Embedding theorems for Sobolev and Hardy–Sobolev spaces and estimates of Fourier transforms

Abstract

We prove embeddings of Sobolev and Hardy–Sobolev spaces into Besov spaces built upon certain mixed norms. This gives an improvement of the known embeddings into usual Besov spaces. Applying these results, we obtain Oberlin-type estimates of Fourier transforms for functions in Sobolev spaces $W_1^1(\mathbb {R}^n).$

1 Introduction

This paper is devoted to the study of some inequalities for functions in the Sobolev spaces $W_p^1(\mathbb {R}^n)$ and Hardy–Sobolev spaces $HW_1^1(\mathbb {R}^n)$.

The Sobolev space $W_p^1(\mathbb {R}^n)$ $(1\le p<\infty )$ is defined as the class of all functions $f\in L^p(\mathbb {R}^n)$ for which every first-order weak derivative exists and belongs to $L^p(\mathbb {R}^n).$ The classical Sobolev theorem (see [26, Ch. V]) states the following.

Theorem 1.1

Let $n\ge 2,$ $1\le p< n,$ and $p^*=np/(n-p).$ Then for any $f\in W_p^1(\mathbb {R}^n)$

$$\begin{aligned} ||f||_{p^*}\le c \Vert \nabla f\Vert _p. \end{aligned}$$

(1.1)

The Lebesgue norm at the left-hand side of (1.1) can be replaced by the stronger Lorentz norm. Namely, for any $f\in W_p^1(\mathbb {R}^n),\,\, n\ge 2,$ $1\le p< n,$

$$\begin{aligned} ||f||_{p^*,p}\le c ||\nabla f||_p \end{aligned}$$

(1.2)

(see [1, 21, 24, 25]).

Let a function f be defined on $\mathbb {R}^n$ and let $k\in \{1,\ldots ,n\}.$ Set

$$\begin{aligned} \Delta _k(h)f(x)= f(x+he_k)-f(x), \quad x\in \mathbb {R}^n,\, h\in \mathbb {R}\end{aligned}$$

(1.3)

($e_k$ is the kth unit coordinate vector).

The following theorem holds.

Theorem 1.2

Let $n\in \mathbb {N}.$ Assume that $1<p<\infty$ and $n\ge 1$, or $p=1$ and $n\ge 2.$ If $p<q<\infty$ and $s=1-n(1/p-1/q)>0,$ then for any $f\in W_p^1(\mathbb {R}^n)$

$$\begin{aligned} \sum _{k=1}^n \left( \int _0^\infty h^{-sp}||\Delta _k(h)f||^p_{q,p}\frac{\mathrm{d}h}{h}\right) ^{1/p}\le c \sum _{k=1}^n||D_k f||_p. \end{aligned}$$

(1.4)

For $p>1$ inequality (1.4) (with the weaker norm $||\Delta _k(h)f||_q$ at the left-hand side) was obtained by Herz [10]. For $p=1, n\ge 2$ Theorem 1.2 was proved in [11] (see also [12]). The case $p=1$ is of special interest; we stress that Theorem 1.2 fails for $p=n=1.$ However, this theorem holds for any function f from the Hardy space $H^1(\mathbb {R})$ such that $f'\in H^1(\mathbb {R})$, if we replace the $L^1$-norm of $f'$ by its $H^1$-norm (see [11, 22]).

One of the main results of this paper is the refinement of the inequality (1.4) given in terms of mixed norms.

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

If $X(\mathbb {R})$ and $Y(\mathbb {R}^{n-1})$ are Banach function spaces, and $k\in \{1,\ldots ,n\}$, we denote by $Y[X]_k$ the mixed norm space obtained by taking first the norm in X with respect to $x_k$, and then the norm in Y with respect to $\widehat{x}_k\in \mathbb {R}^{n-1}.$

We prove the following theorem.

Theorem 1.3

Let $1<p<\infty$ and $n\ge 2$, or $p=1$ and $n\ge 3.$ If $p<q<\infty$ and $\alpha =1-(n-1)(1/p-1/q)>0$, then for any $f\in W_p^1(\mathbb {R}^n)$

$$\begin{aligned} \sum _{k=1}^n \left( \int _0^\infty h^{-\alpha p}||\Delta _k(h)f||^p_{L^{q,p}[L^p]_k}\frac{\mathrm{d}h}{h}\right) ^{1/p}\le c \sum _{k=1}^n||D_k f||_p. \end{aligned}$$

(1.5)

We show that the left-hand side of (1.4) is majorized by the left-hand side of (1.5). Thus, for the indicated values of n and p, Theorem 1.3 provides a refinement of Theorem 1.2. We stress that inequality (1.5) holds for $n=2,$ $p>1$. However, the question of the validity of this inequality for $n=2, p=1$ remains open.

As we have observed above, Theorem 1.2 fails for $p=n=1,$ but in this case there holds a weaker inequality with $L^1$-norm of $f'$ replaced by its $H^1$-norm. Similarly, we supplement Theorem 1.3 by the following result.

As usual, for any $1\le p\le \infty$ we denote $p'=p/(p-1).$

Theorem 1.4

Let $f\in W_1^1(\mathbb {R}^n)$ $(n\ge 2)$ and assume that all partial derivatives $D_jf$ $(j=1,\ldots ,n)$ belong to the Hardy space $H^1(\mathbb {R}^n)$. Then for any $1<q<(n-1)/(n-2)$

$$\begin{aligned} \sum _{k=1}^n \int _0^\infty h^{(n-1)/q'-1}||\Delta _k(h)f||_{L^{q,1}[L^1]_k}\frac{\mathrm{d}h}{h}\le c \sum _{k=1}^n||D_kf||_{H^1}. \end{aligned}$$

(1.6)

That is, inequality (1.5) holds for $p=1, n=2$ if the $L^1$-norms of the derivatives are replaced by the Hardy $H^1$-norms. Of course, for $n\ge 3$ (1.6) follows from (1.5).

We should note that this work was partly inspired by the Oberlin estimate [20] of Fourier transforms of functions in the Hardy space $H^1(\mathbb {R}^n).$ We apply inequality (1.5) to obtain an analogue of this estimate for the derivatives of functions in $W_1^1(\mathbb {R}^n)$. In particular, we prove the following result.

Theorem 1.5

Let $f\in W_1^1(\mathbb {R}^n)$ $(n\ge 3).$ Then

$$\begin{aligned} \sum _{k\in \mathbb {Z}} 2^{k(2-n)}\sup _{2^{k}\le r\le 2^{k+1}}\int _{S_r} |\widehat{f}(\xi )| \mathrm{d}\sigma (\xi )\le c ||\nabla f||_1, \end{aligned}$$

(1.7)

where $S_r$ is the sphere of the radius r centered at the origin in $\mathbb {R}^n$ and $\mathrm{d}\sigma (\xi )$ is the canonical surface measure on $S_r.$

For $n\ge 3$ this theorem gives a refinement of the Hardy-type inequality

$$\begin{aligned} \int _{\mathbb {R}^n} |\widehat{f}(\xi )||\xi |^{1-n}\,\mathrm{d}\xi \le c ||\nabla f||_1, \end{aligned}$$

which was proved for $f\in W_1^1(\mathbb {R}^n) \,\, (n\ge 2)$ by Bourgain [4] and Pełczyński and Wojciechowski [23].

As in the case $p=1$ in Theorem 1.3, it is an open question whether Theorem 1.5is true for $n=2.$

The paper is organized as follows: We give some definitions and auxiliary results in Sect. 2. In Sect. 3 we prove inequalities between Besov norms built upon the spaces $L^{p,\nu }(\mathbb {R}^n)$ and $L^{p,\nu }(\mathbb {R}^{n-1})[L^r(\mathbb {R})],$ $1\le r, \nu \le p$. In Sect. 4 we prove Theorem 1.3. Section 5 contains the proof of Theorem 1.4. Section 6 is devoted to estimates of Fourier transforms of functions in $W_1^1(\mathbb {R}^n).$

2 Some definitions and auxiliary results

Denote by $S_0(\mathbb {R}^n)$ the class of all measurable and almost everywhere finite functions f on $\mathbb {R}^n$ such that

$$\begin{aligned} \lambda _f (y) = | \{x \in \mathbb {R}^n : |f(x)|>y \}| < \infty \quad \text {for each y>0}. \end{aligned}$$

A nonincreasing rearrangement of a function $f \in S_0(\mathbb {R}^n)$ is a nonnegative and nonincreasing function $f^*$ on $\mathbb {R}_+ = (0, + \infty )$ which is equimeasurable with |f|, that is, $\lambda _{f^*}=\lambda _f.$ The rearrangement $f^*$ can be defined by the equality

$$\begin{aligned} f^*(t) = \sup _{|E|=t} \inf _{x \in E} |f(x)|,\quad 0<t<\infty \end{aligned}$$

(2.1)

(see [5, p. 32]).

The following relation holds [2, p. 53]

$$\begin{aligned} \sup _{|E|=t} \int _E |f(x)| \mathrm{d}x = \int _0^t f^*(u) \mathrm{d}u . \end{aligned}$$

(2.2)

In what follows we denote

$$\begin{aligned} f^{**}(t)= \frac{1}{t} \int _0^t f^*(u) \mathrm{d}u. \end{aligned}$$

(2.3)

For any $t>0$ there is a subset $E\subset \mathbb {R}^n$ with $|E|=t$ such that

$$\begin{aligned} \frac{1}{t} \int _E|f(x)|\mathrm{d}x=f^{**}(t) \end{aligned}$$

(2.4)

(see [2, p. 53]).

Let $0<p,r<\infty .$ A function $f \in S_0(\mathbb {R}^n)$ belongs to the Lorentz space $L^{p,r}(\mathbb {R}^n)$ if

$$\begin{aligned} \Vert f\Vert _{L^{p,r}}=\Vert f\Vert _{p,r} = \left( \int _0^\infty \left( t^{1/p} f^*(t) \right) ^r \frac{\mathrm{d}t}{t} \right) ^{1/r} < \infty . \end{aligned}$$

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 [2, Ch. 4]).

We will use the following Hardy’s inequality (see [2, p. 124]).

Proposition 2.1

Let $\varphi$ be a nonnegative measurable function on $(0,\infty )$ and suppose $-\infty<\lambda <1$ and $1\le p<\infty .$ Then

$$\begin{aligned} \biggl (\int _0^{\infty }\Big (t^{\lambda -1}\int _0^t\varphi (u)\mathrm{d}u\Big )^p\frac{\mathrm{d}t}{t}\biggr ) ^{1/p} \le \frac{1}{1-\lambda }\left( \int _0^\infty \left( t^{\lambda }\varphi (t)\right) ^p\frac{\mathrm{d}t}{t} \right) ^{1/p}. \end{aligned}$$

Applying Hardy’s inequality with $p>1,$ $\lambda =1/p$, we obtain that the operator $f\mapsto f^{**}$ is bounded in $L^p$ for $p>1,$

$$\begin{aligned} ||f^{**}||_p\le \frac{p}{p-1}||f||_p,\quad 1<p\le \infty . \end{aligned}$$

(2.5)

We say that a measurable function $\psi$ on $(0,\infty )$ is quasi-decreasing if there exists a constant $c>0$ such that $\psi (t_1)\le c\psi (t_2),$ whenever $0<t_2<t_1<\infty$.

It is well known that in the case $0<p<~1$ Hardy-type inequalities are true for quasi-decreasing functions. We will use the following proposition (a short proof can be found, e.g., in [17]).

Proposition 2.2

Let $\psi$ be a nonnegative, quasi-decreasing function on $(0,\infty ).$ Suppose also that $\alpha>0, \beta >-1$ and $0<p<1$. Then

$$\begin{aligned} \int _0^{\infty }{u}^{-\alpha -1}\Big (\int _0^{u}\psi (t)t^{\beta }\mathrm{d}t\Big )^p\mathrm{d}u \le c\int _0^{\infty } u^{-\alpha -1}\big (\psi (u)u^{\beta +1}\big )^p\mathrm{d}u. \end{aligned}$$

Let a function $\varphi \in L^p(\mathbb {R})$. Set

$$\begin{aligned} \Delta (h)\varphi (x)=\varphi (x+h)-\varphi (x), \quad h\in \mathbb {R}, \end{aligned}$$

(2.6)

and

$$\begin{aligned} \omega (\varphi ;t)_p=\sup _{|h|\le t}||\Delta (h)\varphi ||_p, \quad t\ge 0. \end{aligned}$$

Ul’yanov [28] proved the following estimate: for any $\varphi \in L^p(\mathbb {R}),\,1\le p <\infty$

$$\begin{aligned} \varphi ^{**}(t)-\varphi (t)\le 2t^{-1/p}\omega (\varphi ;t)_p. \end{aligned}$$

It easily follows that

$$\begin{aligned} \varphi ^*(t)\le 2\int _t^\infty s^{-1/p}\omega (\varphi ;s)_p\frac{\mathrm{d}s}{s} \end{aligned}$$

(2.7)

(see also [14, p. 149], [27]). Using these estimates, Ul’yanov obtained that if $1\le p<q<\infty$ and $\varphi \in L^p(\mathbb {R})$, then

$$\begin{aligned} ||\varphi ||_q\le c \left( \int _0^\infty t^{-q/p}||\Delta (t)\varphi ||_p^q\,\mathrm{d}t\right) ^{1/q} \end{aligned}$$

(2.8)

and

$$\begin{aligned} \omega (\varphi ;\delta )_q\le c\left( \int _0^\delta t^{-q/p}||\Delta (t)\varphi ||_p^q\,\mathrm{d}t\right) ^{1/q} \end{aligned}$$

(2.9)

(some discussions and generalizations of these results can be found in [14] and [16]).

In the next section we consider functions $(x,y)\mapsto f(x,y),$ where $x\in \mathbb {R}, \,\, y\in \mathbb {R}^{n-1},$ and we denote

$$\begin{aligned} \Delta _1(h)f(x,y)=f(x+h,y)-f(x,y), \quad h\in \mathbb {R}. \end{aligned}$$

(2.10)

Let $V=V(\mathbb {R}^n)$ be a Banach function space over $\mathbb {R}^n$ (see [2, Ch. 1]). We shall assume that V is translation invariant, that is, whenever $f\in V,$ then $\tau _h f\in V$ and $||\tau _h f||_V=||f||_V$ for all $h\in \mathbb {R}^n$, where $\tau _h f(x)=f(x-h).$ Let $f\in V$. Set

$$\begin{aligned} \omega _1(f;\delta )_V=\sup _{|h|\le \delta }||\Delta _1(h)f||_V, \quad \delta \ge 0. \end{aligned}$$

In these notations, the subindex 1 indicates that the difference is taken with respect to the first variable x.

We have the following inequality

$$\begin{aligned} \omega _1(f;\delta )_V\le \frac{3}{\delta }\int _0^\delta ||\Delta _1(h)f||_V\mathrm{d}h. \end{aligned}$$

(2.11)

Indeed, if $t, h\in [0,\delta ],$ then

$$\begin{aligned} ||\Delta _1(t)f||_V\le ||\Delta _1(h)f||_V + ||\Delta _1(t-h)f||_V. \end{aligned}$$

Integrating with respect to h in $[0,\delta ]$ (for a fixed $t\in [0,\delta ]$), and then taking supremum over t,  we obtain (2.11).

3 Different norm inequalities

Throughout this paper we use the notation (1.3).

Let $0<\alpha <1,$ $1\le p<\infty$, and $1\le \theta <\infty .$ The Besov space $B^\alpha _{p,\theta }(\mathbb R^n)$ consists of all functions $f\in L^p(\mathbb R^n)$ such that

$$\begin{aligned} \Vert f\Vert _{B^\alpha _{p,\theta }}=||f||_p+ \sum _{k=1}^n\left( \int _0^{\infty }\big (t^{-\alpha }||\Delta _k(t)f||_p\big )^\theta \, \frac{\mathrm{d}t}{t}\right) ^{1/\theta }<\infty . \end{aligned}$$

The classical different norm embedding theorem states that if $1\le p<q<\infty$ and $\alpha >n(1/p-1/q),$ then for any $1\le \theta <\infty$

$$\begin{aligned} B_{p,\theta }^\alpha (\mathbb {R}^n)\subset B_{q,\theta }^\beta (\mathbb {R}^n),\quad \text{ where } \quad \beta = \alpha -n(1/p-1/q), \end{aligned}$$

and for any $f\in B_{p,\theta }^\alpha (\mathbb {R}^n)$

$$\begin{aligned} ||f||_{B^\beta _{q,\theta }}\le c||f||_{B^\alpha _{p,\theta }} \end{aligned}$$

(3.1)

(see [19, Ch. 6]).

Roughly speaking, passing from $L^p$ to $L^q$, we lose $n(1/p-1/q)$ in the smoothness exponent.

We shall be especially interested in the one-dimensional case of this theorem. Note that for $n=1$ (3.1) follows immediately from (2.8), (2.9) and Hardy’s inequality.

In this section we obtain different norm inequalities for the Besov spaces defined in some mixed norms. First of all, we are interested in these results in connection with embeddings of Sobolev spaces (in particular, for the comparison of Theorems 1.3 and 1.2).

We keep notations introduced in Sect. 2. Namely, we use the notation $\Delta (h)\varphi$ for functions of one variable (see (2.6)). The notation $\Delta _1(h)f$ (see (2.10)) is applied to functions $(x,y)\mapsto f(x,y),$ where $x\in \mathbb {R}, \,\, y\in \mathbb {R}^{n-1}\,\,(n\ge 2)$.

Let $1\le \theta<\infty , \,\, 0<\alpha <1.$ Let $V=V(\mathbb {R}^n)\,\,(n\ge 2)$ be a translation invariant Banach function space. Denote by $B^\alpha _{\theta ;1}(V)$ the class of all functions $f\in V$ such that

$$\begin{aligned} ||f||_{B^\alpha _{\theta ;1}(V)}= ||f||_V+\left( \int _0^\infty [h^{-\alpha }\omega _1(f;h)_V]^\theta \frac{\mathrm{d}h}{h}\right) ^{1/\theta }<\infty . \end{aligned}$$

As above, the subindex 1 indicates that the difference is taken with respect to the first variable x. Applying (2.11) and Hardy’s inequality, we obtain that

$$\begin{aligned} \int _0^\infty [h^{-\alpha }\omega _1(f;h)_V]^\theta \frac{\mathrm{d}h}{h}\le c\int _0^\infty [h^{-\alpha }||\Delta _1(h)f||_V]^\theta \frac{\mathrm{d}h}{h}. \end{aligned}$$

(3.2)

As in Sect. 1, if $X(\mathbb {R})$ and $Y(\mathbb {R}^{n-1})$ are Banach function spaces, we denote by $Y[X]_1$ the mixed norm space obtained by taking first the norm in $X(\mathbb {R})$ with respect to the variable x, and then the norm in $Y(\mathbb {R}^{n-1})$ with respect to y. In this section the interior norm will be taken only in variable x. Therefore, in this section we write simply Y[X] (omitting the subindex 1).

First, we have the following simple proposition.

Proposition 3.1

Let $1\le \theta <\infty ,$ $1\le r<p<\infty$, and $1/r-1/p<\alpha <1$. Set $\beta =\alpha -1/r+1/p.$ Then $B^\alpha _{\theta ;1}(L^p[L^r])\subset B^\beta _{\theta ;1}(L^p(\mathbb {R}^n))$; more exactly, for any $f\in B^\alpha _{\theta ;1}(L^p[L^r])$

$$\begin{aligned} ||f||_p\le c ||f||_{B^\alpha _{\theta ;1}(L^p[L^r])} \end{aligned}$$

(3.3)

and

$$\begin{aligned} \int _0^\infty h^{-\theta \beta }||\Delta _1 (h)f||_p^\theta \frac{\mathrm{d}h}{h} \le c\int _0^\infty h^{-\theta \alpha } ||\Delta _1(h)f||^\theta _{L^p[L^r]}\frac{\mathrm{d}h}{h}. \end{aligned}$$

(3.4)

Proof

Denote $V=L^p[L^r].$ Let $f\in B^\alpha _{\theta ;1}(V)$. For a fixed $y\in \mathbb {R}^{n-1}$, set $f_y(x)=f(x,y),\,\,x\in \mathbb {R}.$ By (2.8), we have

$$\begin{aligned} ||f_y||_p^p\le c \int _0^\infty t^{-p/r}||\Delta (t)f_y||_r^p\mathrm{d}t. \end{aligned}$$

Integrating with respect to y gives

$$\begin{aligned} ||f||_p^p\le c \int _0^\infty t^{-p/r}||\Delta _1(t)f||_V^p\mathrm{d}t. \end{aligned}$$

Applying standard reasonings (see, e.g., [2, Ch. 5.4]), we get

$$\begin{aligned} \begin{aligned}&\left( \int _0^\infty t^{-p/r}||\Delta _1(t)f||_{V}^p\mathrm{d}t\right) ^{1/p}\\&\quad \le c \left[ ||f||_{V}+\left( \int _0^1[t^{-\theta \alpha }||\Delta _1(t)f||_{V}]^\theta \frac{\mathrm{d}t}{t}\right) ^{1/\theta }\right] . \end{aligned} \end{aligned}$$

These estimates imply (3.3).

Further, inequality (2.9) gives that

$$\begin{aligned} ||\Delta (h)f_y||_p^p\le c\int _0^h ||\Delta (t)f_y||_r^p t^{-p/r}\mathrm{d}t. \end{aligned}$$

Integrating with respect to y, we get

$$\begin{aligned}&\int _{\mathbb {R}^{n}}|\Delta _1 (h)f(x,y)|^p(x,y)\mathrm{d}x\mathrm{d}y= \int _{\mathbb {R}^{n-1}}||\Delta (h)f_y||_p^p\mathrm{d}y\\&\quad \le c \int _{\mathbb {R}^{n-1}}\int _0^h ||\Delta (t)f_y||_r^p t^{-p/r}\mathrm{d}t\mathrm{d}y=c\int _0^h ||\Delta _1(t)f||^p_{V}t^{-p/r}\mathrm{d}t. \end{aligned}$$

This implies that

$$\begin{aligned}&\int _0^\infty h^{-\theta \beta }||\Delta _1 (h)f||_p^\theta \frac{\mathrm{d}h}{h}\\&\quad \le c \int _0^\infty h^{-\theta \beta }\left( \int _0^h ||\Delta _1(t)f||^p_{V}t^{-p/r}\mathrm{d}t\right) ^{\theta /p} \frac{\mathrm{d}h}{h}. \end{aligned}$$

If $\theta \ge p$, then we apply Proposition 2.1 and we obtain (3.4). Let $\theta <p$. Observe that the function $\psi (t)=\omega _1(f;t)_V/t$ is quasi-decreasing. Hence, applying Proposition 2.2 and inequality (3.2), we get

$$\begin{aligned} \begin{aligned}&\int _0^\infty h^{-\theta \beta }||\Delta _1 (h)f||_p^\theta \frac{\mathrm{d}h}{h}\\&\quad \le c \int _0^\infty h^{-\theta \beta }\left( \int _0^h \omega _1(f;t)^p_{V}t^{-p/r}\mathrm{d}t\right) ^{\theta /p} \frac{\mathrm{d}h}{h}\\&\quad \le c'\int _0^\infty h^{-\theta \alpha }\omega _1(f;h)^\theta _{V}\frac{\mathrm{d}h}{h}\le c''\int _0^\infty h^{-\theta \alpha }||\Delta _1(h)f||^\theta _{V}\frac{\mathrm{d}h}{h}. \end{aligned} \end{aligned}$$

This implies (3.4). $\square$

Note that, in contrast to (3.1), the loss in the smoothness exponent given by (3.4) is only $1/r-1/p.$ It is natural because the integrability exponent changes in only one variable.

Now, we replace the $L^p$-norm in (3.3) and (3.4) by the $L^{p,\nu }$-Lorentz norm. In this case simple arguments similar to those given above cannot be applied. Indeed, it was shown by Cwikel [6] that if $p\not = \nu ,$ then neither of the spaces $L^{p,\nu }(\mathbb {R}^2)$ and $L^{p,\nu }(\mathbb {R})[L^{p,\nu }(\mathbb {R})]$ is contained in the other. Therefore, we apply different methods; namely, we shall use iterated rearrangements.

Let $g\in S_0(\mathbb {R}^n), \,\, n\ge 2.$ For a fixed $y\in \mathbb {R}^{n-1},$ denote by $\mathcal R_1 g(s,y)$ the nonincreasing rearrangement of the function $g_y(x)=g(x,y), \,\, x\in \mathbb {R}.$ Further, for a fixed $s>0$, let $\mathcal R_{1,2}g(s,t)$ be the nonincreasing rearrangement of the function $y\mapsto \mathcal R_1 g(s,y),\,\,y\in \mathbb {R}^{n-1}.$

The iterated rearrangement $\mathcal R_{1,2} g$ is defined on $\mathbb {R}_+^2.$ It is nonnegative, nonincreasing in each variable, and equimeasurable with |g| function (see [3, 15, 16]).

Let $0<p, \nu <\infty ,$ and $n\ge 2$. For a function $g\in S_0(\mathbb {R}^n),$ denote

$$\begin{aligned} \Vert g\Vert _{\mathcal L^{p,\nu }}=\left( \int _{\mathbb R _+^2} (st)^{\nu /p-1}\mathcal R_{1,2}g(s,t)^\nu \,\mathrm{d}s \mathrm{d}t\right) ^{1/\nu } \end{aligned}$$

(see [3]). The following inequalities hold [29]:

$$\begin{aligned} \Vert g\Vert _{p,\nu }\le c\Vert g\Vert _{\mathcal L^{p,\nu }}\quad \text {if } 0<\nu \le p<\infty \end{aligned}$$

(3.5)

and

$$\begin{aligned} \Vert g\Vert _{\mathcal L^{p,\nu }}\le c'\Vert g\Vert _{p,\nu }\quad \text {if }0<p\le \nu <\infty . \end{aligned}$$

(3.6)

Proposition 3.2

Let $1\le \theta <\infty ,$ $1\le \nu \le p<\infty$, $1\le r<p$, and $1/r-1/p<\alpha <1$. Set $\beta =\alpha -1/r+1/p.$ Then $B^\alpha _{\theta ;1}(L^{p,\nu }[L^r])\subset B^\beta _{\theta ;1}(L^{p,\nu })$; more exactly, for any $f\in B^\alpha _{\theta ;1}(L^{p,\nu }[L^r])$

$$\begin{aligned} ||f||_{L^{p,\nu }}\le c||f||_{B^\alpha _{\theta ;1}(L^{p,\nu }[L^r])} \end{aligned}$$

(3.7)

and

$$\begin{aligned} \int _0^\infty h^{-\theta \beta }||\Delta _1 (h)f||_{L^{p,\nu }}^\theta \frac{\mathrm{d}h}{h}\le c\int _0^\infty h^{-\theta \alpha }||\Delta _1(h)f||_{L^{p,\nu }[L^r]}^\theta \frac{\mathrm{d}h}{h}. \end{aligned}$$

(3.8)

Proof

Let $f\in B_{\theta ,1}^\alpha (L^{p,\nu }[L^r]).$ Set $\varphi _h(x,y)=|\Delta _1 (h)f(x,y)|.$ Let s and h be fixed positive numbers. We consider the function $y\mapsto \mathcal R_1 \varphi _h(s,y),\,\, y\in \mathbb {R}^{n-1}.$ As in Sect. 2 above (see (2.4)), we can state that for any $t>0$ there exists a set $E=E_{s,t,h}\subset \mathbb {R}^{n-1}$ with ${\text {mes}}_{n-1}E=t$ such that

$$\begin{aligned} \mathcal R_{1,2}\varphi _h(s,t)\le \frac{1}{t} \int _{E} \mathcal R_1 \varphi _h(s,y)\,\mathrm{d}y. \end{aligned}$$

(3.9)

By (2.7), for any $s>0$

$$\begin{aligned} \mathcal R_1 \varphi _h(s,y)\le 2\int _s^\infty \omega (\varphi _h(\cdot ,y);u)_r\frac{\mathrm{d}u}{u^{1+1/r}}. \end{aligned}$$

(3.10)

Set $g_{u,h}(y)= \omega (\varphi _h(\cdot ,y);u)_r.$ By (2.2), we have

$$\begin{aligned} \frac{1}{t}\int _{E} g_{u,h}(y)\,\mathrm{d}y\le g_{u,h}^{**}(t). \end{aligned}$$

(3.11)

Applying inequalities (3.9), (3.10), and (3.11), we obtain

$$\begin{aligned} \mathcal R_{1,2}\varphi _h(s,t)\le & {} \frac{2}{t}\int _s^\infty \int _{E} g_{u,h}(y)\,\mathrm{d}y\frac{\mathrm{d}u}{u^{1+1/r}}\\\le & {} 2\int _s^\infty g_{u,h}^{**}(t)\frac{\mathrm{d}u}{u^{1+1/r}}. \end{aligned}$$

Further, we shall estimate

$$\begin{aligned} ||\Delta _1 (h)f||_{\mathcal L^{p,\nu }}^\nu =\int _0^\infty \int _0^\infty (st)^{\nu /p-1} \mathcal R_{1,2}\varphi _h(s,t)^\nu \mathrm{d}s\mathrm{d}t. \end{aligned}$$

Fix $t>0$. Applying Hardy’s inequality, we have

$$\begin{aligned} \int _0^\infty s^{\nu /p-1} \mathcal R_{1,2}\varphi _h(s,t)^\nu \mathrm{d}s\le & {} 2^\nu \int _0^\infty s^{\nu /p-1} \left( \int _s^\infty g_{u,h}^{**}(t)\frac{\mathrm{d}u}{u^{1+1/r}}\right) ^\nu \mathrm{d}s\\\le & {} c\int _0^\infty s^{\nu /p-\nu /r-1}g_{s,h}^{**}(t)^\nu \mathrm{d}s. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} ||\Delta _1 (h)f||_{\mathcal L^{p,\nu }}^\nu&=\int _{\mathbb R _+^2} (st)^{\nu /p-1}\mathcal R_{1,2}\varphi _h(s,t)^\nu \,\mathrm{d}s\mathrm{d}t\\&\le c\int _0^\infty s^{\nu /p-\nu /r-1} \int _0^\infty t^{\nu /p-1}g_{s,h}^{**}(t)^\nu \mathrm{d}t\mathrm{d}s\\&\le c'\int _0^\infty s^{\nu /p-\nu /r-1}||g_{s,h}||_{L^{p,\nu }}^\nu \mathrm{d}s. \end{aligned} \end{aligned}$$

By (2.11), we have

$$\begin{aligned} g_{s,h}(y)= \omega (\varphi _h(\cdot ,y);s)_r\le \frac{c}{s}\int _0^s ||\Delta (u)\varphi _h(\cdot ,y)||_r\mathrm{d}u. \end{aligned}$$

Thus, by the Minkowski inequality,

$$\begin{aligned} ||g_{s,h}||_{L^{p,\nu }}\le \frac{c}{s}\int _0^s ||\Delta _1(u)\varphi _h||_V \mathrm{d}u, \quad \text{ where }\quad V=L^{p,\nu }[L^r]. \end{aligned}$$

Using this estimate and applying Hardy’s inequality, we obtain

$$\begin{aligned} ||\Delta _1 (h)f||_{\mathcal L^{p,\nu }}^\nu\le & {} c \int _0^\infty s^{\nu /p-\nu /r-1}\left( \frac{1}{s}\int _0^s ||\Delta _1(u)\varphi _h||_V \mathrm{d}u\right) ^\nu \mathrm{d}s\\\le & {} c'\int _0^\infty s^{\nu /p-\nu /r-1}||\Delta _1(s)\varphi _h||_V^\nu \mathrm{d}s. \end{aligned}$$

Obviously,

$$\begin{aligned} ||\Delta _1(s)\varphi _h||_V\le 2||\Delta _1(\min (s,h))f||_V. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned}&\int _0^\infty h^{-\theta \beta }||\Delta _1 (h)f||_{\mathcal L^{p,\nu }}^\theta \frac{\mathrm{d}h}{h}\\&\quad \le c \int _0^\infty h^{-\theta \beta }\left( \int _0^\infty s^{\nu /p-\nu /r-1}||\Delta _1(\min (s,h))f||_V^\nu \mathrm{d}s\right) ^{\theta /\nu }\frac{\mathrm{d}h}{h}\\&\quad \le c'\left[ \int _0^\infty h^{-\theta \beta }\left( \int _0^h s^{\nu /p-\nu /r-1}||\Delta _1(s)f||_V^\nu \mathrm{d}s\right) ^{\theta /\nu }\frac{\mathrm{d}h}{h}\right. \\&\quad \quad +\left. \int _0^\infty h^{-\theta \beta }||\Delta _1(h)f||_V^\theta \left( \int _h^\infty s^{\nu /p-\nu /r-1} \mathrm{d}s\right) ^{\theta /\nu }\frac{\mathrm{d}h}{h}\right] \equiv c(I_1+I_2). \end{aligned} \end{aligned}$$

First,

$$\begin{aligned} I_2=c\int _0^\infty h^{-\theta \alpha }||\Delta _1(h)f||_V^\theta \frac{\mathrm{d}h}{h}. \end{aligned}$$

(3.12)

Further, if $\theta >\nu$, then by Proposition 2.1 we obtain

$$\begin{aligned} I_1\le c\int _0^\infty h^{-\theta \alpha }||\Delta _1(h)f||_V^\theta \frac{\mathrm{d}h}{h}. \end{aligned}$$

(3.13)

If $\theta \le \nu ,$ we obtain estimate (3.13) exactly as in Proposition 3.1. Namely, using the fact that the function $\psi (t)=\omega _1(f;t)_V/t$ is quasi-decreasing, we apply Proposition 2.2 and inequality (3.2). Estimates (3.12) and (3.13) give that

$$\begin{aligned} \int _0^\infty h^{-\theta \beta }||\Delta _1 (h)f||_{\mathcal L^{p,\nu }}^\theta \frac{\mathrm{d}h}{h}\le c\int _0^\infty h^{-\theta \alpha }||\Delta _1(h)f||_V^\theta \frac{\mathrm{d}h}{h}. \end{aligned}$$

Since $\nu \le p$, the latter inequality implies (3.8) (see (3.5)).

Inequality (3.7) follows by similar arguments; we omit the details. $\square$

Remark 3.3

In this work we apply Proposition 3.2 only for $\nu =r<p$. It would be interesting to consider other cases and further generalizations in this direction.

4 Embeddings of Sobolev spaces $W_p^1(\mathbb {R}^n)$

In this section we prove a refinement of Theorem 1.2. For $1\le p,q<\infty$ and $k=1,\ldots ,n$, denote by $V_{q,p,k}(\mathbb {R}^n)$ the mixed norm space $L^{q,p}(\mathbb {R}^{n-1})[L^p(\mathbb {R})]_k$ obtained by taking first the norm in $L^p(\mathbb {R})$ with respect to the variable $x_k$, and then the norm in $L^{q,p}(\mathbb {R}^{n-1})$ with respect to $\widehat{x}_k.$

We shall use the following simple fact.

Proposition 4.1

Let a function $\varphi$ be defined on $\mathbb {R}$ and assume that $\varphi$ is locally absolutely continuous (that is, $\varphi$ is absolutely continuous in each bounded interval $[a,b]\subset \mathbb {R}).$ Let $\psi =|\varphi |$. Then, $\psi$ also is locally absolutely continuous and

$$\begin{aligned} |\psi '(x)|\le |\varphi '(x)| \quad \text{ for } \text{ almost }\;x\in \mathbb {R}. \end{aligned}$$

Indeed, this statement follows immediately from the inequality

$$\begin{aligned} |\psi (x+h)-\psi (x)|\le |\varphi (x+h)-\varphi (x)|. \end{aligned}$$

Theorem 4.2

Let $1<p<\infty$ and $n\ge 2,$ or $p=1$ and $n\ge 3.$ If $p<q<\infty$ and $\alpha =1-(n-1)(1/p-1/q)>0$, then for any $f\in W_p^1(\mathbb {R}^n)$

$$\begin{aligned} \sum _{k=1}^n \left( \int _0^\infty h^{-\alpha p}||\Delta _k(h)f||^p_{V_{q, p,k}}\frac{\mathrm{d}h}{h}\right) ^{1/p}\le c ||\nabla f||_p. \end{aligned}$$

(4.1)

Proof

We estimate the last term of the sum in (4.1). Set

$$\begin{aligned} \varphi _h(\widehat{x}_n)=\left( \int _\mathbb {R}|\Delta _n(h)f(x)|^p\mathrm{d}x_n\right) ^{1/p} \end{aligned}$$

and

$$\begin{aligned} \psi _j(\widehat{x}_n)=\left( \int _\mathbb {R}|D_j f(x)|^p\mathrm{d}x_n\right) ^{1/p},\quad j=1,\ldots ,n. \end{aligned}$$

We consider the integral

$$\begin{aligned} J=\int _0^\infty h^{-\alpha p}K(h)\frac{\mathrm{d}h}{h}, \end{aligned}$$

(4.2)

where

$$\begin{aligned} K(h)=||\Delta _n(h)f||_{V_{q,p,n}}^p=\int _0^\infty t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t. \end{aligned}$$

Set

$$\begin{aligned} K_1(h)=\int _{h^{n-1}}^\infty t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t, \quad K_2(h)=\int _0^{h^{n-1}} t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t. \end{aligned}$$

(4.3)

For any $h>0$

$$\begin{aligned} |\Delta _n(h)f(x)|\le \int _0^h |D_n f(x+u e_n)| \mathrm{d}u. \end{aligned}$$

Raising to the power p, integrating over $x_n$ in $\mathbb {R},$ and applying Hölder’s inequality, we obtain

$$\begin{aligned} \varphi _h(\widehat{x}_n)^p\le \int _\mathbb {R}\left( \int _0^h |D_n f(x+u e_n)| \mathrm{d}u \right) ^p d x_n\le h^p\psi _n(\widehat{x}_n)^p. \end{aligned}$$

Thus,

$$\begin{aligned} \varphi _h^*(t)\le h\psi _n^*(t). \end{aligned}$$

(4.4)

From here (see (4.3))

$$\begin{aligned} K_1(h)\le h^p\int _{h^{n-1}}^\infty t^{p/q-1}\psi _n^*(t)^p \mathrm{d}t \end{aligned}$$

and therefore

$$\begin{aligned} \begin{aligned} J_1&= \int _0^\infty h^{-\alpha p}K_1(h)\frac{\mathrm{d}h}{h}\le \int _0^\infty h^{(1-\alpha )p}\int _{h^{n-1}}^\infty t^{p/q-1}\psi _n^*(t)^p \mathrm{d}t\frac{\mathrm{d}h}{h}\\&=\int _0^\infty t^{p/q-1}\psi _n^*(t)^p \int _0^{t^{1/(n-1)}} h^{(1-\alpha ) p}\frac{\mathrm{d}h}{h}\mathrm{d}t\\&=((1-\alpha )p)^{-1}\int _0^\infty \psi _n^*(t)^p \mathrm{d}t=c||D_nf||_p^p. \end{aligned} \end{aligned}$$

This estimate holds for all $p\ge 1$ and $n\ge 2.$

Estimating $K_2(h)$, we first assume that $p=1$ and $n\ge 3.$ Set

$$\begin{aligned} g(\widehat{x}_n)=\int _\mathbb {R}|f(x)|\,\mathrm{d}x_n. \end{aligned}$$

Then $||g||_{L^1(\mathbb {R}^{n-1})}=||f||_{L^1(\mathbb {R}^n)}.$ Moreover, $g\in W_1^1(\mathbb {R}^{n-1})$ and

$$\begin{aligned} ||D_j g||_{L^1(\mathbb {R}^{n-1})}\le ||D_{j} f||_{L^1(\mathbb {R}^n)},\quad j=1,\ldots ,n-1. \end{aligned}$$

(4.5)

Indeed, since $f\in W_p^1(\mathbb {R}^n)$, then for any $j=1,\ldots ,n$ and almost all $\widehat{x}_j\in \mathbb {R}^{n-1}$ the function f is locally absolutely continuous with respect to $x_j$ (see, e.g., [30, 2.1.4]). Thus, we can apply Proposition 4.1.

We have

$$\begin{aligned} \varphi _h(\widehat{x}_n)\le \int _\mathbb {R}|f(x)|\mathrm{d}x_n+\int _\mathbb {R}|f(x+h e_n)|\mathrm{d}x_n = 2g(\widehat{x}_n). \end{aligned}$$

Thus (see (4.3)),

$$\begin{aligned} K_2(h)\le 2\int _0^{h^{n-1}} t^{1/q-1}g^*(t)\mathrm{d}t \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} J_2&= \int _0^\infty h^{-\alpha }K_2(h)\frac{\mathrm{d}h}{h}\le 2\int _0^\infty h^{-\alpha }\int _0^{h^{n-1}} t^{1/q-1}g^*(t)\mathrm{d}t\frac{\mathrm{d}h}{h}\\&=2\int _0^\infty t^{1/q-1}g^*(t)\int _{t^{1/(n-1)}}^\infty h^{(1-1/q)(n-1)-1}\frac{\mathrm{d}h}{h}\\&=c\int _0^\infty t^{-1/(n-1)}g^*(t)\mathrm{d}t=c||g||_{(n-1)',1}. \end{aligned} \end{aligned}$$

Taking into account (4.5) and applying inequality (1.2), we get

$$\begin{aligned} J_2\le c||g||_{(n-1)',1}\le c'\sum _{j=1}^{n-1}||D_{j}f||_1. \end{aligned}$$

Together with the estimate $J_1\le c ||D_nf||_1$ obtained above, this gives (4.1) for $p=1, n\ge 3.$

Let now $p>1,$ $n\ge 2.$ In what follows we write $x=(u,x_n),\, u=\widehat{x}_n\in \mathbb {R}^{n-1}.$

For a fixed $u\in \mathbb {R}^{n-1}$ and $t>0,$ denote by $Q_u(t)$ the cube in $\mathbb {R}^{n-1}$ centered at u with the side length $(4t)^{1/(n-1)}.$ Let

$$\begin{aligned} A_{u,t,h} =\{v\in Q_u(t): \varphi _h(v)\le \varphi _h^*(2t)\}. \end{aligned}$$

Then ${\text {mes}}_{n-1}A_{u,t,h}\ge 2t.$ Thus, we have

$$\begin{aligned} \varphi _h(u)-\varphi _h^*(2t)\le & {} \varphi _h(u)-\frac{1}{{\text {mes}}_{n-1}A_{u,t,h}}\int _{A_{u,t,h}}\varphi _h(v)\mathrm{d}v\nonumber \\\le & {} \frac{1}{2t}\int _{Q_u(t)} |\varphi _h(u)-\varphi _h(v)|\mathrm{d}v. \end{aligned}$$

(4.6)

Further,

$$\begin{aligned} |\varphi _h(u)-\varphi _h(v)|= & {} \left| \left( \int _\mathbb {R}|f(u,x_n+h)-f(u,x_n)|^p\mathrm{d}x_n\right) ^{1/p}\right. \\&-\left. \left( \int _\mathbb {R}|f(v,x_n+h)-f(v,x_n)|^p\mathrm{d}x_n\right) ^{1/p}\right| \\\le & {} 2\left( \int _\mathbb {R}|f(u,x_n)-f(v,x_n)|^p\mathrm{d}x_n\right) ^{1/p}. \end{aligned}$$

We have (see [18, p. 143])

$$\begin{aligned} |f(u,x_n)-f(v,x_n)|\le |u-v|\sum _{j=1}^{n-1}\int _0^1|D_jf(u+\tau (v-u), x_n)|\mathrm{d}\tau . \end{aligned}$$

If $v\in Q_u(t),$ then $|u-v|\le \sqrt{n-1}(2t)^{1/(n-1)}.$ Thus, by the Minkowski inequality, for any $v\in Q_u(t)$

$$\begin{aligned} \begin{aligned}&|\varphi _h(u)-\varphi _h(v)|\\&\quad \le c t^{1/(n-1)}\sum _{j=1}^{n-1}\left( \int _\mathbb {R}\left( \int _0^1|D_jf(u+\tau (v-u), x_n)|\mathrm{d}\tau \right) ^p\mathrm{d}x_n\right) ^{1/p}\\&\quad \le ct^{1/(n-1)}\sum _{j=1}^{n-1}\int _0^1\left( \int _\mathbb {R}|D_jf(u+\tau (v-u), x_n)|^p\mathrm{d}x_n\right) ^{1/p}\mathrm{d}\tau \\&\quad =ct^{1/(n-1)}\sum _{j=1}^{n-1}\int _0^1\psi _j(u+\tau (v-u))\mathrm{d}\tau . \end{aligned} \end{aligned}$$

From here and (4.6),

$$\begin{aligned} \varphi _h(u)-\varphi _h^*(2t)\le ct^{1/(n-1)-1}\sum _{j=1}^{n-1}\int _{Q_0(t)}\int _0^1\psi _j(u+\tau z)\mathrm{d}\tau \mathrm{d}z. \end{aligned}$$

(4.7)

Taking into account that

$$\begin{aligned} \varphi _h^*(t)\le \sup _{{\text {mes}}_{n-1}E=t}\frac{1}{t}\int _E\varphi _h(u)\mathrm{d}u, \end{aligned}$$

and applying (4.7), we get

$$\begin{aligned} \varphi _h^*(t)-\varphi _h^*(2t)\le & {} \sup _{{\text {mes}}_{n-1}E=t}\frac{1}{t}\int _E[\varphi _h(u)-\varphi _h^*(2t)]\mathrm{d}u\\\le & {} ct^{1/(n-1)-1}\sum _{j=1}^{n-1}\sup _{{\text {mes}}_{n-1}E=t}\int _{Q_0(t)}\int _0^1\frac{1}{t}\int _E\psi _j(u+\tau z)\mathrm{d}u\mathrm{d}\tau \mathrm{d}z. \end{aligned}$$

Let $E\subset \mathbb {R}^{n-1}$, ${\text {mes}}_{n-1} E=t$. Then for all $\tau \in [0,1]$ and $z\in Q_0(t)$

$$\begin{aligned} \frac{1}{t}\int _E \psi _j(u+\tau z)\mathrm{d}u\le \psi ^{**}_j(t). \end{aligned}$$

Thus, we have that

$$\begin{aligned} \varphi _h^*(t)-\varphi _h^*(2t)\le ct^{1/(n-1)}\sum _{j=1}^{n-1}\psi ^{**}_j(t). \end{aligned}$$

(4.8)

Now, for any $\varepsilon >0,$ we have

$$\begin{aligned} J_2(\varepsilon )^{1/p}= & {} \left( \int _\varepsilon ^{1/\varepsilon } h^{-\alpha p}\int _{\varepsilon ^{n-1}}^{h^{n-1}}t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\\\le & {} \left( \int _0^\infty h^{-\alpha p}\int _0^{h^{n-1}}t^{p/q-1}[\varphi _h^*(t)-\varphi _h^*(2t)]^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\\&+\left( \int _\varepsilon ^{1/\varepsilon } h^{-\alpha p}\int _{\varepsilon ^{n-1}}^{h^{n-1}}t^{p/q-1}\varphi _h^*(2t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\equiv I'+I''(\varepsilon ). \end{aligned}$$

By (4.8) and (2.5),

$$\begin{aligned} I'\le & {} c\sum _{j=1}^{n-1}\left( \int _0^\infty h^{-\alpha p}\int _0^{h^{n-1}}t^{p/q+p/(n-1)-1}\psi ^{**}_j(t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\\= & {} c\sum _{j=1}^{n-1}\left( \int _0^\infty t^{p/q+p/(n-1)-1}\psi ^{**}_j(t)^p\int _{t^{1/(n-1)}}^\infty h^{-\alpha p}\frac{\mathrm{d}h}{h}\mathrm{d}t\right) ^{1/p}\\= & {} c'\sum _{j=1}^{n-1}\left( \int _0^\infty \psi ^{**}_j(t)^p \mathrm{d}t\right) ^{1/p}\le c''\sum _{j=1}^{n-1}||\psi _j||_p=c''\sum _{j=1}^{n-1}||D_jf||_p. \end{aligned}$$

Further,

$$\begin{aligned} I''(\varepsilon )= & {} \left( 2^{-p/q}\int _\varepsilon ^{1/\varepsilon } h^{-\alpha p}\int _{2\varepsilon ^{n-1}}^{2h^{n-1}}t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\\\le & {} 2^{-1/q}\left( \int _\varepsilon ^{1/\varepsilon } h^{-\alpha p}\int _{\varepsilon ^{n-1}}^{h^{n-1}}t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\\&+\,2^{-1/q}\left( \int _0^\infty h^{-\alpha p}\int _{h^{n-1}}^\infty t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\\= & {} 2^{-1/q}\left( J_2(\varepsilon )^{1/p}+J_1^{1/p}\right) . \end{aligned}$$

As we have proved above, $J_1^{1/p}\le c ||D_nf||_p$. Thus,

$$\begin{aligned} J_2(\varepsilon )^{1/p}\le I'+I''(\varepsilon )\le 2^{-1/q}J_2(\varepsilon )^{1/p}+ c\sum _{j=1}^{n}||D_jf||_p \end{aligned}$$

and

$$\begin{aligned} J_2(\varepsilon )^{1/p}\le c'\sum _{j=1}^{n}||D_jf||_p. \end{aligned}$$

This implies that

$$\begin{aligned} J_2^{1/p}=\left( \int _0^\infty h^{-\alpha p}\int _0^{h^{n-1}}t^{p/q-1}\varphi _h^*(t)^p\mathrm{d}t\frac{\mathrm{d}h}{h}\right) ^{1/p}\le c'\sum _{j=1}^{n}||D_jf||_p. \end{aligned}$$

Thus, we have (see notations (4.2) and (4.3))

$$\begin{aligned} J^{1/p}\le J_1^{1/p}+J_2^{1/p}\le c''\sum _{j=1}^{n}||D_jf||_p. \end{aligned}$$

In turn, this yields (4.1) for $p>1,\,n\ge 2.$ $\square$

Remark 4.3

By Proposition 3.2, inequality (4.1) gives a refinement of (1.4).

We stress that (4.1) is true for $p>1, n=2$. As it was already observed, we do not know whether this inequality is true for $p=1, \,n=2.$ However, we shall show that similar inequality holds for $p=1, \,n=2$ if we replace the $L^1$-norms of derivatives by the Hardy $H^1$-norms.

5 Embeddings of Hardy–Sobolev spaces

For a function $f\in L^1(\mathbb {R}^n)$ the Fourier transform is defined by

$$\begin{aligned} \widehat{f}(\xi )=\int _{\mathbb {R}^n}f(x)\mathrm{e}^{-i2\pi x\cdot \xi }\,\mathrm{d}x, \quad \xi \in \mathbb {R}^n. \end{aligned}$$

Let $f\in L^1(\mathbb {R}^n).$ The Riesz transforms $R_jf\,\, (j=1,\ldots ,n)$ of f are defined by the equality

$$\begin{aligned} R_jf(x)=\lim _{\varepsilon \rightarrow 0+} c_n\int _{|y|\ge \varepsilon } \frac{y_j}{|y|^{n+1}}f(x-y)\mathrm{d}y,\quad c_n=\frac{\Gamma ((n+1)/2)}{\pi ^{(n+1)/2}}. \end{aligned}$$

The space $H^1(\mathbb {R}^n)$ is the class of all functions $f\in L^1(\mathbb {R}^n)$ such that $R_jf\in L^1(\mathbb {R}^n)$ $(j=1,\ldots ,n)$. The $H^1$-norm is defined by

$$\begin{aligned} ||f||_{H^1}=||f||_1+\sum _{j=1}^n||R_jf||_1 \end{aligned}$$

(see [7, p. 144], [8, Ch. III.4]).

If $f\in H^1(\mathbb {R}^n)$, then we have (see [8, p. 197])

$$\begin{aligned} (R_jf)^\wedge (\xi )=-\frac{i\xi _j}{|\xi |}\widehat{f}(\xi ). \end{aligned}$$

Let $P_t$ be the Poisson kernel in $\mathbb {R}^n$. We consider $n+1$ harmonic functions in $\mathbb {R}^{n+1}_+=\mathbb {R}^n\times (0,+\infty )$

$$\begin{aligned} u_0(x,t)=(P_t*f)(x), \quad u_j(x,t)=(P_t*R_jf)(x) \quad (j=1,\ldots ,n). \end{aligned}$$

(5.1)

These functions satisfy the equations of conjugacy

$$\begin{aligned} \frac{\partial u_j}{\partial x_k}=\frac{\partial u_k}{\partial x_j}\quad (0\le j,k\le n), \quad \sum _{j=0}^n\frac{\partial u_j}{\partial x_j}=0 \quad (x_0=t) \end{aligned}$$

(5.2)

(see [8, Ch. III.4]).

For any $x\in \mathbb {R}^n$, denote by $\Gamma (x)$ the cone

$$\begin{aligned} \Gamma (x)=\left\{ (y,t)\in \mathbb {R}^{n+1}_+: |x-y|\le t\right\} . \end{aligned}$$

Let $f\in L^1(\mathbb {R}^n)$. The nontangential maximal function Nf is defined by

$$\begin{aligned} Nf(x)=\sup _{(y,t)\in \Gamma (x)} |(P_t*f)(y)|. \end{aligned}$$

A function $f\in L^1(\mathbb {R}^n)$ belongs to $H^1(\mathbb {R}^n)$ if and only if $Nf\in L^1(\mathbb {R}^n)$. In this case

$$\begin{aligned} c||f||_{H^1}\le ||Nf||_1\le c'||f||_{H^1}\quad (c>0) \end{aligned}$$

(5.3)

(see [8, Ch. III.4], [9, Th. 6.7.4]).

The nontangential maximal function Nf is controlled by the vertical maximal function

$$\begin{aligned} N_vf(x)=\sup _{t>0}|(P_t*f)(x)|. \end{aligned}$$

Namely, $Nf\in L^1(\mathbb {R}^n)$ if and only if $N_vf\in L^1(\mathbb {R}^n)$, and in this case

$$\begin{aligned} ||N_vf||_1\le ||Nf||_1\le c||N_vf||_1 \end{aligned}$$

(5.4)

(see [7, p.170], [9, Th. 6.4.4]).

Furthermore, if $f\in H^1(\mathbb {R}^n)$, then

$$\begin{aligned} \sum _{j=0}^n||N_vf_j||_1\le c||f||_{H^1}, \end{aligned}$$

(5.5)

where $f_0=f,$ $f_j=R_jf\,\,\,(j=1,\ldots ,n)$ (see [26, Ch. VII.3.2])).

Inequalities (5.3)–(5.5) imply that for any $f\in H^1(\mathbb {R}^n)$ its Riesz transforms $R_jf$ ($j=1,\ldots ,n)$ belong to $H^1(\mathbb {R}^n)$ and

$$\begin{aligned} ||R_j f||_{H^1}\le c ||f||_{H^1}\quad (j=1,\ldots ,n) \end{aligned}$$

(5.6)

(see also [8, pp. 288, 322]).

Denote by $HW_1^1(\mathbb {R}^n)$ the space of all functions $f\in H^1(\mathbb {R}^n)$ for which all weak partial derivatives $D_jf$ exist and belong to $H^1(\mathbb {R}^n)$.

Lemma 5.1

Let $f\in HW_1^1(\mathbb {R}^n)$ and let $u(x,t)=(P_t*f)(x),$ $t>0$. Set

$$\begin{aligned} \widetilde{N}f(x)= \sup _{(y,t)\in \Gamma (x)}\left| \frac{\partial u}{\partial {t}}(y,t)\right| . \end{aligned}$$

(5.7)

Then

$$\begin{aligned} \widetilde{N}f(x)\le \sum _{j=1}^n N(R_j(D_jf))(x) \end{aligned}$$

(5.8)

and

$$\begin{aligned} ||\widetilde{N}f||_1\le c\sum _{j=1}^n||D_jf||_{H^1}. \end{aligned}$$

(5.9)

Proof

Let $u_j(x,t)=P_t*(R_jf)(x)$ $(j=1,\ldots ,n)$. By the Fourier inversion,

$$\begin{aligned} u_j(x,t)=-\int _{\mathbb {R}^n}\widehat{f}(\xi )\frac{i\xi _j}{|\xi |}\mathrm{e}^{2\pi i\xi \cdot x}\mathrm{e}^{-2\pi |\xi |t}\mathrm{d}\xi . \end{aligned}$$

Further,

$$\begin{aligned} \frac{\partial u_j}{\partial {x_j}}(x,t)=-\int _{\mathbb {R}^n}2\pi i\xi _j\widehat{f}(\xi )\frac{i\xi _j}{|\xi |}\mathrm{e}^{2\pi i\xi \cdot x}\mathrm{e}^{-2\pi |\xi |t}\mathrm{d}\xi . \end{aligned}$$

Indeed, differentiation under the integral sign is justified by the convergence of the integral

$$\begin{aligned} \int _{\mathbb {R}^n}|\xi ||\widehat{f}(\xi )|\mathrm{e}^{-2\pi |\xi |t}\mathrm{d}\xi , \quad t>0. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\partial u_j}{\partial {x_j}}(x,t)=(P_t*(R_j(D_jf)))(x) \quad (j=1,\ldots ,n). \end{aligned}$$

(5.10)

By (5.2),

$$\begin{aligned} \left| \frac{\partial u}{\partial {t}}(x,t)\right| \le \sum _{j=1}^n \left| \frac{\partial u_j}{\partial {x_j}}(x,t)\right| . \end{aligned}$$

(5.11)

Applying (5.11) and (5.10), we get (5.8). By (5.3) and (5.6), this implies (5.9). $\square$

As it was mentioned above, the following theorem holds.

Theorem 5.2

Assume that $f\in HW_1^1(\mathbb {R}^n)\,\,(n\in \mathbb {N})$ and $1<q<n'.$ Then

$$\begin{aligned} \sum _{k=1}^n\int _0^\infty h^{n/q'-1}||\Delta _k(h)f||_{q}\frac{\mathrm{d}h}{h}\le c \sum _{k=1}^n||D_kf||_{H^1}. \end{aligned}$$

For $n\ge 2$ this result follows from Theorem 4.2; for $n=1$ it was proved in [22] (see also [11]).

In this section we obtain a refinement of Theorem 5.2 for $n\ge 2.$ For $1<q<\infty$ and $k=1,\ldots ,n$, denote by $V_{q,k}$ the mixed norm space $L^{q,1}(\mathbb {R}^{n-1})[L^1(\mathbb {R})]_k$ obtained by taking first the norm in $L^1(\mathbb {R})$ with respect to the variable $x_k$, and then the norm in $L^{q,1}(\mathbb {R}^{n-1})$ with respect to $\widehat{x}_k.$

Theorem 5.3

Assume that $f\in HW_1^1(\mathbb {R}^n)\,\,(n\ge 2).$ Let $1<q<(n-1)'$ and $\alpha =1-(n-1)/q'.$ Then

$$\begin{aligned} \sum _{k=1}^n\int _0^\infty h^{-\alpha }||\Delta _k(h)f||_{V_{q,k}}\frac{\mathrm{d}h}{h}\le c \sum _{k=1}^n||D_kf||_{H^1}. \end{aligned}$$

(5.12)

Proof

For $n\ge 3$ (5.12) follows from the stronger inequality (4.1). We assume that $n=2.$ Set

$$\begin{aligned} \varphi _h(x)=\int _\mathbb {R}|f(x,y+h)-f(x,y)|\mathrm{d}y,\quad h>0. \end{aligned}$$

We consider the integral

$$\begin{aligned} J=\int _0^\infty h^{-1/q-1}\int _0^\infty s^{1/q-1}\varphi _h^*(s)\mathrm{d}s\mathrm{d}h. \end{aligned}$$

(5.13)

We have

$$\begin{aligned} \begin{aligned} J&= \int _0^\infty h^{-1/q-1}\int _h^\infty s^{1/q-1}\varphi _h^*(s)\mathrm{d}s\mathrm{d}h\\&\quad + \int _0^\infty h^{-1/q-1}\int _0^h s^{1/q-1}\varphi _h^*(s)\mathrm{d}s\mathrm{d}h\equiv J_1+J_2. \end{aligned} \end{aligned}$$

As in Theorem 4.2, we have the estimate

$$\begin{aligned} \varphi _h^*(s)\le hg^*(s), \quad \text{ where }\quad g(x)=\int _\mathbb {R}|D_2f(x,y)|\mathrm{d}y. \end{aligned}$$

(5.14)

Applying this estimate, we immediately get that

$$\begin{aligned} J_1\le c ||D_2f||_1. \end{aligned}$$

(5.15)

Further, for the simplicity, we may assume that $J_2<\infty$ (otherwise we can apply the same arguments as ones given at the final part of the proof of Theorem 4.2 for estimation of $J_2$). We first consider the difference $\varphi _h^*(s)-\varphi _h^*(2s).$ Denote

$$\begin{aligned} \psi (x)= & {} \int _\mathbb {R}N(D_1f)(x,y)\mathrm{d}y,\\ \psi _1(x)= & {} \int _\mathbb {R}N(R_1(D_1f))(x,y)\mathrm{d}y, \quad \psi _2(x)=\int _\mathbb {R}N(R_2(D_2f))(x,y)\mathrm{d}y, \end{aligned}$$

and $\Psi =\psi +\psi _1+\psi _2.$

Let $x\in \mathbb {R}$ and $s>0.$ There exists $\tau =\tau (x,s)\in (0,2s)$ such that

$$\begin{aligned} \varphi _h(x+2\tau )\le \varphi _h^*(2s) \quad \text{ and }\quad \Psi (x+2\tau )\le \Psi ^*(s). \end{aligned}$$

(5.16)

Indeed, let A be the set of all $u\in (0, 4s)$ such that at least one of the inequalities

$$\begin{aligned} \varphi _h(x+u)> \varphi _h^*(2s) \quad \text{ or }\quad \Psi (x+u)> \Psi ^*(s) \end{aligned}$$

(5.17)

holds. Then ${\text {mes}}_1 A\le 3s$ and therefore there exists $u\in (0, 4s)$ for which both the inequalities (5.17) fail.

Further, we have

$$\begin{aligned} \varphi _h(x)-\varphi _h^*(2s)\le & {} \varphi _h(x)-\varphi _h(x+2\tau )\nonumber \\\le & {} 2 \int _\mathbb {R}|f(x+2\tau ,y)-f(x,y)|\mathrm{d}y. \end{aligned}$$

(5.18)

For fixed x,  y, and s, consider the cones

$$\begin{aligned} \Gamma _1=\Gamma (x,y)\quad \text{ and }\quad \Gamma _2=\Gamma (x+2\tau ,y). \end{aligned}$$

The point $(x+\tau ,y,\tau )$ belongs to both of them. Let $u=P_t*f.$ Then

$$\begin{aligned} \begin{aligned}&|f(x+2\tau ,y)-f(x,y)|\\&\quad \le |f(x,y)-u(x+\tau ,y,\tau )|+|f(x+2\tau ,y)-u(x+\tau ,y,\tau )|\\&\quad \le \int _0^\tau \left( \left| \frac{\partial u}{\partial x}(x+t,y,t)\right| +\left| \frac{\partial u}{\partial t}(x+t,y,t)\right| \right) \mathrm{d}t\\&\quad \quad +\int _0^\tau \left( \left| \frac{\partial u}{\partial x}(x+\tau +s,y,\tau -s)\right| +\left| \frac{\partial u}{\partial t}(x+\tau +s,y,\tau -s)\right| \right) \mathrm{d}s\\&\quad \le \tau \sup _{(x',y',t)\in \Gamma _1}\left( \left| \frac{\partial u}{\partial x}(x',y',t)\right| +\left| \frac{\partial u}{\partial t}(x',y',t)\right| \right) \\&\quad \quad +\tau \sup _{(x',y',t)\in \Gamma _2}\left( \left| \frac{\partial u}{\partial x}(x',y',t)\right| +\left| \frac{\partial u}{\partial t}(x',y',t)\right| \right) \\&\quad \le \tau \left[ N(D_1f)(x,y)+N(D_1f)(x+2\tau ,y)+\widetilde{N}f(x,y)+\widetilde{N}f(x+2\tau ,y)\right] \end{aligned} \end{aligned}$$

(we have used the notation (5.7)). Applying (5.8), we have

$$\begin{aligned} \begin{aligned}&|f(x+2\tau ,y)-f(x,y)|\le \tau \left( N(D_1f)(x,y) + N(D_1f)(x+2\tau ,y)\right. \\&\left. +\,N(R_1(D_1f))(x,y)+N(R_1(D_1f))(x+2\tau ,y)\right. \\&\left. +\,N(R_2(D_2f))(x,y)+N(R_2(D_2f))(x+2\tau ,y)\right) . \end{aligned} \end{aligned}$$

By (5.18), this implies that

$$\begin{aligned} \varphi _h(x)-\varphi _h^*(2s) \le 2\tau (\Psi (x)+\Psi (x+2\tau )), \end{aligned}$$

where $\Psi =\psi +\psi _1+\psi _2.$ Taking into account (5.16), we obtain

$$\begin{aligned} \varphi _h^*(s)-\varphi _h^*(2s)\le 8s\Psi ^*(s). \end{aligned}$$

From here

$$\begin{aligned} J_2'= & {} \int _0^\infty h^{-1/q-1}\int _0^h s^{1/q-1}[\varphi _h^*(s)-\varphi _h^*(2s)]\mathrm{d}s\mathrm{d}h\\\le & {} 8\int _0^\infty h^{-1/q-1}\int _0^h s^{1/q}\Psi ^*(s)\mathrm{d}s\mathrm{d}h=8q\int _0^\infty \Psi ^*(s)\mathrm{d}s=8q||\Psi ||_1. \end{aligned}$$

Applying (5.3) and (5.6), we get

$$\begin{aligned} \begin{aligned} ||\Psi ||_1&=||N(D_1f)||_1+||N(R_1(D_1f))||_1+||N(R_2(D_2f))||_1\\&\le c(||D_1f||_{H^1}+||D_2f||_{H^1}). \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} J_2'\le c'(||D_1f||_{H^1}+||D_2f||_{H^1}). \end{aligned}$$

(5.19)

Further, we consider

$$\begin{aligned} J_2''=\int _0^\infty h^{-1/q-1}\int _0^h s^{1/q-1}\varphi _h^*(2s)\mathrm{d}s\mathrm{d}h. \end{aligned}$$

We have (see (5.13))

$$\begin{aligned} J_2''= 2^{-1/q}\int _0^\infty h^{-1/q-1}\int _0^{2h} s^{1/q-1}\varphi _h^*(s)\mathrm{d}s\mathrm{d}h \le 2^{-1/q}J. \end{aligned}$$

(5.20)

Using estimates (5.15), (5.19), and (5.20), we obtain

$$\begin{aligned} J\le 2^{-1/q}J + c(||D_1f||_{H^1}+||D_2f||_{H^1}). \end{aligned}$$

(5.21)

We assumed that $J_2<\infty$ and hence $J=J_1+J_2<\infty .$ Thus, (5.21) implies (5.12) for $n=2.$ $\square$

6 Estimates of Fourier transforms

By Hardy’s inequality, for any $f\in H^1(\mathbb {R}^n)$ $(n\in \mathbb {N})$

$$\begin{aligned} \int _{\mathbb {R}^n} \frac{|\widehat{f}(\xi )|}{|\xi |^n}\,\mathrm{d}\xi \le c||f||_{H^1}. \end{aligned}$$

(6.1)

It was first discovered by Bourgain [4] that for $n\ge 2$ the Fourier transforms of the derivatives of functions in the Sobolev space $W_1^1(\mathbb {R}^n)$ satisfy Hardy’s inequality. More exactly, Bourgain considered the periodic case. His studies were continued by Pełczyński and Wojciechowski [23]. The following theorem holds (Bourgain; Pełczyński and Wojciechowski).

Theorem 6.1

Let $f\in W_1^1(\mathbb {R}^n)$ $(n\ge 2).$ Then

$$\begin{aligned} \int _{\mathbb {R}^n} |\widehat{f}(\xi )||\xi |^{1-n}\,\mathrm{d}\xi \le c ||\nabla f||_1. \end{aligned}$$

(6.2)

Equivalently,

$$\begin{aligned} \sum _{k=1}^n\int _{\mathbb {R}^n}\frac{|(D_k f)^\wedge (\xi )|}{|\xi |^n}\mathrm{d}\xi \le c \sum _{k=1}^n\Vert D_k f\Vert _1. \end{aligned}$$

(6.3)

This is Hardy-type inequality. These results were extended in [13, 15].

In contrast to (6.1), inequalities (6.2) and (6.3) fail to hold for $n=1.$

Oberlin [20] proved the following refinement of Hardy’s inequality (6.1) valid for $n\ge 2$.

Theorem 6.2

Let $f\in H^1(\mathbb {R}^n)$ $(n\ge 2).$ Then

$$\begin{aligned} \sum _{k\in \mathbb {Z}} 2^{k(1-n)}\sup _{2^k\le r\le 2^{k+1}}\int _{S_r} |\widehat{f}(\xi )|\,\mathrm{d}\sigma (\xi )\le c||f||_{H^1}, \end{aligned}$$

(6.4)

where $S_r$ is the sphere of the radius r centered at the origin in $\mathbb {R}^n$ and $\mathrm{d}\sigma (\xi )$ is the canonical surface measure on $S_r.$

Inequality (6.4) was used in [20] to obtain the description of radial Fourier multipliers for $H^1(\mathbb {R}^n)$ $(n\ge 2).$ Observe that these results fail for $n=1.$

In this section we prove some estimates of Fourier transforms of functions in $W_1^1(\mathbb {R}^n)$ $(n\ge 3).$ In particular, these estimates provide Oberlin-type inequalities for the Fourier transforms of the derivatives of functions in $W_1^1(\mathbb {R}^n)$.

We shall use the notation (2.3).

Theorem 6.3

Let $f\in W_1^1(\mathbb {R}^n)$ $(n\ge 3).$ Then

$$\begin{aligned} \sum _{j=1}^n\int _0^\infty F_{t,j}^{**}(t^{n-1})\,\mathrm{d}t\le c ||\nabla f||_1, \end{aligned}$$

(6.5)

where

$$\begin{aligned} F_{t,j}(\widehat{\xi }_j)=\sup _{|\xi _j|\ge t}|\widehat{f}(\xi )|\quad (t>0). \end{aligned}$$

(6.6)

Proof

We estimate the first term of the sum in (6.5). Set $\varphi _h(x)=\Delta _1(h)f(x).$ Then

$$\begin{aligned} \widehat{\varphi _h}(\xi )=\widehat{f}(\xi )(\mathrm{e}^{2\pi ih\xi _1}-1). \end{aligned}$$

Let $t>0$ and $\tau =1/t.$ Assume that $|\xi _1|\ge t.$ Then

$$\begin{aligned} \frac{1}{\tau }\int _0^\tau |\mathrm{e}^{2\pi ih\xi _1}-1|\mathrm{d}h\ge & {} \frac{1}{\tau }\int _0^\tau (1-\cos (2\pi \xi _1 h))\mathrm{d}h\\= & {} 1-\frac{\sin (2\pi \xi _1 \tau )}{2\pi \xi _1\tau }\ge 1-\frac{1}{2\pi |\xi _1|\tau }\ge 1-\frac{1}{2\pi }. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{2}{\tau }\int _0^\tau |\widehat{\varphi _h}(\xi )|\mathrm{d}h\ge |\widehat{f}(\xi )|\quad \text{ if }\quad |\xi _1|\ge t \end{aligned}$$

and

$$\begin{aligned} \frac{2}{\tau }\sup _{|\xi _1|\ge t}\int _0^\tau |\widehat{\varphi _h}(\xi )|\mathrm{d}h\ge F_{t,1}(\widehat{\xi }_1). \end{aligned}$$

By (2.2), we have

$$\begin{aligned} F_{t,1}^{**}(t^{n-1})\le & {} \frac{2t^{1-n}}{\tau }\sup _{{\text {mes}}_{n-1}E=t^{n-1}}\sup _{|\xi _1|\ge t}\int _E\int _0^\tau |\widehat{\varphi _h}(\xi )|\mathrm{d}h\mathrm{d}\widehat{\xi }_1\\\le & {} \frac{2t^{1-n}}{\tau }\sup _{|\xi _1|\ge t}\int _0^\tau \sup _{{\text {mes}}_{n-1}E=t^{n-1}}\int _E|\widehat{\varphi _h}(\xi )|\mathrm{d}\widehat{\xi }_1\mathrm{d}h. \end{aligned}$$

Let $1<q<(n-1)';$ then $q<2$. By Hölder’s inequality, for any set $E\subset \mathbb {R}^{n-1}$ with ${\text {mes}}_{n-1}E=t^{n-1}$ and any fixed $\xi _1\in \mathbb {R}$

$$\begin{aligned} t^{1-n}\int _E|\widehat{\varphi _h}(\xi )|\mathrm{d}\widehat{\xi }_1 \le t^{-(n-1)/q'}\left( \int _{\mathbb {R}^{n-1}}|\widehat{\varphi _h}(\xi )|^{q'}\mathrm{d}\widehat{\xi }_1\right) ^{1/q'}. \end{aligned}$$

Observe that for fixed $h>0$ and $\xi _1\in \mathbb {R}$, $\widehat{\varphi _h}(\xi )=(\widehat{\varphi _h})_{\xi _1}(\widehat{\xi }_1)$ is the Fourier transform of the function

$$\begin{aligned} \widehat{x}_1\mapsto \int _\mathbb {R}\Delta _1(h)f(x)\mathrm{e}^{-2\pi i\xi _1 x_1}\,\mathrm{d}x_1. \end{aligned}$$

Applying the Hausdorff–Young inequality, we obtain

$$\begin{aligned}&\left( \int _{\mathbb {R}^{n-1}}|(\widehat{\varphi _h})_{\xi _1}(\widehat{\xi }_1)|^{q'}\mathrm{d}\widehat{\xi }_1\right) ^{1/q'}\le \left( \int _{\mathbb {R}^{n-1}} \left| \int _\mathbb {R}\Delta _1(h)f(x)\mathrm{e}^{-2\pi i\xi _1 x_1}\,\mathrm{d}x_1\right| ^q \mathrm{d}\widehat{x}_1\right) ^{1/q}\\&\quad \le \left( \int _{\mathbb {R}^{n-1}} \left( \int _\mathbb {R}|\Delta _1(h)f(x)|\,\mathrm{d}x_1\right) ^q\mathrm{d}\widehat{x}_1\right) ^{1/q}. \end{aligned}$$

Thus, we have

$$\begin{aligned}&\frac{2t^{1-n}}{\tau }\sup _{{\text {mes}}_{n-1}E=t^{n-1}}\sup _{|\xi _1|\ge t}\int _E\int _0^\tau |\widehat{\varphi _h}(\xi )|\mathrm{d}h\mathrm{d}\widehat{\xi }_1\\&\quad \le 2t^{1-(n-1)/q'}\int _0^{1/t}||\Delta _1(h)f||_{L^q[L^1]}\mathrm{d}h. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} \int _0^\infty F_{t,1}^{**}(t^{n-1})\,\mathrm{d}t&\le 2\int _0^\infty t^{1-(n-1)/q'}\int _0^{1/t}||\Delta _1(h)f||_{L^q[L^1]}\mathrm{d}t\mathrm{d}t\\&= c \int _0^\infty h^{(n-1)/q'-1}||\Delta _1(h)f||_{L^q[L^1]}\frac{\mathrm{d}h}{h}. \end{aligned} \end{aligned}$$

Applying Theorem 4.2, we obtain that

$$\begin{aligned} \int _0^\infty F_{t,1}^{**}(t^{n-1})\,\mathrm{d}t\le c ||\nabla f||_1. \end{aligned}$$

$\square$

Similarly, we have the following theorem.

Theorem 6.4

Let $f\in HW_1^1(\mathbb {R}^2).$ Then

$$\begin{aligned} \int _0^\infty [F_{t,1}^{**}(t)+F_{t,2}^{**}(t)]\mathrm{d}t\le c (||D_1f||_{H^1}+||D_2f||_{H^1}), \end{aligned}$$

where

$$\begin{aligned} F_{t,1}(\xi )=\sup _{|\eta |\ge t}|\widehat{f}(\xi ,\eta )|,\quad F_{t,2}(\eta )=\sup _{|\xi |\ge t}|\widehat{f}(\xi ,\eta )|. \end{aligned}$$

Applying Theorem 6.3, we obtain the following Oberlin-type estimate.

Theorem 6.5

Let $f\in W_1^1(\mathbb {R}^n)$ $(n\ge 3).$ Then

$$\begin{aligned} \sum _{k\in \mathbb {Z}} 2^{k(2-n)}\sup _{2^{k}\le r\le 2^{k+1}}\int _{S_r} |\widehat{f}(\xi )| \mathrm{d}\sigma (\xi )\le c ||\nabla f||_1, \end{aligned}$$

(6.7)

where $S_r$ is the sphere of the radius r centered at the origin in $\mathbb {R}^n$ and $\mathrm{d}\sigma (\xi )$ is the canonical surface measure on $S_r.$

Proof

Let $B'_r$ be the ball in $\mathbb {R}^{n-1}$ of the radius $r/\sqrt{n'}$ centered at the origin. Set

$$\begin{aligned} S_{r,j}^+=\left\{ \xi \in S_r:\xi _j\ge \frac{r}{\sqrt{n}}\right\} \quad \text{ and }\quad S_{r,j}^-=\left\{ \xi \in S_r:\xi _j\le -\frac{r}{\sqrt{n}}\right\} . \end{aligned}$$

Clearly,

$$\begin{aligned} S_{r,j}^+\cup S_{r,j}^-=\{\xi \in S_r: \widehat{\xi }_j\in B_r'\}\quad \text{ and }\quad S_r=\bigcup _{j=1}^n(S_{r,j}^+\cup S_{r,j}^-). \end{aligned}$$

(6.8)

The surface $S^+_{r,j}$ is given by the equation

$$\begin{aligned} \xi _j=\sqrt{r^2-|\widehat{\xi }_j|^2}, \quad \widehat{\xi }_j\in B'_r. \end{aligned}$$

Using notation (6.6), we have

$$\begin{aligned} \begin{aligned} \int _{S_{r,j}^+}|\widehat{f}(\xi )| \mathrm{d}\sigma (\xi )&=\int _{B_r'}\left| \widehat{f}\left( \sqrt{r^2-|\widehat{\xi }_j|^2}, \widehat{\xi }_j\right) \right| \frac{r}{\sqrt{r^2-|\widehat{\xi }_j|^2}}\mathrm{d}\widehat{\xi }_j\\&\le \sqrt{n}\int _{B_r'} F_{r/\sqrt{n},j}(\widehat{\xi }_j)\mathrm{d}\widehat{\xi }_j. \end{aligned} \end{aligned}$$

Further, ${\text {mes}}_{n-1}B'_r=c_n r^{n-1}.$ If $2^k\le r\le 2^{k+1},$ then ${\text {mes}}_{n-1}B'_r\asymp 2^{k(n-1)}.$ It easily follows that

$$\begin{aligned} \begin{aligned}&2^{k(1-n)}\sup _{2^{k}\le r\le 2^{k+1}}\int _{S^+_{r,j}} |\widehat{f}(\xi )| \mathrm{d}\sigma (\xi )\\&\le c 2^{k(1-n)} \int _0^{2^{k(n-1)}}F^*_{t_k,j}(u)\mathrm{d}u\le c'F^{**}_{t_k,j}(t_k^{n-1}), \end{aligned} \end{aligned}$$

where $t_k=2^k/\sqrt{n}.$ Similar estimates hold for integrals over $S^-_{r,j}.$ Taking into account (6.8), we obtain

$$\begin{aligned} \begin{aligned}&\sum _{k\in \mathbb {Z}} 2^{k(2-n)}\sup _{2^{k}\le r\le 2^{k+1}}\int _{S_r} |\widehat{f}(\xi )| \mathrm{d}\sigma (\xi )\\&\quad \le c\sum _{j=1}^n\sum _{k\in \mathbb {Z}} 2^k F^{**}_{t_k,j}(t_k^{n-1})\le c'\sum _{j=1}^n\int _0^\infty F_{t,j}^{**}(t^{n-1})\mathrm{d}t. \end{aligned} \end{aligned}$$

By Theorem 6.3, this implies (6.7). $\square$

We observe that (6.7) is equivalent to the inequality

$$\begin{aligned} \sum _{j=1}^n\sum _{k\in \mathbb {Z}} 2^{k(1-n)}\sup _{2^{k}\le r\le 2^{k+1}}\int _{S_r} |(D_jf)^\wedge (\xi )| \mathrm{d}\sigma (\xi )\le c \sum _{j=1}^n||D_j f||_1 \end{aligned}$$

which is a direct analogue of the Oberlin inequality (6.4).

Clearly, Theorem 6.3 can be used to derive other Oberlin-type estimates. For example, one can replace spheres by the surfaces of cubes. For $k\in \mathbb {Z}$ and $1\le j\le n,$ denote

$$\begin{aligned} Q_k^{(j)}=\{\widehat{\xi }_j: |\xi _m|\le 2^k, \,\, 1\le m\le n,\,\, m\not =j\}. \end{aligned}$$

Applying Theorem 6.3, we obtain the following

Corollary 6.6

Let $f\in W_1^1(\mathbb {R}^n)$ $(n\ge 3).$ Then

$$\begin{aligned} \sum _{j=1}^n \sum _{k\in \mathbb {Z}} 2^{k(2-n)}\sup _{2^{k}\le |\xi _j|\le 2^{k+1}}\int _{Q_k^{(j)}} |\widehat{f}(\xi )| \mathrm{d}\widehat{\xi }_j\le c ||\nabla f||_1. \end{aligned}$$

(6.9)

Let $Q_k=[-2^k, 2^k]^n$ and $P_k=Q_k\setminus Q_{k-1}\,\,(k\in \mathbb {Z})$. We have

$$\begin{aligned} \sum _{j=1}^n\sup _{2^{k-1}\le |\xi _j|\le 2^{k}}\int _{Q_k^{(j)}} |\widehat{f}(\xi )| \mathrm{d}\widehat{\xi }_j\ge 2^{1-k}\int _{P_k} |\widehat{f}(\xi )| \mathrm{d}\xi . \end{aligned}$$

Thus, (6.9) gives the strengthening of the inequality (6.2) (for $n\ge 3).$