F (s) f (a ν+ )} = ν Z = mes {u > : f (u) f (a ν+ )} = a ν+. = a s/p ν Z a s/p ν Z A ν π(t) s/p F (t) s dt ν Z a ν+ a ν(s/p ) f (u) s du a ν+ a ν a ν a (s/p )( ν+) A ν F (t) s dt u s/p f (u) s du = a s/p f s L p,s. Since a > is arbitrary, this implies inequality (2.4).

6 498 Robert Algervi and Vitor I. Kolyada Remar 2.2. Observe that embedding (2.2) is strict (see [8]). Moreover, if E = {(x, y): < y, < x }, x(ln(2/x)) p/s and < s < p <, then E <, but the characteristic function of the set E does not belong to L p,s {,2} (R2 ) L p,s {2,} (R2 ) (see [, p. 55]). In what follows we set R σ f V (R n + ) R σ f(ˆt, ) dˆt (σ P n, =,..., n). R n + Lemma 2.3. Let f S (R n ). Then (2.6) R σ f V (R n + ) f V (R n ) ( =,..., n) for any σ P n and any =,..., n. Proof. Let σ P n. We have f(x) f(ˆx, ) ψ (ˆx ) ( =,..., n) for almost all x R n. Hence, R σ f(t) R σ ψ (ˆt ), where σ is obtained from σ by removing. This gives R σ f(ˆt, ) R σ ψ (ˆt ), ˆt R n +. Integrating this inequality over R n +, and taing into account that R σ ψ (ˆt ) dˆt = ψ (ˆx ) dˆx = f V (R n ), R n we obtain (2.6). R n + 3. Projections and spaces V Let E R n be a measurable set and let n. For a point ˆx R n, denote by E(ˆx ) the ˆx -section of the set E, E(ˆx ) = {x R: (x, ˆx ) E}. By Fubini s theorem, for any n and almost all ˆx R n, the sections E(ˆx ) are measurable in R, and the functions m (ˆx ) = mes E(ˆx ), =,..., n, defined a.e. on R n, are measurable. The essential projection of E into the coordinate hyperplane x = is defined to be the set Π (E) of all points ˆx R n such that E(ˆx ) is measurable and m (ˆx ) >. Since the function m is measurable, the essential projection Π (E) is measurable. Let now f be a measurable function on R n. Let n. By Fubini s theorem, for almost all ˆx R n the sections fˆx are measurable functions on R. Moreover, the function (3.) ψ (ˆx ) = fˆx L (R) = ess sup x R f(x, ˆx ) (defined a.e. on R n ) is measurable. It suffices to prove the latter statement in the case when f is a bounded function with the compact support. In this case we have ψ (ˆx ) = lim ν fˆx L ν (R),

7 On Fournier Gagliardo mixed norm spaces 499 and the functions ˆx fˆx L ν (R) are measurable by Fubini s theorem. Thus, the definition of the space V (see (.5)) is correct. Now we shall show that the norm in V has a simple geometric interpretation. Let f be a non-negative measurable function on R n and let U f denote the region under the graph of f, U f = {(x, y) R n [; ): y f(x)}. Theorem 3.. Let f be a nonnegative measurable function on R n and let Π (U f ) be the essential projection of U f into the hyperplane x = ( n). Then f V (R n ) if and only if mes n Π (U f ) <. Moreover, in this case f V = mes n Π (U f ). Proof. We consider the case = n and set A = Π n (U f ). The set A consists of all points (ˆx n, y) such that ˆx n R n, y <, the function fˆxn is measurable on R, and (3.2) mes {x n R: f(x n, ˆx n ) y} >. Let a point ˆx n R n be such that fˆxn is measurable on R. First, assume that ψ n (ˆx n ) < (see (3.)) and ψ n (ˆx n ) < y <. Then (3.2) does not hold and (ˆx n, y) A. Now, let ψ n (ˆx n ) > and y < ψ n (ˆx n ). Then, by the definition of essential supremum, mes {x n R: f(x n, ˆx n ) > y} > and hence (ˆx n, y) A. We obtain that Thus, ψ n (ˆx n ) = mes {y > : (ˆx n, y) A}. ψ n (ˆx n ) dˆx n = mes n A. R n On the other hand, by the definition, the latter integral is equal to f Vn. proves the theorem. Fournier [6, Theorem 3.] proved the following theorem (see (.6)). This Theorem 3.2. Let f V (R n ) (n 2) be a nonnegative function. Then f S (R n ). Assume that a function g defined on R n is equimeasurable with f and has the property that for each y > the set {x R n : g(x) > y} is essentially a cube in R n with edges parallel to the coordinate axes. Then g V f V. The proof of this theorem employed the following Loomis Whitney isoperimetric inequality [] (this inequality follows also from (.2)). Theorem 3.3. Let E R n be a measurable set. Then (mes n E) n mes n Π (E). We observe that Theorems 3. and 3.3 combined give a shorter proof of Theorem 3.2.

8 5 Robert Algervi and Vitor I. Kolyada 4. Iterative rearrangement inequalities As we observed in Section, Theorem.3 implies Sobolev-type inequality (.8) for p =. It was proved in [8] that the L q,p -norm on the left-hand side in (.8) can be replaced by the stronger L q,p -norm. Theorem 4.. Let n 2 and p < n. Set q = np/(n p). If f Wp (R n ), then f L q,p (R n ) and (4.) f L q,p c D f p. We obtain a similar refinement for mixed norm spaces V. Denote by M dec (R n +) the class of all nonnegative functions on R n + which are nonincreasing in each variable. For any f M dec (R n +) and any σ P n we have that f = R σ f a.e. on R n +. Our main result is the following: Theorem 4.2. Let n 2, p,..., p n <, and (4.2) =. R+ n p Assume that f V (R n ). Then (4.3) t /p ( ) R σ f(t)dt p p /p f V for any σ P n. Proof. By virtue of Lemma 2.3, it is sufficient to prove inequality (4.3) for a function f M dec (R n +). Set { } A j = t R n + : t j (j =,..., n). Then (4.4) Indeed, set τ(t) = t /p n A j = R n +. j= t /p, t R n +. Assume that there is a point t R n + such that t j > τ(t) for all j =,..., n. Then, applying (4.2), we obtain τ(t) = > τ(t) /p +...+/p n = τ(t), j= t /p j j which is false. Let f M dec (R n +). Observe that (4.5) f(t) f(ˆt j, ) ψ j (ˆt j ), t R n +, j =,..., n.

9 Set On Fournier Gagliardo mixed norm spaces 5 ( ) pj. τ j (ˆt j ) = t /p Then A j = {t R n + : ˆt j R n +, t j τ j (ˆt j )}. Applying (4.5), we have A j From here and (4.4), (4.6) set t /p f(t) dt R n + R n + = p j j R n + j t /p f(t) dt t /p ψ j (ˆt j ) τj (ˆt j ) ψ j (ˆt j ) dˆt j = p j f Vj. p j f Vj. j= t /p j j dt j dˆt j Now we derive the multiplicative inequality (4.3) from (4.6). For ε,..., ε n >, ε = ε and g(t) = f(ε t,..., ε n t n ), t R n +. Then g M dec (R n +). Furthermore, (4.7) g V = ε ε f V ( =,..., n) and (4.8) R n + t /p g(t) dt = ε /p Applying (4.6) to the function g, we have t /p g(t) dt R n + R n + Further, using (4.7) and (4.8), we get (4.9) t /p f(t) dt R n + p g V. ε /p t /p f(t) dt. p ε f V. Now set ε = (p p f V ), =,..., n. Using (4.9) and taing into account (4.2), we obtain (4.3). Corollary 4.3. Let n 2 and < p < (n ). Set r = p /(n ). Assume that f V (R n ). Then n n (4.) t /r R σ f(t) dt (rr f Vn ) /r (pp f V ) /p R n + t /p n

10 52 Robert Algervi and Vitor I. Kolyada for any σ P n. In particular, f L n, σ (4.) f L n, σ for any σ P n. (R n ) and nn n f /n V. Indeed, setting p = = p n = p and p n = r, we have that = n + = p r r + =. r p Thus, the condition (4.2) in Theorem 4.2 is satisfied. Applying this theorem, we get (4.). Taing p = n in (4.), we obtain (4.). By (2.4), f L n, f L n,. Thus, σ estimate (4.) implies Theorem.3 (although with a worse constant coefficient). We emphasize that the norm in L n, σ (R n ) is stronger than the norm in L n, (R n ), and therefore (4.) gives a refinement of Theorem.3. Applying Theorem 4.2 and (.7), we obtain the following embedding for Sobolev spaces. Corollary 4.4. Let n 2, p,..., p n <, and n /p W (R n ), then (4.2) t /p ( ) R σ f(t) dt p p /p D f, for all σ P n. R n + For p = = p n = n inequality (4.2) becomes f L n, σ nn n This estimate coincides with (4.) (for p = ). D f /n. 5. Mixed norm spaces V p =. If f In this section we introduce the scale of intermediate mixed norm spaces V p (R n ). First we recall some definitions. For a function f S (R n ), set It is clear that f (t) = t t f (u) du. (5.) lim t + f (t) = f L. For < p <, the space L,p (R n ) consists of all f S (R n ) such that ( [ ] p f,p f (t) f dt ) /p (t) < t (see [], [3]). Applying the equality d dt f (t) = t (f (t) f (t)) (t > )

11 On Fournier Gagliardo mixed norm spaces 53 and using (5.), we obtain that L, (R n ) = L (R n ) and (5.2) f, = f for any f S (R n ). Further, let < r <. Assume that f L r, (R n ). Applying Fubini s theorem, we have t /r f (t) dt t = t /r f (t) dt r t. This implies that (5.3) r t /r [f (t) f (t)] dt t = r f r,. Applying (5.2), (5.3), and the monotone convergence theorem (for both increasing and decreasing sequences), we easily obtain that (5.4) lim r r f r, = f if f L r, (R n ) for some r >. The spaces V are defined as mixed norm spaces in which the interior norm is L -norm (see (.5)). That is, for a function f V (R n ) almost all linear sections fˆx are essentially bounded on R. Basing on equalities (5.2) and (5.3), it is natural to impose growth conditions on linear sections of functions in terms of L r, (R)-norms ( r ). Then Corollary 4.3 suggests that the corresponding scale of exterior norms should be formed by the spaces L p, (R n ), where r + n =. p These reasonings together with observations given in Introduction (see (.2)) lead us to the following definition. Let n 2 and p (n ). Set r p = p /(n ). For a function f S (R n ) and =,..., n, set ψ (p) (ˆx ) = f(ˆx, ) L rp, (R) and f V p = ψ(p) L p, (R n ). By V p (Rn ) we denote the class of all functions f S (R n ) such that f V p <. As usual, we write Set also V p (R n ) = V p (Rn ) = L p, ˆx (R n )[L rp, (R)]. n x V p (Rn ), f V p = f V p. In this section we study embeddings of the space V into the spaces V p. Observe that V = V, and the norms coincide. Indeed, if p =, then r p =, and by (5.2) f V (R n ) = f(ˆx, ), dˆx = f(ˆx, ) dˆx = f V (R n ). R n R n Further, we note that the case n = 2 and p = also is included to the definition of V p. In this case r = p =, and by (5.2), and similarly V (R 2 ) = L, (R)[L x (R)] = L x 2 (R)[L x (R)] x 2 V 2 (R 2 ) = L x (R)[L x 2 (R)];

12 54 Robert Algervi and Vitor I. Kolyada moreover, the corresponding norms coincide. Further, we have that (5.5) f V f V2 and f V 2 f V. Indeed, the first inequality in (5.5) is equivalent to the obvious estimate ess sup x2 R f(x, x 2 ) dx ess sup x2 R f(x, x 2 ) dx. R The second inequality in (5.5) is obtained similarly. Theorem 5.. Let n 2, < p (n ), and r = p /(n ). If f V (R n ), then f V p (R n ), and for every j =,..., n it holds that (5.6) f V p c j n,p f /r V j f /p V, where c n,p = if n = 2 and p =, and c n,p = (rr ) /r (pp ) (n )/p R j otherwise. Proof. In the case n = 2 and p = inequality (5.6) coincides with (5.5). Assume that either n = 2 and < p <, or n 3 and < p (n ). We will prove (5.6) for j = n. Let R n f(y, η) (y R n, η R + ) be the rearrangement of f with respect to the nth variable. Next, for a fixed η R +, let F (ξ, η) (ξ R + ) be the rearrangement of the function (5.7) y R n f(y, η), y R n, with respect to y. It follows from [4, Theorem 4.5. I] that (5.8) f V p n ξ /p η /r F (ξ, η) dξ dη. On the other hand, let σ = (n, n,..., ). For a fixed η R +, we tae the iterative rearrangement of the function (5.7) successively with respect to the variables y n,..., y. We obtain the rearrangement R σ f(s, η), s R+ n. By Theorem 2., for any fixed η, we have ( n ) /p (5.9) ξ /p F (ξ, η) dξ s R σ f(s, η) ds. Further, by Corollary 4.3, (5.) R n + R + ( n R n + ) /p s η /r R σ f(s, η) ds dη n (rr f Vn ) /r (pp f V ) /p. Applying inequalities (5.8), (5.9), and (5.), we obtain (5.6). Remar 5.2. If n 3, then the spaces V p (Rn ) formally can be defined for p > (n ), too (with r p = p /(n ) < ). However, in this case Theorem 5. fails to hold. Indeed, tae (n )/p < α < n 2 and set f(x) = ˆx j α χ (,) n(x), x R n,

13 On Fournier Gagliardo mixed norm spaces 55 A straightforward computation shows that f V (R n ) and for any r > f L p ˆx j (R n )[L r x j (R)] = (j =,..., n). Applying (5.6) and the theorem on the arithmetic and geometric means, we easily obtain the following: Corollary 5.3. Let either n = 2 and < p <, or n 3 and < p (n ). Set r r p = p /(n ). If f V (R n ), then f V p (R n ) and (5.) f V p r p f V + p j f Vj ( =,..., n). We shall show below that (5.) (divided by r p ) becomes equality as p. Remar 5.4. Let n = 2 and f V (R 2 ). By (5.6), we have that (5.2) f V p pp f /p V and f /p V 2 ( < p < ) (5.3) f V f V2. Observe that for f = χ (,) 2 we have equalities in (5.2) and (5.3). Hence, the constants in these inequalities are optimal. However, we notice that the constant in (5.2) tends to as p, but for p = we have inequality (5.3) with the constant. It is easy to explain this fact. Indeed, it follows directly from (5.2) that (5.4) lim sup p Besides, we show below that p f V p f V 2. (5.5) f V lim inf p p f V p. By virtue of relations (5.4) and (5.5), (5.3) follows from (5.2) as a limiting case as p. To prove (5.5), we observe that for any y R and any µ >, where Thus, f(, y) p, µ F µ (y) = s /p R f(s, y) ds µ /p F µ (y), µ R f(s, y) ds, y R. (5.6) f V p µ/p F µ p,. On the other hand, F µ p, pτ /p Fµ(τ) for any τ >, and therefore lim inf p p f V p F µ (by virtue of (5.6), it follows also from (5.4)). lim µ F µ. Thus, we obtain (5.5). Inequalities (5.) and (.7) yield the following: It is easy to see that f V =

14 56 Robert Algervi and Vitor I. Kolyada Corollary 5.5. If f W (R n ) (n 2), then f V p (R n ) for any < p (n ), and f V p c D f, where c depends only on p and n. Remar 5.6. The results of this section have been derived from the iterative rearrangement inequality (4.3) (see the proof of Theorem 5.). However, these results are expressed in terms of mixed norm spaces and therefore they give a more explicit description of the behaviour of linear sections of functions. In particular, taing p = n in Corollary 5.5, we obtain: For n = 2 we have that W (R n ) L n, ˆx (R n )[L n, x (R)], =,..., n. W (R 2 ) L 2, x 2 (R)[L 2, x (R)] L 2, x (R)[L 2, x 2 (R)]. This inclusion does not follow from the strong type Sobolev inequality (.8), which states that W (R 2 ) L 2, (R 2 ). Indeed, it was shown by Cwiel [5] that L 2, (R 2 ) L 2, x 2 (R)[L 2, x (R)]. At the same time, the results in terms of iterative rearrangements are stronger. In particular, if f W (R 2 ), then f L 2, (R 2 ) (see Theorem 4.). Mixed norm inequalities follow from here, since L 2, {,2} (R2 ) L 2, x 2 (R)[L 2, x (R)], L 2, {2,} (R2 ) L 2, x (R)[L 2, x 2 (R)] (see [4, Theorem 4.5. I]). Finally, we return to inequality (5.) and we shall study the limiting behaviour of f V p as p +. We observe that for any s j (5.7) s f s, = s t /s f (t) dt = (note that (5.4) can be also easily derived from (5.7)). f (u s ) du Theorem 5.7. Let n 2 and let {,..., n}. If f S (R n ) and (5.8) f j V j (R n ), then (5.9) lim p + n p f V p = f V. Proof. We prove (5.9) for = n. Let {p ν } be a decreasing sequence of numbers such that p ν > and p ν. Let s ν = p ν/(n ). Then {s ν } increases and s ν. Set F ν (y) = s ν f(y, ) L sν, (R), y R n.

15 By (5.7), Set also On Fournier Gagliardo mixed norm spaces 57 F ν (y) = G ν (y) = R n f(y, u s ν ) du. R n f(y, u s ν ) du. For a fixed y R n, the functions u R n f(y, u s ν ) form an increasing sequence on (, ) and lim R nf(y, u s ν ) = R n f(y, +) = f(y, ) L ν (R), u (, ). Setting f(y, ) L (R) = ϕ(y), and applying the monotone convergence theorem, we obtain that lim ν G ν(y) = ϕ(y) for any y R n. Moreover, {G ν (y)} is an increasing sequence on R n and thus (5.2) lim ν G ν(t) = ϕ (t) for any t > (see [2, p. 4]). We shall prove that (5.2) lim f ν s V pν = f n Vn. ν We have Fν (t) G ν(t) and therefore (5.22) s ν f V pν n = t /p ν F ν (t) dt t /p ν G ν(t) dt. For any fixed t (, ) the sequence {t /pν G ν(t)} increases. Thus, by (5.2) and the monotone convergence theorem, we have that (5.23) lim ν t /p ν G ν(t) dt = The monotone convergence theorem implies also that (5.24) lim ν G ν(t) dt = ϕ (t) dt. Now, applying (5.22), (5.23), and (5.24), we obtain that (5.25) f Vn = ϕ lim inf ν ϕ (t) dt. t /p ν G ν(t) dt lim inf ν s ν f V pν n. First, this implies (5.2) if f Vn =. Suppose now that f Vn <. Then, by virtue of assumption (5.8), we have that f V (R n ). Thus, by inequality (5.), f s V p n f Vn + p ν f Vj. ν s ν Since p ν /s ν as ν, it follows that (5.26) lim sup ν j s ν f V pν n f Vn. Inequalities (5.25) and (5.26) combined give (5.2). This proves the theorem.

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