On some Fournier–Gagliardo type inequalities

In this paper we consider nonnegative functions f on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} which are defined either by f(x)=min(f1(x1),…,fn(xn))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)=\min \,(f_1(x_1),\ldots ,f_n(x_n))$$\end{document} or by f(x)=min(f1(x^1),…,fn(x^n)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)=\min \,(f_1(\hat{x}_1),\ldots ,f_n(\hat{x}_n)).$$\end{document} Such minimum-functions are useful, in particular, in embedding theorems. We prove sharp estimates of rearrangements and Lorentz type norms for these functions, and we find the link between their Lorentz norms and geometric properties of their level sets.


Introduction
Let x = (x 1 , . . . , x n ). Denote byx k the (n − 1)-dimensional vector obtained from the n-tuple x by removal of its k-th coordinate. We shall write x = (x k ,x k ).
Theorem 1.2 Assume that n ≥ 2 and ψ k ∈ L 1 (R n−1 ) (ψ k ≥ 0, k = 1, . . . , n). Then ∈ L n ,1 (R n ) and It is important to note that there are normalizing factors in the definitions of Lorentz norms given in Sect. 2. Theorem 1.2 implies the embedding of W 1 1 (R n ) into the Lorentz space L n ,1 (R n ). Observe that this theorem was obtained by Fournier [11] in an equivalent form in terms of mixed norm spaces L 1 x k ] (see Sect. 3). Different extensions of Theorem 1.2 and their applications have been studied in the works [1,7,17,19,23].
As in [1,19], in this paper we consider Lorentz spaces defined in terms of iterated rearrangements. Let n be the collection of all permutations of the set {1, . . . , n}. For each σ ∈ n , and 0 < p, r < ∞, we define a Lorentz space L p,r σ (R n ) (see Sect. 2). The relations between L p,r σ -spaces and the classical Lorentz L p,r -spaces are the following: L p,r σ ⊂ L p,r (r ≤ p), L p,r ⊂ L p,r σ ( p ≤ r ), (1.4) and for p = r these embeddings are strict (see [26]). We proved in [16] that Sobolev embeddings can be strengthened using the L-norms. In particular, we have that for any f ∈ W 1 1 (R n ) (1.5) By virtue of (1.4), (1.5) implies embedding W 1 1 (R n ) ⊂ L n ,1 (R n ). In [1], there were proved estimates of generalized Lorentz norms in terms of mixed norm spaces L 1 x k [L ∞ x k ]. These estimates provided a strengthening of Fournier's estimates of classical Lorentz norms [11], but the constants were not optimal. Optimal constants were obtained in [19], where we proved the following extension of Theorem 1.2. Theorem 1. 3 Assume that ψ k ∈ L 1 (R n−1 ) (ψ k ≥ 0, k = 1, . . . , n). Let be defined by (1.2). Then ∈ L n ,1 σ (R n ) and || || L n ,1 σ ≤ n k=1 ||ψ k || L 1 (R n−1 ) 1/n (1.6) for any σ ∈ n .
We see that estimates of functions defined as minimum provide sharp embeddings of Sobolev spaces W 1 1 (R n ). The main objective of this paper is to study another type of such functions. Let f j be nonnegative rearrangeable functions on R. Set (1.7) We observe that in the case, when the functions f j vanish outside R + and decrease on R + , this definition and the expression for the rearrangement f * (t) (see Sect. 4) are closely related to the concept of average modulus of continuity (see [15], [18, p. 51]). This concept was useful in the study of embeddings of some anisotropic function classes.
We prove that ). This implies that estimate (1.8) (with f j = ψ * j ) gives a strengthening of the estimate (1.3). Our main results are related to generalized Lorentz norms (defined in terms of iterated rearrangements). A crucial role here is played by the following equality involving iterated rearrangements of the function (1.7): for any α j > 0 and any σ ∈ n First, this equality immediately implies that 1 for any σ ∈ n and any p > 0. (1.11) The phenomenon exhibited in (1.11) is closely connected with an important geometric property of functions f defined by (1.7). Denote by P n the class of all measurable nonnegative functions on R n such that for any y > 0 the Lebesgue set {x ∈ R n : f (x) > y} is a cartesian product of measurable sets E i (y) ⊂ R. It is obvious that any function f defined by (1.7) belongs to P n . We prove that the inverse statement also is true, that is, any function f ∈ P n can be represented as in (1.7). Thus, for any function f ∈ P n its classical and generalized Lorentz norms coincide. Another immediate consequence of (1.10) is the following inequality: if n k=1 α k = 1 and f j ∈ L 1 (R), then for any σ ∈ n (1.12) By (1.11), this inequality is the same as inequality (1.8). We show that inequality (1.13) (with f j = ψ * j ) is stronger than inequality (1.6). Note also that (1.13) has an equivalent form in terms of mixed norm spaces L 1 We observe that our interest to these questions is inspired by their connections with embedding theorems and Loomis-Whitney type inequalities. Recall that Loomis-Whitney inequality [20] states that for any F σ -set E ⊂ R n mes n E ≤ n k=1 (mes n−1 E k ) 1/(n−1) , (1.14) where E k is the orthogonal projection of E onto the coordinate hyperplane x k = 0. Actually (1.14) can be immediately derived from inequality (1.1). Successively applying (1.14), one can obtain a more general inequality (see [14,Chapter 4,4.4.2]) . The case m = 1 corresponds to the function (1.2). The case m = n − 1 (in which (1.15) is obvious) corresponds to the function (1.7). Probably, it would be interesting to consider similar minimum-functions for any 1 < m < n − 1. The paper is organized as follows. In Sect. 2 we give the basic definitions of rearrangements and Lorentz spaces. In Sect. 3 we define mixed norm spaces and study some of their properties. In Sect. 4 we obtain some simple results concerning estimates of rearrangements and classical Lorentz norms of functions (1.2) and (1.7). Section 5 is devoted to the main results of this paper related to generalized Lorentz norms of the function (1.7). In Sect. 6 we show that L n ,1 σ -norms of the function (1.2) are majorized by L n,1 σ -norms of the function (1.7) (where f j = ψ * j ).

Rearrangements and Lorentz spaces
A measurable and almost everywhere finite real-valued function f on R n is said to be rearrangeable if A nonincreasing rearrangement of a rearrangeable function f defined on R n is a nonnegative and nonincreasing function f * on R + ≡ (0, +∞) which is equimeasurable with | f |, that is, λ f * = λ f . We shall assume in addition that the rearrangement f * is left-continuous on R + . Under this condition it is defined uniquely by Besides, we have the equality (see [9, p. 32]) Let 0 < p, r < ∞. A rearrangeable function f on R n belongs to the Lorentz space We emphasize that the latter integral is multiplied by r / p (as in the original definitions of Lorentz [21,22]).
We have that || f || p, p = || f || p . For a fixed p, the Lorentz spaces L p,r strictly increase as the secondary index r increases; that is, the strict embedding L p,r ⊂ L p,s (r < s) holds (see [5,Ch. 4]).
The Lorentz quasinorm can be given in an alternative form. Namely, (see [13,Proposition 1.4.9]). Let f be a rearrangeable function on R n and let 1 ≤ k ≤ n. We fixx k ∈ R n−1 and consider thex k -section of the function f For almost allx k ∈ R n−1 the function fx k is rearrangeable on R. We set Stress that the k-th argument of the function R k f is equal to u. The function R k f is defined almost everywhere on R + × R n−1 ; we call it the rearrangement of f with respect to the k-th variable. It is easy to show that R k f is a measurable function equimeasurable with | f |. As above, let n be the collection of all permutations of the set {1, . . . , n}. For each σ = {k 1 , . . . , k n } ∈ n we call the function the R σ -rearrangement of f . Thus, we obtain R σ f by "rearranging" f in nonincreasing order successively with respect to the variables It is nonnegative, nonincreasing in each variable, and equimeasurable with | f | (see [3,4,8,16,18,19]).
Let 0 < p, r < ∞ and let σ ∈ n (n ≥ 2). We denote by L p,r (see [8]). The choice of a permutation σ is essential. The relations between L p,r -and L p,r σ -norms are described by embeddings (1.4). Moreover, the following proposition gives sharp constants in these relations (see [3,19]).

Proposition 2.1 Let f be a rearrangeable function on
These inequalities are optimal.
We stress again that for p = r the norms || · || p,r and || · || p,r ;σ may be essentially different. We consider the following simple example which will be also used in the sequel.
As usual, by χ E we denote the characteristic function of a set E, which is equal to 1 on E and 0 outside E.
On the other hand,

Mixed norm spaces
Let f be a measurable function on R n and let 1 ≤ k ≤ n. By Fubini's theorem, for almost allx k ∈ R n−1 the sections fx k are measurable functions on R. Moreover, the function (defined a.e. on R n−1 ) is measurable. It suffices to prove the latter statement in the case when f is a bounded function with compact support. In this case we have and the functionsx k → || fx k || L ν (R) are measurable by Fubini's theorem. Denote by M dec (R n + ) the class of all nonnegative functions on R n which vanish off R n + and are nonincreasing in each variable on R n Thus, in this case ψ k is the trace of f on the hyperplane x k = 0 in the sense of almost everywhere convergence (see [24,Chapter 6]). Let be the space of measurable functions on R n with the finite mixed norm where ψ k is defined by (3.1). As it was observed above, ψ k is measurable on R n−1 , and thus this definition is correct. Denote also An equivalent form of Theorem 1.2 proved by Fournier [11,Theorem 4.1] is the following.
The constant is optimal.
and let D k f denote the first order weak partial derivative of f with respect to x k . Then (see [11, p. 57]). Thus, by (3.4), The following equivalent form of Theorem 1.3 was proved in [19].
The constant is optimal.
Now we introduce another type of mixed norm spaces (related to the traces on coordinate straight lines). Let be the space of measurable functions on R n with the finite mixed norm As above, it is easy to see that these definitions are correct.
Thus, in this case f k is the trace of f on the coordinate straight linex k = 0 in the sense of almost everywhere convergence.

Remark 3.3
Clearly, for n = 2 the spaces U and V coincide. It is also easy to see that for n ≥ 3 neither of the spaces U and V is contained in the other. Indeed, let n = 3. The function belongs to U but does not belong to V. On the other hand, the function belongs to V but does not belong to U.
By (3.5), V k -norms are estimated by L 1 -norms of the first order weak partial derivatives. Similarly, U k -norms can be estimated by L 1 -norms of the pure (non-mixed) partial derivatives of the order n − 1.
Assume that for a function f ∈ L 1 (R m ) all pure weak partial derivatives of the order m exist and belong to L 1 (R m ). Sobolev's theorem asserts that in this case the function f can be modified on a set of measure zero so as to become uniformly continuous and bounded on R m and ||D m k f || 1 (3.9) (see [6, §10]).
Assume that a function f ∈ L 1 (R n ) (n ≥ 2) has all pure weak partial derivatives of the order n − 1. Then for any 1 ≤ j ≤ n and almost every fixed x j ∈ R the function f x j (x j ) = f (x) satisfies the conditions of Sobolev's theorem for m = n − 1 and therefore by (3.9) This implies that

Embeddings into classical Lorentz spaces
We begin with the following simple theorem. If, in addition, f j ∈ L 1 (R), then f ∈ L n,1 (R n ) and

The constant in (4.3) is optimal.
Proof Let y > 0. Then f (x) > y if and only if f j (x j ) > y for all j = 1, . . . , n. That is, , and we have equality in (4.3). Thus, the constant is optimal.
Similarly to Theorems 3.1, 4.1 can be expressed in terms of mixed norms defined by (3.8). Namely, the following theorem is equivalent to Theorem 4.1.
We observe that without loss of generality we can assume that in (4.1) f j are nonincreasing functions on R + with f j (+∞) = 0 that vanish off R + ( j = 1, . . . , n).
Indeed, let f be defined by (4.1). Let σ ∈ n (recall that by n we denote the collection of all permutations of the set {1, . . . , n}). We consider the iterated rearrangement for any σ ∈ n . To show this, it is sufficient to use the following observation: if g(x) ≥ 0 (x ∈ R), a ≥ 0, and h(x) = min(g(x), a), then h * (t) = min(g * (t), a). It is easy to obtain the rearrangement of the function = R σ f in an explicit form.

Theorem 4.3 Let ϕ j be nonnegative nonincreasing and left-continuous functions on
Then for any t > 0.
Proof Fix t > 0. Assume that n j=1 u j = t, u j > 0.

(4.15)
Proof Let t > 0 and let E ⊂ R n be an F σ -set with mes n E = t. Let E j be the projection of E onto the hyperplane x j = 0 ( j = 1, . . . , n). Then for any x ∈ E and any j = 1, . . . , n inf

Estimates of generalized Lorentz norms
In this section we obtain our main results. As above, by n we denote the collection of all permutations of the set {1, . . . , n}. First, we prove the following theorem. 1, . . . , n). Then for any σ ∈ n R n By (4.6), for any σ ∈ n Thus, (5.2) is equivalent to the equality We may assume that all functions ϕ j are continuously differentiable, ϕ j (+0) = +∞, and ϕ j (t) > 0, ϕ j (t) < 0 for all t > 0 and all j = 1, . . . , n.
Indeed, the first equality is obvious. Further, let i = j. Then x ∈ A i ∩ A j if and only if x j = η j (ϕ i (x i )). Thus, the projection of the set A i ∩ A j onto the 2-dimensional plane (x i , x j ) has the 2-dimensional measure zero and therefore mes n ( Indeed, x ∈ A j if and only ifx j ∈ R n−1 + and x j ≥ γ j,k (x k ) for any k = j. Thus, by Fubini's theorem, we have Change of variable y = ϕ j (x j ) gives Since η k (y) = λ ϕ k (y), this implies (5.5).
Then for any σ ∈ n Indeed, applying Hölder's inequality at the right-hand side of (5.2), we obtain In the case f j (t) = χ [0,1] (t) ( j = 1, . . . , n) we have equality in (5.6). This shows that the constant in (5.6) is optimal. Now we suppose that, in Theorem 5.1, α 1 = · · · = α n = 1/ p. Then equality (5.2) assumes the form R n Hence, using (2.3), we obtain that for any σ ∈ n . Thus, we have the following statement. We have emphasized in Sect. 2that, in general, the spaces L p,1;σ (R n ) are strictly smaller than L p,1 (R n ) (see Example 2.2). We shall now analyse the situation that we have in Theorem 5.3.
First, denote by Q n the class of all measurable functions on R n such that for any y > 0 the set is essentially a cube in R n (that is, for any y > 0 there exists a cube Q ⊂ R n with edges parallel to coordinate axes which differs from E f (y) by a set of measure 0). It was shown in [19] that for any function f ∈ Q n (n ≥ 2) f L n ,1 σ = || f || n ,1 for any σ ∈ n . (5.12) We shall show that equality (5.12) is true for a much wider class of functions. Denote by P n the class of all measurable functions on R n such that for any y > 0 the set (5.11) is a cartesian product of measurable sets E i (y) ⊂ R, E(y) = E 1 (y) × · · · × E n (y). (5.13) By virtue of equality (4.4), any function f defined by (4.1) belongs to P n . We shall prove the inverse statement.

Proposition 5.4
Let f ∈ P n (n ≥ 2) be a nonnegative rearrangeable function on R n . Set Then E(y) is a cartesian product (5.13), where E i (y) ⊂ R. Since x / ∈ E(y), there exists j ∈ {1, . . . , n} such that x j / ∈ E j (y). Whatever be u ∈ R n−1 , we have (x j , u) / ∈ E y and therefore . Taking into account (5.15), we obtain (5.14).
Applying Proposition 5.4 and Theorem 5.3, we get the following result. Remark 5. 6 We observe that the coincidence of the classical and generalized Lorentz norms which holds for any function f defined by (5.9), may not hold for the function defined by (4.12). Let N ≥ 1 and let g N be the function defined in Example 2.2.
Similarly to Theorems 4.1, 5.7 can be expressed in terms of mixed norms defined by (3.8). Namely, the following theorem is equivalent to Theorem 5.7.
The equivalence follows by standard arguments. First, assume that f ∈ U(R n ). We have for almost all x ∈ R n . Denote the right-hand side of the above inequality by f k (x k ) and setf We have || f || U k = || f k || 1 and | f (x)| ≤f (x). Thus, (5.18) follows immediately from Theorem 5.7.

Comparison of estimates (1.13) and (1.6)
In Sect. 4we obtained inequality (4.15) between classical Lorentz norms of functions (4.12) and (4.13). In this section we prove a similar inequality between generalized Lorentz norms of these functions. For this, we apply a special bijection of R n + onto R n + . Assume that u ∈ R m + , m ≥ 2. Then Then is a differentiable bijective mapping from R n + onto R n + with the jacobian Proof First we show that is injective. For n = 2 it is obvious. Let n ≥ 3. Assume that there exist x , x ∈ R n + such that By our assumption, x j for all k = 2, . . . , n.
It remains to prove equality (6.4). Fix x ∈ R n + and consider the Jacobi matrix of at the point x. The i-th row of this matrix is formed by the partial derivatives of the function Assume that x k = 0 for all k = 1, . . . , n. The j-th element of the i-th row is equal to and it is equal to 0 if i = j. It is easily verified that the determinant of this matrix is equal to π n (x) n−2 det(γ i j ), where Adding to the first row of the matrix (γ i j ) all other rows, we obtain the matrix (γ i j ), whereγ 1 j = n − 1 andγ i j = γ i j for i ≥ 2 ( j = 1, . . . , n). Next, from all rows of the matrix (γ i j ) beginning from the second, we subtract its first row divided by n − 1.
As above, we denote by M dec (R m + ) the class of all nonnegative functions on R m which vanish off R m + and are nonincreasing in each variable on R m and mes m E u = π m (u). Applying (2.2), we obtain (6.6).

Remark 6.3
The example given above in Remark 4.7 shows that inequality (6.7) cannot be reverted, even by inserting an arbitrarily small constant to the right-hand side. Namely, in this example Theorem 6.2 and Remark 6.3 show that inequality (1.13) (with f j = ψ * j ) is stronger than inequality (1.6).
However, we observe that inequality (6.6) which was used in the proof of Theorem 6.2 is rather rough. Apparently, the constant in inequality (6.7) may be improved. Namely, Theorems 1.3 and 5.7 suggest that the optimal constant should be (n − 1) −n .