## Abstract

In this paper we consider nonnegative functions *f* on \(\mathbb {R}^n\) which are defined either by \(f(x)=\min \,(f_1(x_1),\ldots ,f_n(x_n))\) or by \(f(x)=\min \,(f_1(\hat{x}_1),\ldots ,f_n(\hat{x}_n)).\) 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.

## 1 Introduction

Let \(x=(x_1,\ldots ,x_n).\) Denote by \(\hat{x}_k\) the \((n-1)\)-dimensional vector obtained from the *n*-tuple *x* by removal of its *k*-th coordinate. We shall write \(x=(x_k,\hat{x}_k).\)

Let \(\psi _1,\ldots ,\psi _n\) be nonnegative measurable functions on \(\mathbb {R}^{n-1}~~~~(n\ge 2).\) We consider the geometric mean

Gagliardo [12] proved the following theorem.

### Theorem 1.1

Assume that \(n\ge 2\) and \(\psi _k\in L^1(\mathbb {R}^{n-1})\) \((k=1,\ldots ,n).\) Then \(\mathcal {G}\in L^{n'}(\mathbb {R}^n)~~~(n'=n/(n-1))\) and

This theorem yields the embedding of the Sobolev space \(W_1^1(\mathbb {R}^n)\) into \(L^{n'}(\mathbb {R}^n)\) (see [12]). Later on, it was shown in [2] (see also [10, 11, 25]) that the space \(W_1^1(\mathbb {R}^n)\) is embedded into the Lorentz space \(L^{n',1}(\mathbb {R}^n)\) which is strictly smaller than \(L^{n'}(\mathbb {R}^n)\). We emphasize that a similar refinement of Theorem 1.1 is not true; that is, the function \(\mathcal {G}\) may not belong to \(L^{n',1}(\mathbb {R}^n)\) (see [19, Remark 3.12]). However, set

Clearly, \(\Psi (x)\le \mathcal G(x).\) The following theorem holds [11].

### Theorem 1.2

Assume that \(n\ge 2\) and \(\psi _k\in L^1(\mathbb {R}^{n-1})\) \((\psi _k \ge 0,\,\,k=1,\ldots ,n).\) Then \(\Psi \in L^{n',1}(\mathbb {R}^n)\) 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 \(W_1^1(\mathbb {R}^n)\) into the Lorentz space \(L^{n',1}(\mathbb {R}^n).\) Observe that this theorem was obtained by Fournier [11] in an equivalent form in terms of mixed norm spaces \(L^1_{\hat{x}_k}[L^\infty _{x_k}]\) (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 \(\Omega _n\) be the collection of all permutations of the set \(\{1,\dots ,n\}\). For each \(\sigma \in \Omega _n,\) and \(0<p, r<\infty \), we define a Lorentz space \(\mathcal L^{p,r}_{\sigma }(\mathbb R ^n)\) (see Sect. 2). The relations between \(\mathcal L^{p,r}_{\sigma }\)-spaces and the classical Lorentz \(L^{p,r}\)-spaces are the following:

and for \(p\not =r\) these embeddings are strict (see [26]).

We proved in [16] that Sobolev embeddings can be strengthened using the \(\mathcal L\)-norms. In particular, we have that for any \(f\in W_1^1(\mathbb {R}^n)\)

By virtue of (1.4), (1.5) implies embedding \(W_1^1(\mathbb {R}^n)\subset L^{n',1}(\mathbb {R}^n).\)

In [1], there were proved estimates of generalized Lorentz norms in terms of mixed norm spaces \(L^1_{\hat{x}_k}[L^\infty _{x_k}]\). 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 \(\psi _k\in L^1(\mathbb {R}^{n-1})\) \((\psi _k\ge 0,\,\,k=1,\ldots ,n).\) Let \(\Psi \) be defined by (1.2). Then \(\Psi \in \mathcal L^{n',1}_\sigma (\mathbb {R}^n)\) and

for any \(\sigma \in \Omega _n\).

Since \(||f||_{ L^{n',1}}\le ||f||_{\mathcal L^{n',1}_\sigma }\) (see (2.5) below), inequality (1.6) gives a refinement of inequality (1.3). Inequality (1.6) cannot be improved; it becomes equality if \(\psi _k\) are equal to the characteristic function of the unit cube \([0,1]^{n-1}.\)

It follows from (1.6) that inequality (1.5) holds with \(c=1/2\). This constant is optimal (see [19]).

We see that estimates of functions defined as *minimum* provide sharp embeddings of Sobolev spaces \(W_1^1(\mathbb {R}^n).\) The main objective of this paper is to study another type of such functions. Let \(f_j\) be nonnegative rearrangeable functions on \(\mathbb {R}\). Set

We observe that in the case, when the functions \(f_j\) vanish outside \(\mathbb {R}_+\) and decrease on \(\mathbb {R}_+\), 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

Further, we show that

(where \(\Psi \) is defined by (1.2) and \(f_j=\psi _j^*\) in (1.7)). This implies that estimate (1.8) (with \(f_j=\psi _j^*\)) 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 \(\alpha _j>0\) and any \(\sigma \in \Omega _n\)

First, this equality immediately implies that

The phenomenon exhibited in (1.11) is closely connected with an important geometric property of functions *f* defined by (1.7). Denote by \(\mathcal P_n\) the class of all measurable nonnegative functions on \(\mathbb {R}^n\) such that for any \(y>0\) the Lebesgue set \(\{x\in \mathbb {R}^n: f(x)>y\}\) is a cartesian product of measurable sets \(E_i(y)\subset \mathbb {R}.\) It is obvious that any function *f* defined by (1.7) belongs to \(\mathcal P_n.\) We prove that the inverse statement also is true, that is, any function \(f\in \mathcal P_n\) can be represented as in (1.7). Thus, for any function \(f\in \mathcal P_n\) its classical and generalized Lorentz norms coincide.

Another immediate consequence of (1.10) is the following inequality: if \(\sum _{k=1}^n\alpha _k=1\) and \(f_j\in L^1(\mathbb {R})\), then for any \(\sigma \in \Omega _n\)

Taking \(\alpha _j=1/n\,\, (j=1,\ldots ,n)\) we obtain the inequality

By (1.11), this inequality is the same as inequality (1.8). We show that inequality (1.13) (with \(f_j=\psi _j^*\)) is stronger than inequality (1.6).

Note also that (1.13) has an equivalent form in terms of mixed norm spaces \(L^1_{x_k}[L^\infty _{\hat{x}_k}]\) (see Sect. 3).

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 \(F_\sigma \)-set \(E\subset \mathbb {R}^n\)

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])

where \(E_{i_1,\ldots ,i_m}\) is the projection of *E* onto \((n-m)\)-dimensional coordinate plane \(x_{i_1}=\cdots =x_{i_m}=0\) \((1\le m\le n-1)\). 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 \(\mathcal L_\sigma ^{n',1}\)-norms of the function (1.2) are majorized by \(\mathcal L_\sigma ^{n,1}\)-norms of the function (1.7) (where \(f_j=\psi _j^*\)).

## 2 Rearrangements and Lorentz spaces

A measurable and almost everywhere finite real-valued function *f* on \(\mathbb {R}^n\) is said to be *rearrangeable* if

*A nonincreasing rearrangement* of a rearrangeable function *f* defined on \(\mathbb {R}^n\) is a nonnegative and nonincreasing function \(f^*\) on \(\mathbb {R}_+ \equiv (0, + \infty )\) which is equimeasurable with |*f*|, that is, \(\lambda _{f^*}=\lambda _f.\) We shall assume in addition that the rearrangement \(f^*\) is left-continuous on \(\mathbb {R}_+.\) Under this condition it is defined uniquely by

Besides, we have the equality (see [9, p. 32])

Let \(0<p,r<\infty .\) A rearrangeable function *f* on \(\mathbb {R}^n\) belongs to the Lorentz space \(L^{p,r}(\mathbb {R}^n)\) if

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 \(L^{p,r}\subset L^{p,s}~~~(r<s)\) 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 \(\mathbb {R}^n\) and let \(1\le k \le n.\) We fix \(\hat{x}_k\in \mathbb R^{n-1} \) and consider the \(\hat{x}_k\)-section of the function *f*

For almost all \(\hat{x}_k\in \mathbb R^{n-1}\) the function \(f_{\hat{x}_k}\) is rearrangeable on \(\mathbb {R}\). We set

Stress that the *k*-th argument of the function \(\mathcal R_kf\) is equal to *u*. The function \(\mathcal R_kf\) is defined almost everywhere on \(\mathbb R_+\times \mathbb R^{n-1}\); we call it *the rearrangement of f with respect to the k-th variable.* It is easy to show that \(\mathcal R_kf\) is a measurable function equimeasurable with |*f*|. As above, let \(\Omega _n\) be the collection of all permutations of the set \(\{1,\dots ,n\}\). For each \(\sigma =\{k_1,\dots ,k_n\}\in \Omega _n\) we call the function

the \(\mathcal R_\sigma \)-rearrangement of *f*. Thus, we obtain \(\mathcal R_\sigma f\) by “rearranging” *f* in non-increasing order successively with respect to the variables \(x_{k_1},\dots ,x_{k_n}\). The rearrangement \(\mathcal R_\sigma f\) is defined on \(\mathbb {R}_+^n.\) It is nonnegative, nonincreasing in each variable, and equimeasurable with |*f*| (see [3, 4, 8, 16, 18, 19]).

Let \(0<p, r<\infty \) and let \(\sigma \in \Omega _n\, (n\ge 2).\) We denote by \(\mathcal L^{p,r}_{\sigma }(\mathbb R ^n)\) the class of all rearrangeable functions *f* on \(\mathbb {R}^n\) such that

(see [8]). The choice of a permutation \(\sigma \) is essential.

The relations between \(L^{p,r}\)- and \(\mathcal L^{p,r}_{\sigma }\)-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 \(\mathbb {R}^n,~~n\ge 2.\) Then for any \(\sigma \in \Omega _n\)

and

These inequalities are optimal.

We stress again that for \(p\not =r\) the norms \(||\cdot ||_{p,r}\) and \(||\cdot ||_{p,r;\sigma }\) may be essentially different. We consider the following simple example which will be also used in the sequel.

As usual, by \(\chi _E\) we denote the characteristic function of a set *E*, which is equal to 1 on *E* and 0 outside *E*.

### Example 2.2

Let \(N\ge 1\). Set

Then \(|E_N|\le 1\) and

On the other hand,

where \(\alpha _N\) is determined by the equation \(\alpha _N(\ln (e/\alpha _N))^2=1/N\) and \(\alpha _N\rightarrow 0\) as \(N\rightarrow \infty .\) Thus, \(||f_N||_{2,1;\sigma }\rightarrow \infty .\)

## 3 Mixed norm spaces

For a measurable set \(E\subset \mathbb {R}^d,\) we denote by \({\text {mes}}_d E\) the Lebesgue measure of *E* in \(\mathbb {R}^d.\)

Let *f* be a measurable function on \(\mathbb {R}^n\) and let \(1\le k\le n.\) By Fubini’s theorem, for almost all \(\hat{x}_k\in \mathbb {R}^{n-1}\) the sections \(f_{\hat{x}_k}\) are measurable functions on \(\mathbb {R}.\) Moreover, the function

(defined a.e. on \(\mathbb {R}^{n-1}\)) 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 functions \(\hat{x}_k\mapsto ||f_{\hat{x}_k}||_{L^\nu (\mathbb {R})}\) are measurable by Fubini’s theorem.

Denote by \(\mathcal {M}_{dec}(\mathbb {R}^n_+)\) the class of all nonnegative functions on \(\mathbb {R}^n\) which vanish off \(\mathbb {R}^n_+\) and are nonincreasing in each variable on \(\mathbb {R}^n_+\).

Let \(f\in \mathcal {M}_{dec}(\mathbb {R}^n_+)\). Assume that for almost all \(\hat{x}_k\in \mathbb {R}^{n-1}\) the function *f* is bounded with respect to \(x_k.\) Then

Thus, in this case \(\psi _k\) 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 \(\mathbb {R}^n\) with the finite mixed norm

where \(\psi _k\) is defined by (3.1). As it was observed above, \(\psi _k\) is measurable on \(\mathbb {R}^{n-1}\), 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.

### Theorem 3.1

If \(f\in \mathcal {V}(\mathbb {R}^n) ~~(n\ge 2),\) then \(f\in L^{n',1}(\mathbb {R}^n)\) and

The constant is optimal.

Theorem 3.1 implies the embedding of \(W_1^1(\mathbb {R}^n)\) into the Lorentz space \(L^{n',1}(\mathbb {R}^n).\) Indeed, let \(f\in W_1^1(\mathbb {R}^n)\) and let \(D_kf\) 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].

### Theorem 3.2

If \(f\in \mathcal {V}(\mathbb {R}^n) ~~(n\ge 2),\) then \(f\in \mathcal L^{n',1}_\sigma (\mathbb {R}^n)\) and

for any \(\sigma \in \Omega _n.\) 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 \(\mathbb {R}^n\) with the finite mixed norm

Denote also

As above, it is easy to see that these definitions are correct.

Let \(f\in \mathcal {M}_{dec}(\mathbb {R}^n_+)\). Assume that for almost all \(x_k\in \mathbb {R}\) the function *f* is bounded with respect to \(\hat{x}_k.\) Then

Thus, in this case \(f_k\) is the *trace* of *f* on the coordinate straight line \(\hat{x}_k=0\) in the sense of almost everywhere convergence.

### Remark 3.3

Clearly, for \(n=2\) the spaces \(\mathcal {U}\) and \(\mathcal {V}\) coincide. It is also easy to see that for \(n\ge 3\) neither of the spaces \(\mathcal {U}\) and \(\mathcal {V}\) is contained in the other. Indeed, let \(n=3\). The function

belongs to \(\mathcal {U}\) but does not belong to \(\mathcal {V}.\) On the other hand, the function

belongs to \(\mathcal {V}\) but does not belong to \(\mathcal {U}.\)

By (3.5), \(\mathcal V_k\)-norms are estimated by \(L^1\)-norms of the first order weak partial derivatives. Similarly, \(\mathcal {U}_k\)-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 \(f\in L^1(\mathbb {R}^m)\) all pure weak partial derivatives

of the order *m* exist and belong to \(L^1(\mathbb {R}^m).\) 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 \(\mathbb {R}^m\) and

(see [6, §10]).

Assume that a function \(f\in L^1(\mathbb {R}^n)\) \((n\ge 2)\) has all pure weak partial derivatives of the order \(n-1.\) Then for any \(1\le j\le n\) and almost every fixed \(x_j\in \mathbb {R}\) the function \(f_{x_j}(\hat{x}_j)=f(x)\) satisfies the conditions of Sobolev’s theorem for \(m=n-1\) and therefore by (3.9)

This implies that

## 4 Embeddings into classical Lorentz spaces

We begin with the following simple theorem.

### Theorem 4.1

Let \(f_j\) \((j=1,\ldots ,n)\) be nonnegative rearrangeable functions on \(\mathbb {R}\) and let

Then

If, in addition, \(f_j\in L^1(\mathbb {R})\), then \(f\in L^{n,1}(\mathbb {R}^n)\) 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 \(j=1,\ldots ,n.\) That is,

This implies (4.2).

Now, applying (2.3), (4.2), and using Hölder’s inequality, we get

If \(f_j(u)=\chi _{[0,1]}(u),\) then \(f(x)=\chi _{[0,1]^n}(x),\) \(f^*(t)=\chi _{[0,1]}(t),\) and we have equality in (4.3). Thus, the constant is optimal. \(\square \)

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.

### Theorem 4.2

If \(f\in \mathcal {U}(\mathbb {R}^n) ~~(n\ge 2),\) then \(f\in L^{n,1}(\mathbb {R}^n)\) and

To show the equivalence, we first assume that \(f\in \mathcal {U}(\mathbb {R}^n)\). Let \(f_k\) be defined by (3.8). Then

Since \(||f||_{\mathcal {U}_k}=||f_k||_1,\) (4.5) follows immediately from (4.3). Conversely, assume that \(f_j\in L^1(\mathbb {R})\) \((j=1,\ldots ,n)\) are nonnegative functions on \(\mathbb {R}\) and *f* is defined by (4.1). Then

We observe that without loss of generality we can assume that in (4.1) \(f_j\) are nonincreasing functions on \(\mathbb {R}_+\) with \(f_j(+\infty )=0\) that vanish off \(\mathbb {R}_+\) \((j=1,\ldots ,n).\) Indeed, let *f* be defined by (4.1). Let \(\sigma \in \Omega _n\) (recall that by \(\Omega _n\) we denote the collection of all permutations of the set \(\{1,\dots ,n\}\)). We consider the iterated rearrangement \(\mathcal R_\sigma f(t),\,\,t\in \mathbb {R}^n_+.\) We have

for any \(\sigma \in \Omega _n.\) To show this, it is sufficient to use the following observation: if \(g(x)\ge 0\,\,(x\in \mathbb {R}), \,\,a\ge 0,\) and \(h(x)=\min (g(x),a),\) then \(h^*(t)=\min (g^*(t), a).\)

It is easy to obtain the rearrangement of the function \(\Phi =\mathcal R_\sigma f\) in an explicit form.

### Theorem 4.3

Let \(\varphi _j\) be nonnegative nonincreasing and left-continuous functions on \(\mathbb {R}_+\) with \(\varphi _j(+\infty )=0\) \((j=1,\ldots ,n)\). Set

Then

for any \(t>0.\)

### Proof

Fix \(t>0.\) Assume that

Since the function \(\Phi \) decreases in each variable \(x_j\in \mathbb {R}_+,\) we have \(\Phi (x)\ge \Phi (u)\) for any \(x\in (0,u_1]\times \cdots \times (0,u_n]\), that is, for any *x* from a set of measure *t*. By (2.2), this implies that \(\Phi ^*(t)\ge \Phi (u).\) Denoting

we have that \(\Phi ^*(t)\ge \omega (t)\). Let \(\varepsilon >0\). Denote \(y=\Phi ^*(t)-\varepsilon \). Then \(\lambda _\Phi (y)\ge t.\) Thus, by (4.2),

Therefore, there exist \(\tilde{u}_j\le \lambda _{\varphi _j}(y)\) \((j=1,\ldots ,n)\) such that

Since the functions \(\varphi _j\) are nonincreasing and left-continuous, we have

This implies that \(\Phi (\tilde{u})\ge y\) \((\tilde{u}=(\tilde{u}_1,\ldots ,\tilde{u}_n))\) and, by (4.9), we obtain that \(\omega (t)\ge y=\Phi ^*(t)-\varepsilon \). It follows that \(\omega (t)\ge \Phi ^*(t)\). Thus, \(\Phi ^*(t)=\omega (t).\) \(\square \)

### Remark 4.4

Assume, in addition, that all functions \(\varphi _j\) are continuous and strictly decreasing on \(\mathbb {R}_+\). Then there exist functions \(u_j\) on \(\mathbb {R}_+\) such that for any \(t>0\)

and

Indeed, taking into account that \(\Phi ^*(t)<\varphi _j(+0)\) for any \(t>0\) and any \(j=1,\ldots ,n\), define

Then we have (4.10). Further, \((\Phi ^*)^{-1}(y)=\lambda _\Phi (y)\) and \(\varphi _j^{-1}(y)=\lambda _{\varphi _j}(y).\) Thus, by (4.2),

Setting \(y=\Phi ^*(t)\), we obtain that

and (4.11) holds.

Now we show that inequality (4.3) implies Fournier’s inequality (1.3). First, we have the following theorem.

### Theorem 4.5

Let \(\psi _j\) \((j=1,\ldots ,n,\,\,n\ge 2)\) be nonnegative rearrangeable functions on \(\mathbb {R}^{n-1}.\) Set

Let \(\varphi _j=\psi ^*_j\) and

Then

and

### Proof

Let \(t>0\) and let \(E\subset \mathbb {R}^n\) be an \(F_\sigma \)-set with \({\text {mes}}_nE=t.\) Let \(E_j\) be the projection of *E* onto the hyperplane \(x_j=0 \,\,(j=1,\ldots ,n).\) Then for any \(x\in E\) and any \(j=1,\ldots ,n\)

Set

Further, the functions \(\varphi _j\) are left-continuous because they are defined as nonincreasing rearrangements (see (2.1)). Therefore, by Theorem 4.3, for all \(t>0\) we have equality (4.8). Thus,

By the Loomis–Whitney inequality (1.14),

Hence, by (4.17),

Here \(E\subset \mathbb {R}^n\) is an arbitrary \(F_\sigma \)-set with \({\text {mes}}_n E=t.\) Thus, by (2.2), the latter inequality implies (4.14).

Now, applying inequality (4.14), we obtain

This implies inequality (4.15). \(\square \)

Inequalities (4.15) and (4.3) yield the following strengthening of inequality (1.3).

### Corollary 4.6

Let \(\psi _j\in L^1(\mathbb {R}^{n-1})\) \((j=1,\ldots ,n,\,\,n\ge 2)\) be nonnegative functions. Set

and

Then

.

In comparison with (1.3), inequality (4.18) contains the intermediate term \(||\Phi ||_{n,1}.\) We shall see that this term can be essentially greater that \(||\Psi ||_{n',1}.\)

### Remark 4.7

First, we observe that for \(n=2\) we have \(\Psi ^*(t)=\Phi ^*(t)\) for all \(t>0.\) Indeed, let \(t>0\). Assume that \(u_1,u_2\in \mathbb {R}_+, \,\,u_1u_2=t.\) There exist \(F_\sigma \)-sets \(E_1,E_2\subset \mathbb {R}\) such that

Let \(E=E_1\times E_2.\) Then \({\text {mes}}_2E=t\) and

By (4.8), this implies that \(\Phi ^*(t)\le \Psi ^*(t)\), and we obtain that \(\Phi ^*(t)=\Psi ^*(t).\)

Similar arguments fail for \(n\ge 3\). Indeed, let \(n=3\) and assume that \(0<\varepsilon <1.\) Let \(I=[0,\varepsilon ]\times [0,1/\varepsilon ].\) Set \(\psi _j=\chi _I \,\,\,(j=1,2,3)\). Then

where \(P_\varepsilon =[0,\varepsilon ]\times [0,\varepsilon ]\times [0,1/\varepsilon ].\) Thus, \(\Psi ^*(t)=\chi _{[0,\varepsilon ]}(t).\) On the other hand, \(\psi _j^*(u)=\chi _{[0,1]}(u)\) \((j=1,2,3)\), and

We have \(||\Phi ||_{3,1}=1\) and \(||\Psi ||_{3/2,1}=\varepsilon ^{2/3}.\) Thus, for \(n\ge 3\) inequality (4.3) (with \(f_j=\psi _j^*\)) is essentially stronger than (1.3).

## 5 Estimates of generalized Lorentz norms

In this section we obtain our main results.

As above, by \(\Omega _n\) we denote the collection of all permutations of the set \(\{1,\ldots ,n\}.\)

First, we prove the following theorem.

### Theorem 5.1

Let \(f_j\in L^1(\mathbb {R})\) \((j=1,\ldots ,n)\) be nonnegative rearrangeable functions on \(\mathbb {R}\) and let

Let \(\alpha _j>0\,\,(j=1,\ldots ,n).\) Then for any \(\sigma \in \Omega _n\)

### Proof

Set

By (4.6), for any \(\sigma \in \Omega _n\)

Thus, (5.2) is equivalent to the equality

We may assume that all functions \(\varphi _j\) are continuously differentiable, \(\varphi _j(+0)=+\infty ,\) and

Then we have \(\varphi _j(\mathbb {R}_+)=\mathbb {R}_+\) \((j=1,\ldots ,n).\) Denote

Let \(A_j\) be the set of all \(x\in \mathbb {R}_+^n\) such that \(\Phi (x)=\varphi _j(x_j),\,\,j=1,\ldots ,n\). Then

Indeed, the first equality is obvious. Further, let \(i\not = j.\) Then \(x\in A_i\cap A_j\) if and only if \(x_j=\eta _j(\varphi _i(x_i)).\) Thus, the projection of the set \(A_i\cap A_j\) onto the 2-dimensional plane \((x_i,x_j)\) has the 2-dimensional measure zero and therefore \({\text {mes}}_n(A_i\cap A_j)=0\).

Denote \(\gamma _{j,k}(x_k)=\eta _j(\varphi _k(x_k)), \,\, x_k\in \mathbb {R}_+.\) For a fixed \(1\le j\le n\) we have

where \(\sigma _j(\hat{x}_j)=\max _{k\not = j} \gamma _{j,k}(x_k).\) Indeed, \(x\in A_j\) if and only if \(\hat{x}_j\in \mathbb {R}_+^{n-1}\) and \(x_j\ge \gamma _{j,k}(x_k)\) for any \(k\not =j.\) Thus, by Fubini’s theorem, we have

Change of variable \(y=\varphi _j(x_j)\) gives

From here, setting

we obtain

Since \(\eta _k(y)=\lambda _{\varphi _k}(y),\) this implies (5.5). \(\square \)

### Corollary 5.2

Assume, in addition, that

Then for any \(\sigma \in \Omega _n\)

Indeed, applying Hölder’s inequality at the right-hand side of (5.2), we obtain

In the case \(f_j(t)=\chi _{[0,1]}(t)\,\,\,(j=1,\ldots ,n)\) we have equality in (5.6). This shows that the constant in (5.6) is optimal.

Now we suppose that, in Theorem 5.1, \(\alpha _1=\cdots =\alpha _n=1/p.\) Then equality (5.2) assumes the form

Hence, using (2.3), we obtain that

for any \(\sigma \in \Omega _n\). Thus, we have the following statement.

### Theorem 5.3

Let \(f_j\in L^1(\mathbb {R})\) \((j=1,\ldots ,n)\) be nonnegative functions on \(\mathbb {R}\) and let

Then for any \(p>0\)

We have emphasized in Sect. 2that, in general, the spaces \(\mathcal L_{p,1;\sigma }(\mathbb {R}^n)\) are strictly smaller than \(L_{p,1}(\mathbb {R}^n)\) (see Example 2.2). We shall now analyse the situation that we have in Theorem 5.3.

First, denote by \(\mathcal Q_n\) the class of all measurable functions on \(\mathbb {R}^n\) such that for any \(y>0\) the set

is *essentially a cube in* \(\mathbb {R}^n\) (that is, for any \(y>0\) there exists a cube \(Q\subset \mathbb {R}^n\) 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 \(f\in \mathcal Q_n\) \((n\ge 2)\)

We shall show that equality (5.12) is true for a much wider class of functions.

Denote by \(\mathcal P_n\) the class of all measurable functions on \(\mathbb {R}^n\) such that for any \(y>0\) the set (5.11) is a cartesian product of measurable sets \(E_i(y)\subset \mathbb {R},\)

By virtue of equality (4.4), any function *f* defined by (4.1) belongs to \(\mathcal P_n.\) We shall prove the inverse statement.

### Proposition 5.4

Let \(f\in \mathcal P_n\,\, (n\ge 2)\) be a nonnegative rearrangeable function on \(\mathbb {R}^n.\) Set

Then

### Proof

First, we have for all \(x\in \mathbb {R}^n\)

Fix \(x\in \mathbb {R}^n\) and denote \(y=f(x).\) Let

Then *E*(*y*) is a cartesian product (5.13), where \(E_i(y)\subset \mathbb {R}.\) Since \(x\not \in E(y),\) there exists \(j\in \{1,\ldots ,n\}\) such that \(x_j\not \in E_j(y).\) Whatever be \(u\in \mathbb {R}^{n-1}\), we have \((x_j, u)\not \in E_y\) and therefore

Thus, \(f_j(x_j)\le f(x).\) Taking into account (5.15), we obtain (5.14). \(\square \)

Applying Proposition 5.4 and Theorem 5.3, we get the following result.

### Theorem 5.5

Let \(f\in \mathcal P_n\,\, (n\ge 2)\) be a rearrangeable function on \(\mathbb {R}^n.\) Then for any \(p>0\)

### 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 \(\Psi \) defined by (4.12). Let \(N\ge 1\) and let \(g_N\) be the function defined in Example 2.2. Set

Let

Then \(\Psi _N=\chi _{E_N}\), where

We have \(|E_N|\le 1\) and \(||\Psi _N||_{2,1}\le 1.\) At the same time,

Thus, \(||\Psi _N||_{2,1;\sigma }\rightarrow \infty \) as \(N\rightarrow \infty .\)

Finally, we return to Corollary 5.2. Taking \(\alpha _k=1/n \,\,(k=1,\ldots ,n),\) we obtain

### Theorem 5.7

Let \(f_k\in L^1(\mathbb {R})\) \((k=1,\ldots ,n)\) be nonnegative functions on \(\mathbb {R}\) and let *f* be defined by (4.1). Then for any \(\sigma \in \Omega _n\)

and the constant is optimal.

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.

### Theorem 5.8

Assume that \(f\in \mathcal {U}(\mathbb {R}^{n})\) \((n\ge 2).\) Then \(f\in \mathcal L^{n,1}_\sigma (\mathbb {R}^n)\) and

for any \(\sigma \in \Omega _n\).

The equivalence follows by standard arguments. First, assume that \(f\in \mathcal {U}(\mathbb {R}^{n})\). We have

for almost all \(x\in \mathbb {R}^n\). Denote the right-hand side of the above inequality by \(f_k(x_k)\) and set

We have \(||f||_{\mathcal {U}_k}=||f_k||_1\) and \(|f(x)|\le \tilde{f}(x).\) Thus, (5.18) follows immediately from Theorem 5.7.

Conversely, under the conditions of Theorem 5.7,

and therefore \(||f||_{\mathcal {U}_k}\le ||f_k||_1\). Thus, (5.17) follows from Theorem 5.8.

However, there is a difference between inequalities (5.17) and (5.18). Namely, by Theorem 5.3, the left-hand side of (5.17) coincides with \(||f||_{n,1}\). At the same time, the condition \(f\in \mathcal {U}(\mathbb {R}^{n})\) doesn’t imply such coincidence, and the norm \(||f||_{\mathcal {L}^{n,1}_\sigma }\) at the left-hand side of (5.18) may be much greater than \(||f||_{n,1}\) (see Example 2.2).

## 6 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 \(\mathbb {R}_+^n\) onto \(\mathbb {R}_+^n\).

Assume that \(u\in \mathbb {R}^m_+,\,\, m\ge 2.\) Then

In what follows we denote

### Lemma 6.1

Let \(n\ge 2\) and let the mapping \(y=\Lambda (x)\) be defined by

Then \(\Lambda \) is a differentiable bijective mapping from \(\mathbb {R}^n_+\) onto \(\mathbb {R}^n_+\) with the jacobian

### Proof

First we show that \(\Lambda \) is injective. For \(n=2\) it is obvious. Let \(n\ge 3.\) Assume that there exist \(x',x''\in \mathbb {R}^n_+\) such that

By our assumption,

Thus

Multiplying these inequalities and using (6.1), we obtain that

which contradicts (6.5).

Next, we show that \(\Lambda \) is surjective. Let \(y\in \mathbb {R}_+^n.\) Set

Then

Thus,

It remains to prove equality (6.4). Fix \(x\in \mathbb {R}_+^n\) and consider the Jacobi matrix of \(\Lambda \) at the point *x*. The *i*-th row of this matrix is formed by the partial derivatives of the function

Assume that \(x_k\not =0\) for all \(k=1,\ldots ,n.\) 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 \(\pi _n(x)^{n-2}\det (\gamma _{ij}),\) where

Adding to the first row of the matrix \((\gamma _{ij})\) all other rows, we obtain the matrix \((\tilde{\gamma }_{ij}),\) where \(\tilde{\gamma }_{1j}=n-1\) and \(\tilde{\gamma }_{ij}= \gamma _{ij}\) for \(i\ge 2\) \(\,(j=1,\ldots ,n).\) Next, from all rows of the matrix \((\tilde{\gamma }_{ij})\) beginning from the second, we subtract its first row divided by \(n-1\). We get a triangular matrix with the diagonal \(n-1, -1, \ldots ,-1.\) Thus, \(\det (\gamma _{ij})=(-1)^{n-1}(n-1)\), and we obtain equality (6.4).

\(\square \)

As above, we denote by \(\mathcal {M}_{dec}(\mathbb {R}^m_+)\) the class of all nonnegative functions on \(\mathbb {R}^m\) which vanish off \(\mathbb {R}^m_+\) and are nonincreasing in each variable on \(\mathbb {R}^m_+\). If \(f\in \mathcal {M}_{dec}(\mathbb {R}^m_+)\), then

Indeed,

and \({\text {mes}}_m E_u=\pi _m(u).\) Applying (2.2), we obtain (6.6).

### Theorem 6.2

Assume that \(n\ge 2\) and \(\psi _j\in L^1(\mathbb {R}^{n-1})\) \((j=1,\ldots ,n)\). Set

Let \(\varphi _j(t)=\psi _j^*(t),\,\,t\in \mathbb {R}_+\), and

Then for any \(\sigma \in \Omega _n\)

### Proof

Let \(\sigma \in \Omega _n\) and let \(\sigma _k\) be obtained from \(\sigma \) by removal of *k*. Then for any \(1\le k\le n\)

Set

Then \(\mathcal R_\sigma \Psi (u)\le \widetilde{\Psi }(u), \,\,u\in \mathbb {R}^n_+,\) and \(\widetilde{\psi }_k^*(t)=\psi _k^*(t),\,\, t>0.\) Since \(\widetilde{\psi }_k\in \mathcal {M}_{dec}(\mathbb {R}^{n-1}_+)\) and \(\widetilde{\psi }_k^*=\varphi _k,\) we have by (6.6)

We consider the mapping \(y=\Lambda (x)\) from \(\mathbb {R}_+^n\) to \(\mathbb {R}_+^n\) defined in Lemma 6.1,

By Lemma 6.1, \(\Lambda \) is bijective and \(|J\Lambda (x)|=(n-1)\pi _{n}(x)^{n-2}\). For \(y=\Lambda (x)\) we have \(\pi _n(y)=\pi _n(x)^{n-1}\) (see (6.1)). Thus, the inverse mapping \(G=\Lambda ^{-1}\) satisfies the equality

Applying inequality (6.8), performing the change of variables (6.9), and using equality (6.10), we get

Since \(\mathcal R_\sigma \Psi (x)\le \widetilde{\Psi }(x),\) this yields (6.7). \(\square \)

### 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

and

Theorem 6.2 and Remark 6.3 show that inequality (1.13) (with \(f_j=\psi _j^*\)) 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 \((n-1)^{-n}\).

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Kolyada, V.I. On some Fournier–Gagliardo type inequalities.
*Rev Mat Complut* **35**, 573–598 (2022). https://doi.org/10.1007/s13163-021-00396-w

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DOI: https://doi.org/10.1007/s13163-021-00396-w