The Fourier Transform on Rearrangement-Invariant Spaces

Abstract

Let $\rho$ be a rearrangement-invariant (r.i.) norm on the set $M({\mathbb {R}}^n)$ of Lebesgue-measurable functions on ${\mathbb {R}}^n$ such that the space $L_{\rho }({\mathbb {R}}^n) = \left\{ f \in M({\mathbb {R}}^n): \rho (f) < \infty \right\}$ is an interpolation space between $L_{2}({\mathbb {R}}^n)$ and $L_{{\infty }}({\mathbb {R}}^n).$ The principal result of this paper asserts that given such a $\rho ,$ the inequality

$$\begin{aligned} \rho ({\hat{f}}) \le C \sigma (f) \end{aligned}$$

holds for any r.i. norm $\sigma$ on $M({\mathbb {R}}^n)$ if and only if

$$\begin{aligned} {\bar{\rho }} \left( U f^{*} \right) \le C {\bar{\sigma }} (f^{*}). \end{aligned}$$

Here, ${\bar{\rho }}$ is the unique r.i. norm on $M({\mathbb {R}}_+)$, ${\mathbb {R}}_+ = (0, \infty )$, satisfying ${\bar{\rho }}(f^{*})=\rho (f)$ and $U f^{*} (t) = \int _{0}^{1/t} f^{*}$, in which $f^{*}$ is the nonincreasing rearrangement of f on $\mathbb {R_+}$. Further, in this case the smallest r.i. norm $\sigma$ for which $\rho ( {\hat{f}}) \le C \sigma (f)$ holds is given by

$$\begin{aligned} \sigma (f) = {\bar{\sigma }} (f^{*}) = {\bar{\rho }} \left( U f^{*}\right) , \end{aligned}$$

where, necessarily, ${\bar{\rho }} \left( \int _{0}^{1/t} \chi _{(0, a)} \right) = {\bar{\rho }} \left( \min \{1/t, \, a\} \right) < \infty$, for all $a>0$. We further specialize and expand these results in the contexts of Orlicz and Lorentz Gamma spaces.

1 Introduction

Given f an $L_{1}({\mathbb {R}}^n )$ function, its Fourier transform, defined by

$$\begin{aligned} ( {\mathscr {F}}f )(\xi )= {\hat{f}}(\xi ) = \int _{{\mathbb {R}}^{n}} f(x) e^{- 2 \pi i x \cdot \xi } \, dx,\quad \xi \in {\mathbb {R}}^n, \end{aligned}$$

satisfies the inequality

$$\begin{aligned} \Vert {\hat{f}} \Vert _{\infty } \le \Vert f\Vert _{1}. \end{aligned}$$

Plancherel, in 1910, proved the n-dimensional version of the Riesz–Fischer theorem, namely

$$\begin{aligned} \Vert {\hat{f}} \Vert _2=\Vert f\Vert _2. \end{aligned}$$

Standard interpolation theorems yield that $L_{p'} ({\mathbb {R}}^n),$ $p'= \frac{p}{p-1},$ is an interpolation space (defined in Sect. 2) between $L_{2} ({\mathbb {R}}^n)$ and $L_{\infty } ({\mathbb {R}}^n)$ for $1<p<2,$ leading to the Hausdorff–Young inequality (1926),

$$\begin{aligned} \Vert {\hat{f}} \Vert _{p'} \le C_p\Vert f\Vert _p, \end{aligned}$$

in this case.

Inspired by the work of Jodeit and Torchinsky [13], in which the authors have generalized the Hausdorff–Young inequality, replacing the $L_p$ spaces with Orlicz spaces, we prove the following theorem which is central to the rest of the results in this paper.

Theorem 1.1

Let $\rho (f) = {\bar{\rho }}(f^{*})$ be an r.i. norm such that the Banach space $L_{\rho } ({\mathbb {R}}^n)$ is an interpolation space between $L_{2}( {\mathbb {R}}^{n} )$ and $L_{\infty }( {\mathbb {R}}^{n} )$. Then,

$$\begin{aligned} \rho ({\hat{f}}) \le C \sigma (f), \end{aligned}$$

(1.1)

for any r.i. norm $\sigma$ if and only if

$$\begin{aligned} {\bar{\rho }}( U f^{*}) \le C \, {\bar{\sigma }}( f^{*}), \end{aligned}$$

(1.2)

where $C>0$ is independent of $f \in L_{\sigma }({\mathbb {R}}^n)$.

For r.i norms $\rho =\rho _{p'}$ and $\sigma = \rho _{p},$ where $\rho _{p}(f) =\Vert f\Vert _p$, $1< p < \infty$, $\textstyle {\frac{1}{p} + \frac{1}{p'}=1}$, the space $L_{\rho _{p'}} ({\mathbb {R}}^n) = L_{p'} ({\mathbb {R}}^n)$ is an interpolation space between $L_{2} ({\mathbb {R}}^n)$ and $L_{\infty } ({\mathbb {R}}^n)$ when $1<p<2$ and the inequality (1.2), amounts to

$$\begin{aligned} \begin{aligned} C \, {\bar{\rho }}_{p} (f^{*}) \ge {\bar{\rho }}_{p'} ( U f^{*}) = \left[ \int _{ {\mathbb {R}}_+ } \left( \int _{0}^{1/t} f^{*} \right) ^{p'} dt \right] ^{\frac{1}{p'}} = \left[ \int _{ {\mathbb {R}}_+ } \left( \int _{0}^{t} f^{*} \right) ^{p'} \frac{dt}{t^2} \right] ^{\frac{1}{p'}}, \end{aligned} \end{aligned}$$

which is a special case of Hardy’s inequality; see [9, p. 124]. Therefore, Theorem 1.1 leads to the Hausdorff–Young inequality in this case.

The Orlicz spaces, $L_{\rho _{\Phi }}({\mathbb {R}}^n)$, are defined in terms of a nondecreasing convex (Orlicz) function $\Phi$ mapping ${\mathbb {R}}_{+}$ onto itself with the norm being given by

$$\begin{aligned} \rho _{\Phi }(f) = \inf \left\{ \lambda >0: \int _{{\mathbb {R}}^{n}} \Phi \left( \frac{|f(x)|}{\lambda } \right) dx \le 1 \right\} . \end{aligned}$$

Our reformulation of the result in [13] asserts that, given an Orlicz function $\Phi$, one has

$$\begin{aligned} \rho _{\Phi _{1}} ({\hat{f}}) \le C \rho _{\Phi _{2}} (f), \end{aligned}$$

in which

$$\begin{aligned} \rho _{\Phi _{1}}(f) = \rho _{\Phi }(f^{2})^{1/2} \end{aligned}$$

and $\rho _{\Phi _{2}}$ defined in terms of ${\widetilde{\Phi }}_{2}$, with

$$\begin{aligned} {\widetilde{\Phi }}_{2}(t) = \frac{1}{ \Phi _{1}(t^{-1})}, \ \ t \in {\mathbb {R}}_+. \end{aligned}$$

We discuss this and related results on Orlicz spaces in detail in Sect. 4. Theorem 1.1 tells us that, for $1< p \le 2$, the smallest r.i. norm $\sigma$ for which

$$\begin{aligned} \rho _{p'} ( {\hat{f}} ) \le C \sigma (f), \end{aligned}$$

is given by

$$\begin{aligned} \sigma (f)= {\bar{\sigma }}(f^{*})= \rho _{p'} \left( U f^{*} \right) = \left[ \int _{ {\mathbb {R}}_+ } \left( \int _{0}^{t} f^{*} \right) ^{p'} \frac{dt}{t^2} \right] ^{\frac{1}{p'}} = \left[ \int _{ {\mathbb {R}}_+ } \left( t^{1/p} f^{**}(t) \right) ^{p'} \frac{dt}{t} \right] ^{\frac{1}{p'}}, \end{aligned}$$

the so-called Lorentz norm $\rho _{p, p'}$, which is smaller than $\rho _{p}$.

In the next section we provide material on r.i. spaces and interpolation theory. Theorem 1.1 and some of its consequences are proved in Sect. 3. Section 4 deals with the Fourier transform in the context of Orlicz spaces and Sect. 5 considers the boundedness of the Fourier transform between Lorentz Gamma spaces. Section 6 concludes with some remarks on other related work.

Throughout this article, we write $A \simeq B$ to abbreviate $C_1 A\le B \le C_2 A$ for some constants $C_1, C_2 > 0$ independent of A and B.

2 Rearrangement Invariant Spaces and the K-Functional

Definition 2.1

A rearrangement-invariant (r.i.) Banach function norm $\rho$ on $M(\Omega )$, $\Omega = {\mathbb {R}}^n$ or $\mathbb {R_+}$, satisfies

• (1) $\rho (f) \ge 0$, with $\rho (f)=0$ if and only if $f=0$ a.e.;

• (2) $\rho (cf)=c \, \rho (f)$, $c>0$;

• (3) $\rho (f+g) \le \rho (f) + \rho (g)$;

• (4) $0 \le f_n \nearrow f$ implies $\rho (f_n) \nearrow \rho (f)$;

• (5) $\rho (\chi _{E})<\infty$ for all measurable $E \subset \Omega$ such that $|E|<\infty$;

• (6) $\int _{E} f \le C_{E} \, \rho (f)$, with $E \subset \Omega$, $|E| < \infty$ and $C_{E}>0$ independent of $f \in M(\Omega )$;

• (7) $\rho (f)= \rho (g)$ whenever $\mu _{f} = \mu _{g}$. Here, $\mu _{h}$, for $h \in M(\Omega )$, denotes the distribution function of h defined as $\mu _{h}(\lambda ) = |\{ x \in \Omega : |h(x)| > \lambda \}|$, $\lambda \in {\mathbb {R}}_+$.

Corresponding to an r.i. norm $\rho$ on $M(\Omega )$ is the class

$$\begin{aligned} L_{\rho }(\Omega ):= \left\{ f \in M(\Omega ): \rho (f)< \infty \right\} , \end{aligned}$$

which becomes a Banach space of Lebesgue measurable functions under the norm $\rho (f)$, $f \in L_{\rho }(\Omega )$. The space $L_{\rho }(\Omega )$ is then a rearrangement-invariant space.

According to a fundamental result of Luxemberg [9, Chapter 2, Theorem 4.10], there corresponds to every r.i. norm $\rho$ on $M({\mathbb {R}}^n)$ an r.i. norm ${\bar{\rho }}$ on $M(\mathbb {R_+})$ such that

$$\begin{aligned} \rho (f) = {\bar{\rho }}(f^{*}), \ \ \ f \in M({\mathbb {R}}^n). \end{aligned}$$

(2.1)

Here,

$$\begin{aligned} f^{*}(t) = \mu _{f}^{-1}(t) = \inf \left\{ \lambda \in {\mathbb {R}}_+: \mu _{f}(\lambda ) \le t\right\} , \, t \in {\mathbb {R}}_+. \end{aligned}$$

There is only one such ${\bar{\rho }}$ since both ${\mathbb {R}}^{n}$ and ${\mathbb {R}}_+$ are nonatomic and have infinite Lebesgue measure, see [9, p. 64].

A theorem of Hardy and Littlewood asserts that

$$\begin{aligned} \int _{{\mathbb {R}}^n} |f(x)g(x)| \, dx \le \int _{\mathbb {R_+}}f^*(t)g^*(t)dt,\quad \ f,g \in M({\mathbb {R}}^n). \end{aligned}$$

(2.2)

The operation of rearrangement, though not sublinear itself, is sublinear in the average, namely,

$$\begin{aligned} (f+g)^{**}(t) \le f^{**}(t) + g^{**}(t),\quad \ f,g \in M({\mathbb {R}}^n),\quad \ t \in \mathbb {R_+}, \end{aligned}$$

(2.3)

in which

$$\begin{aligned} h^{**}(t)={t}^{-1} \int _0^t h^{*}, \ \ \ 0 \le h \in M({\mathbb {R}}_+),\quad \ t \in \mathbb {R_+}. \end{aligned}$$

A basic technique for working with r.i. norms involves the Hardy–Littlewood–Polya (HLP) Principle which asserts that

$$\begin{aligned} f^{**} \le g^{**}\quad \text {implies}\quad \rho (f) \le \rho (g); \end{aligned}$$

see [9, Chapter 3, Proposition 4.6]. This principle is based on a result of Hardy, a generalized form of which reads

$$\begin{aligned} \int _{0}^{t} f \le \int _{0}^{t} g \end{aligned}$$

(2.4)

implies

$$\begin{aligned} \int _{0}^{t} f h^{*} \le \int _{0}^{t} g h^{*},\quad \ t \in \mathbb {R_+}, \end{aligned}$$

for all $0 \le f, g \in M( {\mathbb {R}}_+)$ and $h \in M({\mathbb {R}}_+)$. The Köthe dual of an r.i. norm $\rho$ on $M(\Omega )$ is another such norm, $\rho '$, with

$$\begin{aligned} \rho '(g):= \sup _{\rho (f) \le 1} \int _{\Omega } |f(x)g(x)| dx,\quad \ f, g \in M(\Omega ). \end{aligned}$$

It obeys the Principle of Duality

$$\begin{aligned} \rho '' = (\rho ')' = \rho . \end{aligned}$$

(2.5)

Further, one has the Hölder inequality

$$\begin{aligned} \int _{\Omega } |f(x) g(x)| dx \le \rho (f) \rho '(g),\quad \ f, g \in M(\Omega ). \end{aligned}$$

Finally,

$$\begin{aligned} \bar{\rho '} = ({\bar{\rho }})'. \end{aligned}$$

The Orlicz and Lorentz Gamma spaces studied in sections 5 and 6, respectively, are examples of such r.i. spaces.

The dilation operator $E_{s}$, $s \in \mathbb {R_+}$, is defined at $f \in M(\mathbb {R_+})$, $t \in \mathbb {R_+}$, by

$$\begin{aligned} (E_s f)(t)= f(st),\quad \ s, t \in \mathbb {R_+}. \end{aligned}$$

The operator $E_{s}$ is bounded on any r.i. space $L_{\rho }(\mathbb {R_+})$. We denote its norm by $h_{\rho }(s)$. Using $h_{\rho }$ we define the lower and upper indices of $L_{\rho }(\mathbb {R_+})$ as

$$\begin{aligned} i_{\rho }= \sup \limits _{s > 1} \frac{- \log h_{\rho }(s)}{\log s}\quad \text {and}\quad \ I_{\rho }= \inf \limits _{0<s < 1} \frac{- \log h_{\rho }(s)}{\log s}, \end{aligned}$$

(2.6)

respectively. One has

$$\begin{aligned} i_{\rho } = \lim \limits _{s \rightarrow \infty } \frac{- \log h_{\rho }(s)}{\log s} \ \ \ \text {and} \ \ \ I_{\rho }= \lim \limits _{s \rightarrow 0^+} \frac{- \log h_{\rho }(s)}{\log s}. \end{aligned}$$

Further, $0 \le i_{\rho } \le I_{\rho } \le 1$ and, moreover,

$$\begin{aligned} i_{\rho '}= 1 - I_{\rho } \ \ \ \text {and} \ \ \ I_{\rho '} = 1 - i_{\rho }. \end{aligned}$$

For all this, see [8, pp. 1250–1252].

If we denote by $k_{\rho }(s)$ the norm of $E_{s}$ on the characteristic functions $\chi _{F}$, $F \subset \mathbb {R_+}$, $|F|< \infty$, and define $j_{\rho }$ and $J_{\rho }$ by replacing $h_{\rho }(s)$ in (2.6) by $k_{\rho }(s)$, we obtain the fundamental indices of $L_{\rho }(\mathbb {R_+})$. It turns out that when $L_{\rho }(\mathbb {R_+})$ is an Orlicz space or Lorentz Gamma space $i_{\rho }= j_{\rho }$ and $I_{\rho }= J_{\rho }$. For $\rho$ an Orlicz norm see [9]; for $\rho$ a Lorentz Gamma norm see [10].

Finally, we describe that part of Interpolation Theory which is relevant to this paper.

Let $X_{1}$ and $X_{2}$ be Banach spaces compatible in the sense that both are continuously imbedded in the same Hausdorff topological space H, written

$$\begin{aligned} X_{i} \hookrightarrow H,\quad \ i = 1,2. \end{aligned}$$

The spaces $X_{1} \cap X_{2}$ and $X_{1} + X_{2}$ are the sets

$$\begin{aligned} X_{1} \cap X_{2}:= \left\{ x: x \in X_{1} \ \ \text {and} \ \ x \in X_{2} \right\} \end{aligned}$$

and

$$\begin{aligned} X_{1} + X_{2}:= \left\{ x: x= x_{1} + x_{2}, \ \ \text {for some} \ x_{1} \in X_{1}, \ x_{2} \in X_{2} \right\} , \end{aligned}$$

with norms

$$\begin{aligned} \Vert x \Vert _{X_{1} \cap X_{2} } = \max \left[ \Vert x \Vert _{X_{1}}, \Vert x \Vert _{X_{2}} \right] \end{aligned}$$

and

$$\begin{aligned} \Vert x \Vert _{X_{1} + X_{2} } = \inf \left\{ \, \Vert x_{1} \Vert _{X_{1}} + \Vert x_{2} \Vert _{X_{2}}: x= x_{1} + x_{2}, \, x_{1} \in X_{1}, \ x_{2} \in X_{2} \right\} . \end{aligned}$$

Recall that given Banach spaces $X_{1}$ and $X_{2}$ imbedded in a common Hausdorff topological vector space, their Peetre K-functional is defined for $x \in X_{1}+ X_{2}$, $t>0$, by

$$\begin{aligned} K(t,x;X_{1},X_{2}) = \inf _{x \, = \, x_{1} + x_{2}} \left[ \Vert x_{1} \Vert _{X_{1}} + t \, \Vert x_{2} \Vert _{X_{2}} \right] . \end{aligned}$$

We observe that, for $\Omega = {\mathbb {R}}^n$ or $\mathbb {R_+}$, $p \in [1, \infty )$, $L_{p}(\Omega )$ and $L_{\infty }(\Omega )$ are compatible, each being continuously imbedded in the Hausdorff topological space $M(\Omega )$ equipped with the topology of convergence in measure. One has

$$\begin{aligned} K(t, f; L_{p}(\Omega ), L_{\infty }(\Omega )) \simeq \left[ \int _{0}^{t^{p}} f^{*}(s)^{p} ds \right] ^{1/p}, \ t>0, \end{aligned}$$

(2.7)

$f \in \left( L_{p} + L_{\infty } \right) (\Omega )$, see [12].

The inequality

$$\begin{aligned} \int _0^t {({\hat{f}})}^{*}(s)^2ds \le C_1 \int _0^t (U{f}^{*})(s)^2ds,\quad t \in {\mathbb {R}}_+, \end{aligned}$$

(2.8)

from [13] reads

$$\begin{aligned} K \left( t, ({\hat{f}})^{*}; L_{2}(\mathbb {R_+}), L_{\infty }(\mathbb {R_+}) \right) \le K \left( t, C_{2} \, U f^{*}; L_{2}(\mathbb {R_+}), L_{\infty }(\mathbb {R_+}) \right) . \end{aligned}$$

(2.9)

Definition 2.2

A Banach space Y is said to be intermediate between $X_{1}$ and $X_{2}$ if

$$\begin{aligned} X_{1} \cap X_{2} \hookrightarrow Y \hookrightarrow X_{1} + X_{2}. \end{aligned}$$

Definition 2.3

A Banach space Y intermediate between the compatible spaces $X_{1}$ and $X_{2}$ is said to be an interpolation space between $X_{1}$ and $X_{2}$ if every linear operator T on $X_{1} + X_{2}$ satisfying

$$\begin{aligned} T: X_{i} \rightarrow X_{i},\quad i =1,2, \end{aligned}$$

also satisfies $T: Y \rightarrow Y$.

Suppose now that $\mu$ is an r.i. norm on $M(\mathbb {R_+})$ satisfying $\mu \left( \frac{1}{1+t} \right) < \infty$. Denote by $X_{\mu }$ the set of all $x \in X_{1}+ X_{2}$ for which

$$\begin{aligned} \rho _{\mu } (x)= \mu \left( \frac{ K(t,x;X_{1},X_{2}) }{t} \right) < \infty . \end{aligned}$$

Then, $X_{\mu }$, with the norm $\rho _{\mu }$, is an interpolation space between $X_{1}$ and $X_{2}$, see [4]. Therefore, from (2.7), we have that the space $X_{\rho _{\mu , p}}$, with the norm

$$\begin{aligned} \rho _{\mu , p} (f) = \overline{\rho _{\mu , p}} (f^{*}) = {\bar{\mu }} \left( t^{-1} \left[ \int _{0}^{t^{p}} f^{*}(s)^{p} ds \right] ^{1/p} \right) , \ \ \ f \in M(\Omega ), \end{aligned}$$

(2.10)

is an interpolation space between $L_{p}(\Omega )$ and $L_{\infty }(\Omega )$.

Definition 2.4

A Banach space Y intermediate between the compatible spaces $X_{1}$ and $X_{2}$ is said to be monotone if, given $x, y \in X_{1} + X_{2}$, with

$$\begin{aligned} K(t, x; X_{1}, X_{2}) \le K(t, y; X_{1}, X_{2}), \ \ \ t \in \mathbb {R_+}, \end{aligned}$$

(2.11)

one has $y \in Y$ implies $x \in Y$ and $\Vert x \Vert _{Y} \le \Vert y \Vert _{Y}$.

The result of Lorentz–Shimogaki in [17, Theorem 2 and Lemma 3] asserts that the r.i. interpolation spaces between $L_{p}(\Omega )$ and $L_{\infty }(\Omega )$ are precisely the monotone spaces in that context. Further, the inequality (2.9) is a special case of (2.11). Thus, for $L_{\rho }({\mathbb {R}}^{n})$ between $L_{2}({\mathbb {R}}^{n})$ and $L_{\infty }({\mathbb {R}}^{n})$, there holds

$$\begin{aligned} \begin{aligned} \rho ({\hat{f}}) = {\bar{\rho }} \left( ({\hat{f}})^{*} \right)&\le C_{2} {\bar{\rho }} \left( U f^{*} \right) \\&\le M C_{2} {\bar{\sigma }} ( f^{*} )\\&= M C_{2} \sigma ( f ), \end{aligned} \end{aligned}$$

whenever the r.i. norms $\rho$ and $\sigma$ on $M({\mathbb {R}}^{n})$ satisfy

$$\begin{aligned} {\bar{\rho }}(U f^{*}) \le M \, {\bar{\sigma }}(f^{*}),\quad f \in M({\mathbb {R}}^{n}). \end{aligned}$$

(2.12)

Remark 2.1

We have, for simplicity, chosen to restrict attention to functions $f \in L_{1}({\mathbb {R}}^n)$, since then ${\hat{f}}$ is defined as a classical Lebesgue integral. Again, it is well known that for $f \in L_{2}({\mathbb {R}}^n)$

$$\begin{aligned} \lim _{R \rightarrow \infty } \int _{ \left\{ x \in {\mathbb {R}}^n \,: \, |x| \le R \right\} } f(x) \, e^{ - 2 \pi i x \cdot \xi } \, dx, \ \ \ \xi \in {\mathbb {R}}^n, \end{aligned}$$

exists in the norm of $L_{2}({\mathbb {R}}^n)$, which can be used to define ${\hat{f}}$. Thus, the Fourier transform can be defined as a function for all $f \in \left( L_{1} + L_{2}\right) ({\mathbb {R}}^n)$. Indeed, it is shown in [3] that $\left( L_{1} + L_{2}\right) ({\mathbb {R}}^n)$ is the largest r.i. space of functions that is mapped by ${\mathscr {F}}$ into a space of locally integrable functions.

The Editor has referred us to the paper [28], among others, where it is shown that, essentially the set of functions f for which ${\hat{f}}$ is defined as a function is the amalgam space $\ell _{2} \left( L_{1} ({\mathbb {R}}^n) \right)$, which in the case $n=1$ has the norm

$$\begin{aligned} \left( \sum _{k= - \infty }^{\infty } \left( \int _{k}^{k+1} |f(x)| \, dx \, \right) ^{2} \right) ^{1/2}. \end{aligned}$$

This is a Banach function norm on $M({\mathbb {R}}^n)$ that is not rearrangement-invariant, namely, it satisfies (1)–(6) in Definition 2.1, but not (7). Thus, we need spaces other than the r.i. ones to study the Fourier transform in the context of this space.

3 Proof of Theorem 1.1

Proof

The “if” part was proved towards the end of the Sect. 2. For the “only if” part, let B be the unit ball in ${\mathbb {R}}^n$ centered at the origin. Then $\widehat{ \chi _{B} }$ is real-valued, radial and continuous, with $\widehat{\chi _{B}}(0)=|B|.$ Also, $0 \le \chi _B *\chi _{B} \le |B| \chi _{2B} \in L^1$ and $\widehat{(\chi _B *\chi _{B})}=({{\widehat{\chi }}}_{B})^2.$

Choose $r > 0$ such that $\widehat{\chi _B} \ge |B|/2$ on rB. Let $0 \le f \in L^1({\mathbb {R}}^n)$ be radial and radially decreasing. For $t \in \mathbb {R_+}$, choose $s>0$ such that $|sB|=t^{-1}.$ Then

$$\begin{aligned} \begin{aligned} (U f^{*})(t)&= \int _{0}^{1/t} f^{*} = \int _{sB}f(y)dy \\&= \left( r^{-1}s\right) ^{n} \int _{rB} f\left( r^{-1}sy\right) dy\\&\le \left( r^{-1}s\right) ^{n} \int _{rB} \frac{4}{|B|^2} \left( \widehat{\chi _{B}}(y)\right) ^2 f\left( r^{-1}sy\right) dy \\&\le \left( r^{-1}s\right) ^{n} \frac{4}{|B|^2} \int _{{\mathbb {R}}^n} \widehat{\left( \chi _B *\chi _{B}\right) } (y) f\left( r^{-1}sy\right) dy \\&= \left( r^{-1}s\right) ^{n} \frac{4}{|B|^2} (rs^{-1})^n\int _{{\mathbb {R}}^n} {\left( \chi _B *\chi _{B}\right) } (\xi ){\hat{f}}\left( rs^{-1}\xi \right) d\xi \\&\le \frac{4}{|B|^2} |B| \int _{2B}|{\hat{f}}\left( rs^{-1}\xi \right) | d\xi \\&=\frac{4}{|B|} \left( r^{-1}s\right) ^{n} \int _{|\eta |\le 2 r s^{-1}}|{\hat{f}}(\eta )| d\eta \\&= \frac{4}{|B|} \left( r^{-1}s\right) ^{n} \int _0^{ {\left( 2r s^{-1}\right) }^n |B|} \left( {\hat{f}}\right) ^* \\&= \frac{4}{|B|^2} r^{-n} \frac{1}{t} \int _0^{2^n{r}^n |B|^2 t} \left( {\hat{f}}\right) ^* \\&\le \frac{4}{|B|^2} r^{-n} \frac{1}{t} \int _0^{t} \left( {\hat{f}}\right) ^* = C_n \, \frac{1}{t} \int _0^{t} \left( {\hat{f}}\right) ^*, \end{aligned} \end{aligned}$$

where we further shrink r to be such that $2^nr^n|B|^2<1$ and the constant $C_n>0$ depends only on n.

Therefore, for $f \in (L_{1} \cap L_{\sigma } )({\mathbb {R}}^n)$ such that $f(x)= g(|x|)$, $x \in {\mathbb {R}}^n$, with $g \downarrow$ on $\mathbb {R_+},$

$$\begin{aligned} \begin{aligned} {\bar{\rho }} (U f^{*})&\le C_{n} \, {\bar{\rho }} \left( \frac{1}{t} \int _{0}^{t} \left( {\hat{f}}\right) ^{*} \right) \\&\le C_{n}' \, {\bar{\rho }} \left( \left( {\hat{f}}\right) ^{*}\right) \\&= C_{n}' \, \rho \left( {\hat{f}}\right) \\&\le C \, C_{n}' \, \sigma (f), \ \ \text {by assumption,}\\&= C \, C_{n}' \, {\bar{\sigma }}\left( f^{*}\right) , \end{aligned} \end{aligned}$$

where the second inequality is the boundedness of the averaging operator, $P: g \mapsto \frac{1}{t} \int _{0}^{t} g$, on $L_{{\bar{\rho }}}({\mathbb {R}}_+)$, which follows from our hypothesis on $L_{{\bar{\rho }}}({\mathbb {R}}_+)$ that it is the interpolation space between $L_{2}({\mathbb {R}}_+)$ and $L_{\infty }({\mathbb {R}}_+)$, and the Hardy’s inequality.

Given $h \in \left( L_{1} \cap L_{ {\bar{\sigma }} } \right) (\mathbb {R_+})$, let $g(t) = h^{*} ( |B| t^{n} )$ and set $f(x)= g(|x|)$. Then, the rearrangement of f with respect to n-dimensional Lebesgue measure is equal to the rearrangement of h with respect to 1-dimensional Lebesgue measure. The foregoing argument then yields

$$\begin{aligned} {\bar{\rho }} (U h) \le {\bar{\rho }} \left( U h^{*} \right) = {\bar{\rho }} \left( U f^{*} \right) \le C {\bar{\sigma }} (f^{*}) = C {\bar{\sigma }} (h^{*})= C {\bar{\sigma }} (h). \ \ \end{aligned}$$

The space $\left( L_{1} \cap L_{ {\bar{\sigma }} } \right) (\mathbb {R_+})$ includes all bounded functions of compact support whence the monotone convergence theorem and the Fatou property of ${\bar{\rho }}$ and ${\bar{\sigma }}$ completes the proof. $\square$

Boyd in [7, pp. 92–98] associates to each r.i. norm $\rho$ on $M(\Omega )$, $\Omega = {\mathbb {R}}^{n}$ or ${\mathbb {R}}_+$, and each $p>1$ the functional

$$\begin{aligned} \rho ^{(p)} (f) = \rho (|f|^{p})^{\frac{1}{p}}, \ \ \ f \in M(\Omega ). \end{aligned}$$

(3.1)

He shows that $\rho ^{(p)}$ is an r.i. norm on $M(\Omega )$ and that ${\bar{\rho }}(f^{**}) \le C {\bar{\rho }}(f^{*})$ holds with ${\bar{\rho }} = \overline{\rho ^{(p)}} = {\bar{\rho }}^{(p)}$.

The space defined by the norm $\rho ^{(p)}$ is now referred to as the p-convexification of $L_{\rho }({\mathbb {R}}^n)$. It was studied in a series of papers by G. Lozanovskiĭ about the time Boyd, independently, introduced his spaces. See the references to G. Lozanovskiĭ’s work in [19]. This latter paper treats the K-functional of p-convexifications, as does the paper [1]. These papers should shed light on the work involving $\rho ^{(2)}$ in this and the next two sections.

Theorem 3.1

Let $\rho$ be an r.i. norm on $M(\Omega )$. For fixed $p>1$, define $\rho ^{(p)}$ as in (3.1). Then, $L_{\rho ^{(p)}}(\Omega )$ is an interpolation space between $L_{p}(\Omega )$ and $L_{\infty }(\Omega )$.

Proof

Suppose the linear operator T satisfies

$$\begin{aligned} T: L_{p}(\Omega ) \rightarrow L_{p}(\Omega ) \ \ \ \text {and} \ \ \ T: L_{\infty }(\Omega ) \rightarrow L_{\infty }(\Omega ). \end{aligned}$$

Then, according to [9, Theorem 1.11, pp. 301–304], there exists $C>0$, such that

$$\begin{aligned} \int _{0}^{t} (Tf)^{*}(s)^{p} ds \le C \int _{0}^{t} f^{*}(s)^{p} ds, \ \ \ f \in (L_{p} + L_{\infty })(\Omega ), \ t \in \mathbb {R_+}. \end{aligned}$$

(3.2)

The HLP Principle involving ${\bar{\rho }}$ yields

$$\begin{aligned} \begin{aligned} \rho \left( (Tf)^{p} \right)&= {\bar{\rho }} \left( \left[ (Tf)^{*} \right] ^{p} \right) \\&\le C {\bar{\rho }} \left( \left[ f^{*} \right] ^{p} \right) \\&= C \rho \left( |f|^{p} \right) , \ \ \ f \in L_{\rho ^{(p)}}(\Omega ), \end{aligned} \end{aligned}$$

and hence

$$\begin{aligned} \rho ^{(p)} (Tf) \le C^{1/p} \rho ^{(p)} (f), \ \ \ f \in L_{\rho ^{(p)}}(\Omega ). \end{aligned}$$

$\square$

Theorem 3.2

Let $\rho$ and $\sigma$ be r.i. norms on $M({\mathbb {R}}^n)$ determined, respectively, by the r.i. norms ${\bar{\rho }}$ and ${\bar{\sigma }}$ on $M(\mathbb {R_+})$ by $\rho (f) = {\bar{\rho }}(f^{*})$ and $\sigma (f) = {\bar{\sigma }}(f^{*})$, $f \in M({\mathbb {R}}^n)$. Then,

$$\begin{aligned} \rho ^{(2)} ( {\hat{f}}) \le C \sigma (f), \ \ \ f \in (L_{\sigma } \, \cap \, L_{1})({\mathbb {R}}^n), \end{aligned}$$

if and only if

$$\begin{aligned} \overline{\rho ^{(2)}} (Ug) \le C {\bar{\sigma }} (g), \ \ \ g \in M(\mathbb {R_+}). \end{aligned}$$

Proof

The result is a consequence of Theorems 3.1 and 1.1. $\square$

From our discussion on the spaces $X_{\rho _{\mu , p}}$, with the norm $\rho _{\mu , p}$ given by (2.10), Theorem 1.1 guarantees

Theorem 3.3

Let $\mu$ and $\sigma$ be r.i. norms on $M({\mathbb {R}}^n)$ determined, respectively, by the r.i. norms ${\bar{\mu }}$ and ${\bar{\sigma }}$ on $M(\mathbb {R_+})$. Suppose ${\bar{\mu }} \left( \frac{1}{1+t} \right) < \infty$. Set

$$\begin{aligned} \rho _{\mu , 2} (f) = \overline{\rho _{\mu , 2}} (f^{*}) = {\bar{\mu }} \left( t^{-1} \left[ \int _{0}^{t^{2}} f^{*}(s)^{2} ds \right] ^{1/2} \right) . \end{aligned}$$

Then,

$$\begin{aligned} \rho _{\mu , 2} ( {\hat{f}}) \le C \sigma (f), \ \ \ f \in (L_{\sigma } \, \cap \, L_{1})({\mathbb {R}}^n), \end{aligned}$$

if and only if

$$\begin{aligned} \overline{ \rho _{\mu , 2} } (Ug) \le C {\bar{\sigma }} (g), \ \ \ g \in M(\mathbb {R_+}). \end{aligned}$$

Finally, consider an r.i. norm $\rho$ on $M({\mathbb {R}}^n)$ determined by the r.i. norm $\bar{\rho }$ on $M(\mathbb {R_+})$ and set

$$\begin{aligned} \rho _{U} (f):= \left( \bar{\rho } \, \circ U \right) (f^{*}) = {\bar{\rho }} (U f^{*}), \ \ \ f \in M({\mathbb {R}}^n). \end{aligned}$$

One has $\rho _{U}$ an r.i. norm if $\left( \bar{\rho } \, \circ U \right) ( \chi _{(0,t)} ) < \infty$ for all $t>0$, or, equivalently, ${\bar{\rho }} \left( \frac{1}{1+t} \right) < \infty$. In that case, $L_{{\bar{\rho }} \, \circ U } (\mathbb {R_+})$ is the largest r.i. space to be mapped into $L_{{\bar{\rho }}}(\mathbb {R_+})$ by U.

With this background we now have

Theorem 3.4

Let $\rho$ be an r.i. norm on $M({\mathbb {R}}^n)$ defined in terms of an r.i. norm ${\bar{\rho }}$ on $M(\mathbb {R_+})$ such that

$$\begin{aligned} {\bar{\rho }} \left( \frac{1}{1+t} \right) < \infty . \end{aligned}$$

Assuming $L_{ {\bar{\rho }}}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+})$, one has that $L_{\rho _{U}} ({\mathbb {R}}^n)$ is the largest r.i. space of functions on ${\mathbb {R}}^n$ to be mapped into $L_{\rho }({\mathbb {R}}^n)$ by ${\mathscr {F}}$.

4 ${\mathscr {F}}$ in the Context of Orlicz Spaces

An Orlicz gauge norm is given in terms of an N-function

$$\begin{aligned} \Phi (x)= \int _{0}^{x} \phi ,\quad x \in \mathbb {R_+}; \end{aligned}$$

here $\phi$ is a nondecreasing function mapping $\mathbb {R_+}$ onto itself. These N-functions are convex functions of the type from [13] referred to in the Introduction. Specifically, the gauge norm $\rho _{\Phi }$ is defined at $f \in M(\Omega )$, $\Omega = {\mathbb {R}}^n$ or ${\mathbb {R}}_+$, by

$$\begin{aligned} \rho _{\Phi }(f) = \inf \left\{ \lambda >0: \int _{\Omega } \Phi \left( \frac{|f(x)|}{\lambda } \right) dx \le 1 \right\} . \end{aligned}$$

One can show $\rho _{\Phi }(f)= {\bar{\rho }}_{\Phi }(f^{*})$, so that the Orlicz space

$$\begin{aligned} L_{\Phi }(\Omega )= \left\{ f \in M(\Omega ): \rho _{\Phi }(f) < \infty \right\} \end{aligned}$$

is an r.i. space. The norm $\left( \rho _{\Phi } \right) '$ dual to $\rho _{\Phi }$ is equivalent to the gauge norm $\rho _{{\tilde{\Phi }}}$, where ${\tilde{\Phi }}(t) = \int _{0}^{t} \phi ^{-1}$, $t \in {\mathbb {R}}_+$, see [9].

The definitive work on ${\mathscr {F}}$ between Orlicz spaces is due to Jodeit and Torchinsky. See, in particular, [13, Theorem 2.16]. This theorem asserts that if A and B are N-functions with $L_{A}({\mathbb {R}}^n) \subset (L_{1} + L_{2})({\mathbb {R}}^n)$, $L_{B}({\mathbb {R}}^n) \subset (L_{2} + L_{\infty })({\mathbb {R}}^n)$ and ${\mathscr {F}}: L_{A}({\mathbb {R}}^n) \rightarrow L_{B}({\mathbb {R}}^n)$, then there exist N-functions $A_{1}$ and $B_{1}$ with $L_{A_{1}}({\mathbb {R}}^n) \supset L_{A}({\mathbb {R}}^n)$ and $L_{B_{1}}({\mathbb {R}}^n) \subset L_{B}({\mathbb {R}}^n)$ for which ${\mathscr {F}}: L_{A_{1}}({\mathbb {R}}^n) \rightarrow L_{B_{1}}({\mathbb {R}}^n)$. Moreover, $B_{1}(t)= 1 / \tilde{A_{1}}(t^{-1})$; $A_{1}(t) / t^{2} \downarrow$ on $\mathbb {R_+}$ and so $B_{1}(t) / t^{2}$ $\uparrow$ on $\mathbb {R_+}$.

Using the results in the previous sections we now show $L_{A_{1}}(\mathbb {R_+})$ is an interpolation space between $L_{1}(\mathbb {R_+})$ and $L_{2}(\mathbb {R_+})$, while $L_{B_{1}}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+})$.

To begin, we observe that $B_{1}(t) / t^{2}$ $\uparrow$ is equivalent to $B_{1}(t) = \Phi (t^{2})$ for some N- function $\Phi$. Indeed, given the latter, one has $B_{1}(t) / t^{2} = \Phi (t^{2}) / t^{2} \uparrow$ on $\mathbb {R_+}$. Again $B_{1}(t) / t^{2} \uparrow$ implies $B_{1}(t^{1/2} ) / t \uparrow$, so that $\Phi (t)=B_{1}(t^{1/2})$ is such that $B_{1}(t)= \Phi (t^{2})$. Next,

$$\begin{aligned} \rho _{\Phi }^{(2)} (f) = \rho _{\Phi } (f^{2})^{\frac{1}{2}}= \rho _{B_{1}} (f). \end{aligned}$$

According to Theorem 3.1, then, $L_{B_{1}}({\mathbb {R}}^n)$ is an interpolation space between $L_{2}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$.

Now, $B_{1}(t)= 1 / \tilde{A_{1}}(t^{-1})$ is equivalent to $\tilde{A_{1}}(t)= 1 / B_{1}(t^{-1})$, whence $\tilde{A_{1}}(t) / t^{2}$ $= (t^{-1})^{2} / B_{1}(t^{-1})$ and so $B_{1}(t) / t^{2} \uparrow$ amounts to $\tilde{A_{1}}(t) / t^{2} \uparrow$, that is, $L_{\tilde{A_{1}}}({\mathbb {R}}^n)$ is an interpolation space between $L_{2}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$. Since $L_{\tilde{A_{1}}}({\mathbb {R}}^n)$ is the Köthe dual of $L_{A_{1}}({\mathbb {R}}^n)$ we conclude that $L_{A_{1}}({\mathbb {R}}^n)$ is an interpolation space between $L_{1}({\mathbb {R}}^n)$ and $L_{2}({\mathbb {R}}^n)$.

The monotonicity conditions on $A_{1}$ and $B_{1}$ translate into conditions on their associated fundamental functions. For example, $L_{B_{1}}({\mathbb {R}}^n)$ has fundamental function $\phi _{B_{1}}(t)= \rho _{B_{1}} \left( \chi _{(0,t)} \right) = 1 / B_{1}^{-1}(t^{-1})$, $t \in \mathbb {R_+}$. Thus setting $t = 1 / B_{1}(y)$ in $\frac{\phi _{B_{1}}(t)}{t^{1/2}} = \frac{1}{B_{1}^{-1}(t^{-1}) t^{1/2}}$ we arrive at $\left( \frac{B_{1}(y)}{ y^{2}} \right) ^{1/2}$, which increases in y and therefore decreases in t, so $\frac{\phi _{B_{1}} (t) }{ t^{1/2}} \downarrow$.

We observe that $L_{A_{1}} ({\mathbb {R}}^n)$ is not the largest r.i. space that ${\mathscr {F}}$ maps into $L_{B_{1}} ({\mathbb {R}}^n)$; that space has norm $\rho _{B_{1}} (U f^{*})$. In the Lebesgue context, in which, say, $B_{1}(t)= t^{p'}$, $1<p<2$,

$$\begin{aligned} \rho _{p'} (U f^{*}) = \left[ \int _{\mathbb {R_+}} (U f^{*})(t)^{p'} dt \right] ^{1/p'} \simeq \left[ \int _{\mathbb {R_+}} \left[ t^{\frac{1}{p}} f^{**}(t) \right] ^{p'} \frac{dt}{t} \right] ^{1/p'}, \end{aligned}$$

which is the so-called Lorentz norm $\rho _{p,p'}$. This norm is smaller than $\rho _{A_{1}} = \rho _{p}$. For more details see the next section.

The foregoing argument can be used to associate a pair of N-functions (A, B) to a given N-function $\Phi$ such that ${\mathscr {F}}: L_{A} ({\mathbb {R}}^{n}) \rightarrow L_{B} ({\mathbb {R}}^{n})$. Moreover, $L_{A} ({\mathbb {R}}^{n})$ is an interpolation space between $L_{1}({\mathbb {R}}^{n})$ and $L_{2}({\mathbb {R}}^{n})$, while $L_{B} ({\mathbb {R}}^{n})$ is an interpolation space between $L_{2}({\mathbb {R}}^{n})$ and $L_{\infty }({\mathbb {R}}^{n})$. Indeed, we have

Theorem 4.1

Let $\Phi$ be an N-function. Set $B(t)= \Phi (t^2)$ and ${\tilde{A}}(t) = 1 / B(t^{-1})$. Then, essentially, $\frac{A(t)}{t^2} \downarrow$,

$$\begin{aligned} {\mathscr {F}}: L_{A} (\mathbb {{\mathbb {R}}}^n) \rightarrow L_{B} (\mathbb {{\mathbb {R}}}^n) \end{aligned}$$

or, equivalently,

$$\begin{aligned} {\mathscr {F}}: L_{ {\tilde{B}}} (\mathbb {{\mathbb {R}}}^n) \rightarrow L_{{\tilde{A}}} (\mathbb {{\mathbb {R}}}^n) \end{aligned}$$

with $L_{A} ({\mathbb {R}}^n)$ an interpolation space between $L_{1}({\mathbb {R}}^n)$ and $L_{2}({\mathbb {R}}^n)$, while $L_{B}({\mathbb {R}}^n)$ is an interpolation space between $L_{2}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$.

Proof

The preceding discussion shows $\rho _{B} = \rho _{\Phi ^{(2)}}$ whence $L_{B} ({\mathbb {R}}^n)$ is between $L_{2}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$. Again $B(t) / t^2 \uparrow$ and ${\tilde{A}} (t) / t^2 \uparrow$ which means $L_{{\tilde{A}}}$ is an interpolation space between $L_{2}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$, whence $L_{A}({\mathbb {R}}^n)$ is an interpolation space between $L_{1}({\mathbb {R}}^n)$ and $L_{2}({\mathbb {R}}^n)$. Finally, Theorem 3.10 in [13] ensures

$$\begin{aligned} {\mathscr {F}}: L_{A}({\mathbb {R}}^n) \rightarrow L_{B}({\mathbb {R}}^n), \end{aligned}$$

since $B(t) = 1 / {\tilde{A}}(t^{-1})$ is equivalent to ${\tilde{A}}(t) = 1 / B(t^{-1})$. $\square$

5 ${\mathscr {F}}$ Between Lorentz Gamma Spaces

In this section, we make use of the operators P and Q defined by

$$\begin{aligned} (P f) (t) = \frac{1}{t} \int _{0}^{t} f\quad \text {and}\quad (Q g )(t) = \int _{t}^{\infty } g(s) \frac{ds}{s}, \ \ \ f,g \in M(\mathbb {R_+}), \ t \in \mathbb {R_+}. \end{aligned}$$

These operators satisfy the equations

$$\begin{aligned} \int _{\mathbb {R_+}} g \, Pf = \int _{\mathbb {R_+}} f \, Qg, \ \ \ f,g \in M(\mathbb {R_+}), \end{aligned}$$

and

$$\begin{aligned} PQ= QP = P + Q. \end{aligned}$$

Fix an index $p \in (1, \infty )$ and a weight $0 \le u \in M({\mathbb {R}}_+)$. The Lorentz Gamma norm $\rho _{p, u}$ defined in terms of the Lorentz norm $\lambda _{p,u}(f) = \lambda _{p,u}(f^{*}) = \left( \int _{{\mathbb {R}}_+} f^{*}(t)^p dt \right) ^{1/p}$ by

$$\begin{aligned} \rho _{p,u} (f) = \lambda _{p,u} (f^{**}), \ \ \ f \in M(\Omega ), \end{aligned}$$

where, once again, $\Omega = {\mathbb {R}}^{n}$ or ${\mathbb {R}}_{+}$.

To guarantee $\rho _{p, u}(\chi _{E}) < \infty$ for all measurable sets $E \subset \Omega$ with $|E|< \infty$, we require

$$\begin{aligned} \int _{{\mathbb {R}}_+} \frac{u(t)}{ 1+ t^p } dt < \infty . \end{aligned}$$

(5.1)

The Lorentz Gamma space

$$\begin{aligned} \Gamma _{p,u}(\Omega )=\left\{ f \in M(\Omega ): \rho _{p,u}(f) <\infty \right\} , \end{aligned}$$

is then an r.i. space. The norm, $\rho _{p',v'}$, dual to $\rho _{p,u}$ is given by

$$\begin{aligned} \rho _{p', v'}(g) = \left( \int _{{\mathbb {R}}_+} g^{**}(t)^p v'(t) dt \right) ^{1/p'}, \ \ \ g \in M(\Omega ) \end{aligned}$$

where, $p' = p / (p-1)$ and

$$\begin{aligned} v'(t)= \frac{t^{p'+p-1}\int _0^t u \ \int _t^\infty u(s)s^{-p} ds}{{\left[ \int _0^t v + t^p \int _t^\infty u(s)s^{-p} ds\right] }^{p'+1}}. \end{aligned}$$

This is shown, for example, in [10].

In this section we study the inequality

$$\begin{aligned} \rho _{p,u}({\hat{f}}) \le C \, \rho _{q,v}(f), \ \ \ f \in (L_1\cap \Gamma _{q,v})({\mathbb {R}}^n). \end{aligned}$$

(5.2)

We begin by assuming $\Gamma _{p,u}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+}),$ then address the question of when this is the case later in the section.

Recall that Theorem 1.1 ensures that (5.2) holds if and only if

$$\begin{aligned} {{\bar{\rho }}}_ {p,u}(Uf^*) \le C {{\bar{\rho }}}_ {q,v}(f^*), \ \ \ f \in M(\mathbb {R_+}). \end{aligned}$$

(5.3)

Theorem 5.1

Let the indices p, q and weights u, v be as described above. Then, given that $\Gamma _{p,u}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+}),$ one has (5.2) if and only if

$$\begin{aligned} {{\bar{\rho }}}_{q',v'}(g^{**}) \le C {{\bar{\rho }}}_{p',{u_p}'}(g^{*}), \ \ \ g \in M(\mathbb {R_+}). \end{aligned}$$

(5.4)

where, as usual, $p'=\frac{p}{p-1}$, $q'=\frac{q}{q-1}$, $\int _{\mathbb {R_+}} v=\infty$,

$$\begin{aligned}{} & {} v'(t)= \frac{t^{q'+q-1}\int _0^t v\int _t^\infty v(s)s^{-q} ds}{{\left[ \int _0^t v+t^q \int _t^\infty v(s)s^{-q} ds\right] }^{q'+1}}\\{} & {} u_p(t)= u(t^{-1})t^{p-2}, \ \ \ \int _{\mathbb {R_+}} u_{p} = \infty , \end{aligned}$$

and

$$\begin{aligned} {u_p}'(t)= \frac{ t^{p'+p-1} \int _{0}^{t} u_{p} \int _{t}^{\infty } u_{p}(s) s^{-p} ds }{ \left[ \int _{0}^{t} u_{p} + t^{p} \int _{t}^{\infty } u_{p}(s)s^{-p} ds \right] ^{p'+1 } }, \ \ \ t \in {{\mathbb {R}}}_{+}. \end{aligned}$$

Proof

The inequality (5.3) tells that the space determined by the r.i. norm ${{\bar{\rho }}}_{p,u} (Uf^*)$ is the largest one mapped into $\Gamma _{p,u}({\mathbb {R}}_+)$ by U. Now,

$$\begin{aligned}{} & {} {\bar{\rho }}_{p,u}(Uf^*)= \left[ \int _{ {\mathbb {R}}_+ } P(Uf^{*})(t)^{p} u(t) \, dt \right] ^{\frac{1}{p}},\\{} & {} \begin{aligned} P \left( Uf^* \right) (t)&= t^{-1} \left( P (Q f^{*}) \right) ( t^{-1}). \end{aligned} \end{aligned}$$

Further,

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R_+}} \left[ t^{-1} \left( P (Q f^{*}) \right) ( t^{-1}) \right] ^{p} u(t) dt&= \int _{\mathbb {R_+}} \left[ t \left( P (Q f^{*}) \right) ( t) \right] ^{p} u(t^{-1}) t^{-2} dt. \end{aligned} \end{aligned}$$

We have shown that

$$\begin{aligned} \begin{aligned} {{\bar{\rho }}}_{p,u}(Uf^*)&=\left[ \int _{\mathbb {R_+}} \left( P (Q f^{*}) \right) (t)^p u_p(t) dt\right] ^{\frac{1}{p}}\\&={{\bar{\rho }}}_{p,u_p}(Qf^*). \end{aligned} \end{aligned}$$

Therefore, any $\Gamma _{q,v}(\mathbb {R_+})$ mapped into $\Gamma _{p,u}(\mathbb {R_+})$ by U must be embedded into this largest domain; that is,

$$\begin{aligned} {{\bar{\rho }}}_ {p,u_p}(Qf^*) \le C {{\bar{\rho }}}_ {q,v}(f^*), \ \ \ f \in M({\mathbb {R}}_+). \end{aligned}$$

(5.5)

But, since (5.5) is equivalent to (5.4), its dual inequality, (5.5) may be tested over any $0 \le f \in M({\mathbb {R}}_+),$ as is seen in

$$\begin{aligned} \int _{\mathbb {R_+}} g^{*} Qf = \int _{\mathbb {R_+}} f P g^{*} \le \int _{\mathbb {R_+}} f^{*} P g^{*}= \int _{\mathbb {R_+}} g^{*} Q f^{*}. \end{aligned}$$

$\square$

The inequality (5.4), and hence (5.2), amounts to

$$\begin{aligned} \left[ \int _{\mathbb {R_+}} (Ph)^{q'}v' \right] ^{\frac{1}{q'}} \le C \left[ \int _{\mathbb {R_+}} h^{p'}{u_p}' \right] ^{\frac{1}{p'}}, \end{aligned}$$

(5.6)

with $h=g^{**}$ belonging to

$$\begin{aligned} \Omega _{0,1}(\mathbb {R_+})= \{ 0 \le h \in M(\mathbb {R_+}): h(t) \downarrow \ \text {and} \ t \, h(t) \uparrow \text {on} \ \mathbb {R_+}\}. \end{aligned}$$

Such inequalities are shown in Theorem 4.4 of [10] to be equivalent to a pair of weighted norm inequalities involving general non-negative measurable functions. In the case of (5.6) this leads to

Theorem 5.2

Let the indices p, q and weights u, v be as in Theorem 5.1. Then, (5.6) holds if and only if

$$\begin{aligned} \left[ \int _{\mathbb {R_+}} \left[ (P+Q)g \right] ^{q'} v'\right] ^{\frac{1}{q'}} \le C \left[ \int _{\mathbb {R_+}} g^{p'}{u_p}^{1-p'}\right] ^{\frac{1}{p'}} \end{aligned}$$

(5.7)

and

$$\begin{aligned} \left[ \int _{\mathbb {R_+}} (P^2g)^{q'} v'\right] ^{\frac{1}{q'}} \le C \left[ \int _{\mathbb {R_+}} g^{p'}{u_p}^{1-p'}\right] ^{\frac{1}{p'}}, \ \ \ 0 \le g \in M( \mathbb {R_+}). \end{aligned}$$

Proof

According to Theorem 4.4 in [10], one has (5.6) if and only if

$$\begin{aligned} \left[ \int _{\mathbb {R_+}} \left[ (P+Q)Qg \right] ^{p} u_p\right] ^{\frac{1}{p}} \le C \left[ \int _{\mathbb {R_+}} g^{q}{v'}^{1-q}\right] ^{\frac{1}{q}} \end{aligned}$$

and

$$\begin{aligned} \left[ \int _{\mathbb {R_+}} \left[ P(P+Q)g \right] ^{q'} v'\right] ^{\frac{1}{q'}} \le C \left[ \int _{\mathbb {R_+}} g^{p'}{u_p}^{1-p'}\right] ^{\frac{1}{p'}}, \end{aligned}$$

holds for all $0 \le g \in M(\mathbb {R_+}).$

These are dual inequalities. We choose the second one, which easily reduces to (5.7). $\square$

To deal with the case $q \le p$ we will use special instances of the following combination of Theorems 1.7 and 4.1 from [6].

Theorem 5.3

Consider $0 \le K(x,y) \in M \left( \mathbb {R_+}\times \mathbb {R_+} \right)$, which, for fixed $y \in \mathbb {R_+}$, increases in x and, for fixed $x \in \mathbb {R_+},$ decreases in y and which, moreover, satisfies the growth condition

$$\begin{aligned} K(x,y) \le K(x,z)+K(z,y),\quad 0<y<z<x. \end{aligned}$$

Let t, u, v and w be nonnegative, measurable (weight) functions on $\mathbb {R_+}$ and suppose $\Phi _1(x)= \int _0^x \phi _1$ and $\Phi _2(x)= \int _0^x \phi _2$ are N-functions having complementary functions $\Psi _1(x)= \int _0^x {\phi _1}^{-1}$ and $\Psi _2(x)= \int _0^x {\phi _2}^{-1}$, respectively, with $\Phi _1\circ {\Phi _2}^{-1}$ convex. Then there exists $c>0$ such that

$$\begin{aligned} {\Phi _1}^{-1} \left( \int _{\mathbb {R_+}} \Phi _1 \left( c w(x) \int _0^x K(x,y) f(y) dy \right) t(x) dx \right) \\ \le {\Phi _2}^{-1} \left( \int _{\mathbb {R_+}} \Phi _2 \left( \, u(y) f(y) \, \right) v(y) dy \right) , \end{aligned}$$

$0 \le f \in M({\mathbb {R_+}})$, if and only if

$$\begin{aligned} \int _0^x \frac{K(x,y)}{u(y)} {\phi _2}^{-1} \left( \frac{c \alpha (\lambda ,x) K(x,y)}{\lambda u(y)v(y)} \right) dy \le c^{-1}\lambda \end{aligned}$$

and

$$\begin{aligned} \int _0^x \frac{1}{u(y)}\phi ^{-1}_2 \left( \frac{ c \beta (\lambda ,x)}{\lambda u(y)v(y)} \right) {dy}\le c^{-1} \lambda , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \alpha (\lambda ,x)&= \Phi _2\circ {\Phi _1}^{-1}\left( \int _x^{\infty } \Phi _1 \left( \lambda w(y) \right) t(y)dy \right) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \beta (\lambda ,x)&= \Phi _2\circ {\Phi _1}^{-1}\left( \int _x^{\infty }\Phi _1 \left( \lambda w(y)K(y,x) \right) t(y)dy \right) . \end{aligned} \end{aligned}$$

Theorem 5.4

Let $p,q,u,u',u_p,v,v'$ be as in the Theorem 5.1, with $1<q \le p<\infty$. Then, given that $\Gamma _{p,u}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+})$, one has (5.2) if and only if

• (1)
  $$\begin{aligned} {\left( \int _0^x u_p(y)dy\right) }^{\frac{1}{p}} {\left( \int _x^{\infty }v'(y)y^{-q'} dy \right) }^{\frac{1}{q'}} \le C \end{aligned}$$
• (2)
  $$\begin{aligned} {\left( \int _0^x v'(y) dy \right) }^{\frac{1}{q'}} {\left( \int _x^{\infty }u_p(y)y^{-p} dy \right) }^{\frac{1}{p}}\le C \end{aligned}$$
• (3)
  $$\begin{aligned} {\left( \int _0^x {\left( \log \frac{x}{y} \right) }^p u_p(y) dy \right) }^{\frac{1}{p}} {\left( \int _x^{\infty }v'(y)y^{-q'} dy \right) }^{\frac{1}{q'}} \le C \end{aligned}$$
• (4)
  $$\begin{aligned} {\left( \int _0^x u_p(y)dy\right) }^{\frac{1}{p}} {\left( \int _x^{\infty } v'(y){\left( \frac{1}{y} \log \frac{y}{x} \right) }^{q'} dy \right) }^{\frac{1}{q'}}\le C. \end{aligned}$$

Indeed (1) and (3) can be combined into

$$\begin{aligned} {\left( \int _0^\infty { \left( \log \left( 1+\frac{x}{y} \right) \right) }^p u_p(y) dy \right) }^{\frac{1}{p}} {\left( \int _x^{\infty }v'(y)y^{-q'} dy \right) }^{\frac{1}{q'}} \le C. \end{aligned}$$

Proof

The first inequality in (5.7) amounts to

$$\begin{aligned} \left[ \int _{\mathbb {R_+}}(P g )^{q'} v'\right] ^{\frac{1}{q'}} \le C \left[ \int _{\mathbb {R_+}} g^{p'} u_{p}^{1-p'}\right] ^{\frac{1}{p'}} \end{aligned}$$

and

$$\begin{aligned} \left[ \int _{\mathbb {R_+}}(Q g )^{q'} v'\right] ^{\frac{1}{q'}} \le C \left[ \int _{\mathbb {R_+}} g^{p'} u_{p}^{1-p'}\right] ^{\frac{1}{p'}}, \ \ \ 0 \le g \in M(\mathbb {R_+}), \end{aligned}$$

the latter inequality being, by duality, equivalent to

$$\begin{aligned} \left[ \int _{\mathbb {R_+}}(Pf)^{p} \, u_p\right] ^{\frac{1}{p}}\le C \left[ \int _{\mathbb {R_+}}{f}^{q} \, {v'}^{1-q}\right] ^{\frac{1}{q}}, \ \ \ 0 \le f \in M(\mathbb {R_+}). \end{aligned}$$

We illustrate the method of proof with the second inequality in (6.7) involving

$$\begin{aligned} (P^2g)(x)= \frac{1}{x} \int _0^x \log \left( \frac{x}{y} \right) g(y) dy. \end{aligned}$$

Thus, taking, in Theorem 6.3, $K(x,y)={\log }_+ \frac{x}{y}$, $\Phi _1(x)=x^{q'}$, $\Phi _2(x)=x^{p'}$ (observe that $(\Phi _1 \circ {\Phi _2}^{-1})(x)= x^{\frac{q'}{p'}},$ which is convex when $q \le p$), $w(y)=y^{-1}$, $t(y)=v'(y)$, $u(y)={u_p(y)}^{-1}$, $v(y)=u_p(y)$ we get

$$\begin{aligned} \alpha (\lambda ,x) = \lambda ^{p'} {\left( \int _{x}^\infty v'(y) y^{-q'} dy \right) }^{\frac{p'}{q'}} \end{aligned}$$

and

$$\begin{aligned} \beta (\lambda ,x) = \lambda ^{p'} {\left( \int _{x}^\infty v'(y) \left( y^{-1}\log \frac{y}{x} \right) ^{q'} dy \right) }^{\frac{p'}{q'}}, \end{aligned}$$

from which the conditions in Theorem 5.3 yields (3) and (4). We point out that $\lambda$ cancels. $\square$

The inequality (5.2) is much easier to deal with when

$$\begin{aligned} \rho _{p,u} (f) \simeq \lambda _{p,u} (f) = \left( \int _{\mathbb {R_+}} f^{*}(t)^{p} \, u(t) \, dt \right) ^{\frac{1}{p}}, \end{aligned}$$

(5.8)

which equivalence is not all that uncommon, as we will see later in this section. Indeed, given (5.8),

$$\begin{aligned} \begin{aligned} \rho _{p,u} (U f^{*})&\simeq \left( \int _{\mathbb {R_+}} \left( U f^{*} \right) (t)^{p} \, u(t) \, dt \right) ^{\frac{1}{p}} \\&= \left( \int _{\mathbb {R_+}} f^{**} (t)^{p} \, u_{p}(t) \, dt \right) ^{\frac{1}{p}}\\&= \rho _{p,u_{p}} ( f ). \end{aligned} \end{aligned}$$

We thus have

Theorem 5.5

Let p, q, u, $u_{p}$ and v be as in Theorem 5.1. Then, given that $\Gamma _{p,u} (\mathbb {R_+})$ is an interpolation space between $L_{2} (\mathbb {R_+})$ and $L_{\infty } (\mathbb {R_+})$, with $\rho _{p,u}$ satisfying (5.8), one has

$$\begin{aligned} \rho _{ p, u} ( {\hat{f}} ) \le C \, \rho _{ p, u_{p}} (f). \end{aligned}$$

(5.9)

Moreover, there is no essentially smaller r.i.-norm that can replace $\rho _{p, u_{p}}$ in (5.9).

Finally, there is a relatively simple condition sufficient to guarantee (5.2). It comes out of working with the inequality (5.5) and involves the norm of the dilation operator $E_s$ as a mapping from $\Gamma _{q,v}(\mathbb {R_+})$ to $\Gamma _{p,u_p}(\mathbb {R_+}),$ namely,

$$\begin{aligned} h(\Gamma _{q,v},\Gamma _{p,u_p})(t)\!=\! \inf \left\{ M>0: \bar{ \rho }_{p, u_{p}}(f(ts))\!=\! {\bar{\rho }}_{p, u_{p}} \!\left( (E_tf)(s) \right) \! \le \! M {\bar{\rho }}_{{q, v}}(f) \!<\! \infty \!\right\} . \end{aligned}$$

The argument in the proof of Theorem 4.1 of [16] ensures (5.5) provided

$$\begin{aligned} \int _1^ \infty h(\Gamma _{q,v},\Gamma _{p,u_p})(t)\, \frac{dt}{t} < \infty . \end{aligned}$$

Again the argument in the proof of Theorem 5.2 in [10] yields

$$\begin{aligned} h(\Gamma _{q,v},\Gamma _{p,u_p})(t)= \sup _{s>0} \frac{ \left[ \int _0^{s/t} u_p + (s/t)^p\int _{s/t}^\infty u_p(y)y^{-p} dy \right] ^{\frac{1}{p}} }{{\left[ \int _0^s v+s^q \int _s^\infty v(y)y^{-q} dy\right] }^{\frac{1}{q}}}, \end{aligned}$$

when $1<q\le p<\infty .$

Altogether, we have

Theorem 5.6

Let p, q, u, $u_{p}$ and v be as in Theorem 5.1, with $1<q\le p<\infty$. Then, given that $\Gamma _{p,u} (\mathbb {R_+})$ is an interpolation space between $L_{2} (\mathbb {R_+})$ and $L_{\infty } (\mathbb {R_+})$, one has (5.9) provided

$$\begin{aligned} \int _{1}^{\infty } \sup _{s>0} \frac{ \left[ \int _0^{s/t} u_p + (s/t)^p\int _{s/t}^\infty u_p(y)y^{-p} dy \right] ^{\frac{1}{p}} }{{\left[ \int _0^s v+s^q \int _s^\infty v(y)y^{-q} dy\right] }^{\frac{1}{q}}} \frac{dt}{t} < \infty . \end{aligned}$$

Proof

The result follows from the preceeding discussion, since (5.9) and (5.5) are equivalent when $\Gamma _{p,u} (\mathbb {R_+})$ is an interpolation space between $L_{2} (\mathbb {R_+})$ and $L_{\infty } (\mathbb {R_+}).$ $\square$

We now consider the question of when $\Gamma _{p,u} (\mathbb {R_+})$ is an interpolation space between $L_{2} (\mathbb {R_+})$ and $L_{\infty } (\mathbb {R_+}).$

To begin, recall that $\rho _{p,u} \simeq \lambda _{p,u}$ was shown in [2] to be equivalent to the $B_{p}$ condition

$$\begin{aligned} t^{p} \int _{t}^{\infty } u(s) \frac{ds}{s^{p}} \le C \int _{0}^{t} u, \ \ \ t \in \mathbb {R_+}. \end{aligned}$$

(5.10)

We have

Theorem 5.7

Fix $p \in (2, \infty )$ and a weight $0 \le u \in M(\mathbb {R_+})$. Suppose u satisfies the $B_{p/2}$ condition. Then, $\Gamma _{p,u}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+})$.

Proof

The $B_{p/2}$ condition is necessary and sufficient in order that $\rho _{p/2,u} \simeq \lambda _{p/2,u}$. Thus,

$$\begin{aligned} \begin{aligned} \rho _{p/2, u}^{(2)} (f^{*})&\simeq \left[ \int _{\mathbb {R_+}} ( f^{*}(t)^2 )^{\frac{p}{2}} \, u(t) \, dt \right] ^{ \frac{2}{p} \cdot \frac{1}{2} }\\&= \lambda _{p,u} (f^{*}). \end{aligned} \end{aligned}$$

But, $B_{p/2}$ condition implies

$$\begin{aligned} t^p \int _{t}^{\infty } u(s) \frac{ds}{s^p} = \int _{t}^{\infty } u(s) \left( \frac{t}{s} \right) ^{p} ds \le \int _{t}^{\infty } u(s) \left( \frac{t}{s} \right) ^{\frac{p}{2}} ds \le C \int _{0}^{t} u, \ \ \ t \in \mathbb {R_+}, \end{aligned}$$

and so

$$\begin{aligned} \lambda _{p,u} (f^{*}) \simeq \rho _{p,u} (f^{*}), \ \ \ f \in M(\mathbb {R_+}). \end{aligned}$$

We conclude $\Gamma _{p,u}(\mathbb {R_+}) = L_{ \rho _{p/2, u}^{(2)} } (\mathbb {R_+})$ and hence, in view of Theorem 3.1, $\Gamma _{p,u}(\mathbb {R_+})$ is an interpolation space between $L_{2}(\mathbb {R_+})$ and $L_{\infty }(\mathbb {R_+})$. $\square$

Remark 5.1

G. Sinnamon in [27] proved that, given $u \in B_{p/2}$ and provided $0<q \le 2 \le p < \infty$, one has (5.2) if and only if

$$\begin{aligned} {\bar{\rho }}_{p, u} (\chi _{(0,t)}) \le C {\bar{\rho }}_{q, v_{q}} (\chi _{(0,t)}), \ \ \ t \in {\mathbb {R}}_+, \end{aligned}$$

with $v_{q}(t)= v(t^{-1}) \, t^{q-2}$, $t \in {\mathbb {R}}_+$. Theorem 5.7 and the fact that ${\bar{\rho }}_{p, u} (U f^{*}) \simeq {\bar{\rho }}_{p, u_{p}}(f^{*})$, ensures that, for $p \in [2, \infty )$ and any $q \in (1, \infty )$, one has (5.2) if and only if

$$\begin{aligned} {\bar{\rho }}_{p, u_{p}} (\chi _{(0,t)}) \le C {\bar{\rho }}_{q, v} (\chi _{(0,t)}), \ \ \ t \in {\mathbb {R}}_+. \end{aligned}$$

In the proof of Theorem 5.10 below we require a corollary of the following result of R. Sharpley from [25, Lemma 3.1, Corollary 3.2]

Theorem 5.8

Let $\rho$ be an r.i. norm on $M({\mathbb {R}}^{n})$. Suppose the fundamental indices of $L_{ {\bar{\rho }}} (\mathbb {R_+})$ lie in (0, 1). Given $p \in (1, \infty )$, set $\mu _{p}(t) = \frac{ {\bar{\rho }} \left( \chi _{(0,t)}\right) ^{p} }{t}$, $t \in \mathbb {R_+}$. Then, ${\bar{\rho }}_{p, \mu _{p}} \left( \chi _{(0,t)}\right) = {\bar{\rho }} \left( \chi _{(0,t)}\right)$, $t \in \mathbb {R_+}$. Moreover,

$$\begin{aligned} {\bar{\rho }}_{p, \mu _{p}}(f^{*}) \simeq {\bar{\lambda }}_{p, \mu _{p}}(f^{*}), \ \ \ f \in M({\mathbb {R}}^n). \end{aligned}$$

Corollary 5.9

Let $\rho = \rho _{p, u}$ be as in Theorem 5.8. Then, $\rho = \rho _{p, \mu _{p}}$, where $\mu _{p}(t) = \frac{ {\bar{\rho }} \left( \chi _{(0,t)}\right) ^{p} }{t},$ $t \in \mathbb {R_+}$.

Proof

The spaces $\Gamma _{p, u}({\mathbb {R}}^n)$ and $\Gamma _{p, \mu _{p}}({\mathbb {R}}^n)$ have ${\bar{\rho }}_{p, \mu _{p}} \left( \chi _{(0,t)}\right) = {\bar{\rho }}_{p,u} \left( \chi _{(0,t)}\right)$, $t \in \mathbb {R_+}$. As such, the spaces are identical, in view of [10, Theorem 5.1]. $\square$

The principal result of this section is

Theorem 5.10

Fix $p \in [2, \infty )$ and $0 \le u \in M(\mathbb {R_+})$, with $\int _{R_+} \frac{u(t)}{1+t^{p}}dt < \infty$. Suppose the fundamental indices of $\Gamma _{p,u}({\mathbb {R}}^{n})$ lie in (0, 1). Then, $\Gamma _{p,u}({\mathbb {R}}^n)$ is an interpolation space between $L_{p}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$ (and hence between $L_{2}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$) if and only if

$$\begin{aligned} \sup _{s \ge t} \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,s)}\right) ^{p}}{s} \le C \, \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,t)}\right) ^{p}}{t}, \end{aligned}$$

(5.11)

for some $C>0$ independent of $t \in \mathbb {R_+}$. Moreover, the optimal r.i. domain for ${\mathscr {F}}$ corresponding to $\Gamma _{p,u}({\mathbb {R}}^n)$ has the norm

$$\begin{aligned} {\bar{\rho }}_{p,u} (U f^{*}) \simeq {\bar{\lambda }}_{p, u} (U f^{*}) = {\bar{\rho }}_{p, u_{p}}(f^{*}). \end{aligned}$$

Proof

Suppose first that $p=2$. Given $T: L_{2}({\mathbb {R}}^n), L_{\infty }({\mathbb {R}}^n) \rightarrow L_{2}({\mathbb {R}}^n), L_{\infty }({\mathbb {R}}^n)$ one has, according to [9, Theorem 1.11, p. 301] and [12],

$$\begin{aligned} \int _{0}^{t} (T f)^{*} (s)^{2} ds \le C' M_{2}^{2} \int _{0}^{Mt} f^{*} (s)^{2} ds = C' M_{2}M_{\infty } \int _{0}^{t} f^{*} (Ms)^{2} ds, \end{aligned}$$

$f \in \left( L_{2} + L_{\infty } \right) (\mathbb {R_+})$, in which $M= M_{\infty } / M_{2}$, $M_{k}$ being the norm of T on $L_{k}(\mathbb {R_+})$, $k=2, \infty$. In view of (5.11), HLP yields

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R_+}} (Tf)^{*}(t)^{2} \, \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,t)}\right) ^{2}}{t} dt&\le \int _{\mathbb {R_+}} (Tf)^{*}(t)^{2} \sup _{s \ge t} \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,s)}\right) ^{2}}{s} dt\\&\le C' M_{2} M_{\infty } \int _{\mathbb {R_+}} f^{*}(Mt)^{2} \sup _{s \ge t} \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,s)}\right) ^{2}}{s} dt\\&\le C C' M_{2} M_{\infty } \int _{\mathbb {R_+}} f^{*}(Mt)^{2} \, \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,t)}\right) ^{2}}{t} dt, \end{aligned} \end{aligned}$$

$f \in \left( L_{2} + L_{\infty } \right) (\mathbb {R_+})$. Theorem 5.8 now ensures the latter is equivalent to

$$\begin{aligned} \int _{\mathbb {R_+}} (Tf)^{**}(t)^{2} \, \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,t)}\right) ^{2}}{t} dt \le C C' M_{2} M_{\infty } h(M)^2 \int _{\mathbb {R_+}} f^{**}(t)^{2} \, \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,t)}\right) ^{2}}{t} dt, \end{aligned}$$

where h(t) is the norm of the dilation operator $E_{t}$ on $\Gamma _{ 2, \mu _{2} } (\mathbb {R_+}) = \Gamma _{2,u}(\mathbb {R_+})$, $\mu _{2}(s)= \frac{{\bar{\rho }}_{2,u} \left( \chi _{(0,s)}\right) ^{2}}{s}$, by Corollary 5.9, that is, $T: \Gamma _{2,u}({\mathbb {R}}^n) \rightarrow \Gamma _{2,u}({\mathbb {R}}^n)$. Thus, $\Gamma _{2,u}({\mathbb {R}}^n)$ is between $L_{2}({\mathbb {R}}^n)$ and $L_{ \infty }({\mathbb {R}}^n)$.

Suppose, next, $p>2$. The “if” part of our theorem will follow in this case if we can show (5.11) implies the weight $w(t)= \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,t)}\right) ^{p}}{t}$ satisfies $B_{p/2}$ condition. But,

$$\begin{aligned} \begin{aligned} t^{p/2} \int _{t}^{\infty } \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,s)}\right) ^{p}}{s} \frac{ds}{s^{p/2}}&\le t^{p/2} \int _{t}^{\infty } \sup _{y \ge s} \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,y)}\right) ^{p}}{y} \frac{ds}{s^{p/2}}\\&\le t^{p/2} \sup _{y \ge t} \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,y)}\right) ^{p}}{y} \int _{t}^{\infty }\frac{ds}{s^{p/2}}\\&\le C \, t \, \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,t)}\right) ^{p}}{t} \\&\le C^2 \int _{0}^{t} \frac{{\bar{\rho }}_{p,u} \left( \chi _{(0,s)}\right) ^{p}}{s} ds, \ \ \ t \in \mathbb {R_+}. \end{aligned} \end{aligned}$$

This completes the proof of “if” part.

As for the “only if” part we rely on a result of L. Maligranda [18] asserting that if $L_{\rho }({\mathbb {R}}^n)$ is an interpolation space between $L_{p}({\mathbb {R}}^n)$ and $L_{\infty }({\mathbb {R}}^n)$, then

$$\begin{aligned} \frac{ {\bar{\rho }} (\chi _{(0,s)}) }{ {\bar{\rho }} (\chi _{(0,t)}) } \le C \max \left[ \left( \frac{s}{t} \right) ^{\frac{1}{p}}, 1 \right] . \end{aligned}$$

(5.12)

Indeed, for $t \le s$, (5.12) yields

$$\begin{aligned} \frac{ {\bar{\rho }} (\chi _{(0,s)}) }{ {\bar{\rho }} (\chi _{(0,t)}) } \le C \left( \frac{s}{t} \right) ^{\frac{1}{p}} \end{aligned}$$

or

$$\begin{aligned} \frac{ {\bar{\rho }} (\chi _{(0,s)})^{p} }{ s } \le C \, \frac{ {\bar{\rho }} (\chi _{(0,t)})^{p} }{t}, \end{aligned}$$

from which (5.11) follows. $\square$

To this point the Lorentz Gamma range norms have been equivalent to functionals of the form

$$\begin{aligned} \lambda _{p,u} (f) = \left[ \int _{\mathbb {R_+}} f^{*}(s)^{p} u(t) dt \right] ^{\frac{1}{p}}. \end{aligned}$$

This need not be the case for the $\rho _{ 2p, u}$ in Theorem 5.12 below.

Lemma 5.11

Fix $p \in (1, \infty )$ and $0 \le u \in M(\mathbb {R_+})$, with

$$\begin{aligned} \int _{\mathbb {R_+}} \frac{ u(t) }{ 1+t^{p} } dt < \infty . \end{aligned}$$

Then,

$$\begin{aligned} \left( \int _{\mathbb {R_+}} f^{**}(t)^{p} \, u(t) \, dt \right) ^{\frac{1}{p}} \le \left( \int _{\mathbb {R_+}} f^{*}(t)^{p} \, u^{(p)}(t) \, dt \right) ^{\frac{1}{p}}, \ \ \ f \in M(\mathbb {R_+}), \end{aligned}$$

(5.13)

where

$$\begin{aligned} u^{(p)}(t) = p \,t^{p-1} \int _{t}^{\infty } u(s) s^{-p} ds, \ \ \ t \in \mathbb {R_+}; \end{aligned}$$

moreover, $u^{(p)}$ is essentially the smallest weight for which (5.13) holds.

Proof

It is shown in [21] that

$$\begin{aligned} \left( \int _{\mathbb {R_+}} f^{**}(t)^{p} \, u(t) \, dt \right) ^{\frac{1}{p}} \le \left( \int _{\mathbb {R_+}} f^{*}(t)^{p} \, v (t) \, dt \right) ^{\frac{1}{p}}, \ \ \ f \in M(\mathbb {R_+}), \end{aligned}$$

if and only if

$$\begin{aligned} \int _{0}^{t} u(s) ds + t^{p} \int _{t}^{\infty } u(s) s^{-p} ds \le C \int _{0}^{t} v, \ \ \ t \in \mathbb {R_+}. \end{aligned}$$

But,

$$\begin{aligned} \begin{aligned} \int _{0}^{t} u^{(p)} (s) ds&= \int _{0}^{t} p \, s^{p-1} \int _{s}^{\infty } u(y) y^{-p} dy \, ds\\&= \int _{0}^{t} p \, s^{p-1} \int _{s}^{t} u(y) y^{-p} dy \, ds + \left[ \int _{0}^{t} p \, s^{p-1} ds \right] \left[ \int _{t}^{\infty } u(s) s^{-p} ds \right] \\&= \int _{0}^{t} \left( \int _{0}^{y} p \, s^{p-1} ds \right) u(y) y^{-p} dy + t^{p} \int _{t}^{\infty } u(s) s^{-p} ds \\&= \int _{0}^{t} u + t^{p} \int _{t}^{\infty } u(s) s^{-p} ds. \end{aligned} \end{aligned}$$

We conclude that

$$\begin{aligned} \left( \int _{\mathbb {R_+}} f^{**}(t)^{p} \, u(t) \, dt \right) ^{\frac{1}{p}} \le C \left( \int _{\mathbb {R_+}} f^{*}(t)^{p} \, u^{(p)}(t) \, dt \right) ^{\frac{1}{p}}, \ \ \ f \in M(\mathbb {R_+}). \end{aligned}$$

$\square$

Theorem 5.12

Let p and u be as in Lemma 5.11. Then,

$$\begin{aligned} \rho _{ 2p, u } ( {\widehat{f}} \, ) = {\bar{\rho }}_{ 2p, u } ( ( {\widehat{f}} )^{*} ) \le C {\bar{\rho }}_{ 2p, u^{(p)}_{2p} } ( f^{*} ) = \rho _{ 2p, u^{(p)}_{2p} } ( f ), \end{aligned}$$

(5.14)

where

$$\begin{aligned} u^{(p)}_{2p} (t)\!=\! u^{(p)} ( t^{-1} ) \, t^{2p -2} \!=\! p(t^{-1})^{p-1}\! \left( \int _{t^{-1}}^{\infty } u(s) s^{-p} ds \right) t^{2p -2} \!=\! p \, t^{p-1}\! \int _{t^{-1}}^{\infty }\! u(s)s^{-p} ds.\end{aligned}$$

Proof

Applying the construction in (3.1) to the functionals in (5.13) yields

$$\begin{aligned} \left( \int _{\mathbb {R_+}} \left( t^{-1} \int _{0}^{t} f^{*}(s)^{2} ds \right) ^{p} \, u(t) \, dt \right) ^{\frac{1}{2p}} \le \left( \int _{\mathbb {R_+}} f^{*}(t)^{2p} \, u^{(p)}(t) \, dt \right) ^{\frac{1}{2p}} = \lambda _{2p, u^{(p)}} (f). \end{aligned}$$

Again,

$$\begin{aligned} \left( t^{-1} \int _{0}^{t} f^{*}(s) ds \right) ^{2p} \le \left( t^{-1} \int _{0}^{t} f^{*}(s)^{2} ds \right) ^{p} \end{aligned}$$

by Hölder’s inequality.

Hence, using HLP in (2.8), yields

$$\begin{aligned} \begin{aligned} \rho _{ 2p, u } ( {\widehat{f}} \, )&\le \rho _{ p, u } ( \, |{\widehat{f}} \, |^{ \ 2} \, )^{1/2} \\&\le C \rho _{ p, u } ( (U f^{*})^{ 2} \, )^{1/2} \\&\le C \lambda _{ 2p, u^{(p)} } ( U f^{*} ) \\&= C \left( \int _{\mathbb {R_+}} (U f^{*})(t)^{2p} \, u^{(p)}(t) \, dt \right) ^{\frac{1}{2p}} \\&= C \left( \int _{\mathbb {R_+}} f^{**}(t)^{2p} \, u^{(p)}_{2p}(t) \, dt \right) ^{\frac{1}{2p}} \\&= C \rho _{ 2p, u^{(p)}_{2p} } ( f ). \end{aligned} \end{aligned}$$

$\square$

Example 5.1

Fix p, $1<p< \infty$, and set

$$\begin{aligned} u(t)= {\left\{ \begin{array}{ll} t^{2p-1} \left( \log \textstyle \frac{1}{t} \right) ^{- \alpha }, &{} 0<t<1,\\ t^{p-1- \alpha }, &{} t>1, \end{array}\right. } \end{aligned}$$

with $0<\alpha <1$. Then, one has

$$\begin{aligned} \rho _{ 2p, u } (f) \not \simeq \lambda _{2p, u} (f), \ \ \ f \in M(\mathbb {R_+}), \end{aligned}$$

or, equivalently,

$$\begin{aligned} t^{2p} \int _{t}^{\infty } u(s) s^{-2p} ds \le C \int _{0}^{t} u, \ \ \ t \in \mathbb {R_+}, \end{aligned}$$

(5.15)

does not hold. Indeed, the left hand side of (5.15) is equal to $C t^{2p} \left( \log \textstyle \frac{1}{t} \right) ^{- \alpha + 1}$, while the right hand side is

$$\begin{aligned} \int _{0}^{t} u = \int _{0}^{t} s^{2p-1} \left( \log \textstyle \frac{1}{s} \right) ^{- \alpha } \simeq t^{2p} \left( \log \textstyle \frac{1}{t} \right) ^{- \alpha }, \ \ \ 0<t<1, \end{aligned}$$

in view of L’Hôspital rule. The ratio of the left side to the right side in (5.15) is, essentially, $\log \textstyle \frac{1}{t}$ which $\rightarrow \infty$ as $t \rightarrow 0^{+}$.

6 Other Work

Inequalities involving Fourier transform other than those considered in this paper are weighted Lebesgue inequalities

$$\begin{aligned} \left( \int _{{\mathbb {R}}^n} |{\hat{f}}(x) \, w(x) |^{q} \, dx \right) ^{\frac{1}{q}} \le C \left( \int _{{\mathbb {R}}^n} |f(x) \, v(x) |^{p} \, dt \right) ^{\frac{1}{p}} \end{aligned}$$

and weighted Lorentz inequalities

$$\begin{aligned} \left( \int _{\mathbb {R_+}} ({\hat{f}} \, )^{*}(t)^{q} \, w(t) \, dt \right) ^{\frac{1}{q}} \le C \left( \int _{\mathbb {R_+}} \left( \int _{0}^{1/t} f^{*} \, \right) ^{p} \, v(t) \, dt \right) ^{\frac{1}{p}}, \end{aligned}$$

in which $0 \le v, w \in M({\mathbb {R}}^n)$ and $1< p, q < \infty$.

In both [15, 20] conditions are for the Lebesgue inequalities that apply not just to w and v but to all weights equimeasurable with them. The extreme cases of these are the decreasing rearrangement, W, of w and the increasing rearrangement, V, of v. This reduces the considerations to the case $w \downarrow$ and $v \uparrow$.

The weighted Lebesgue inequalities are shown in [15, 20] to be equivalent to inequalities of the form, for example when $1 \le p \le q \le \infty$,

$$\begin{aligned} \left( \int _{0}^{t^{-1}} w \right) ^{\frac{1}{q}} \left( \int _{0}^{t} w^{- \frac{1}{p-1}} \right) ^{\frac{q}{p'}} \le B, \ \ \ t \in {\mathbb {R}}_+. \end{aligned}$$

(6.1)

In [15] the sufficiency is proved using the inequality (2.8) from [13]. The necessity comes out of the inequality

$$\begin{aligned} \rho _{q} \left( ({\hat{f}} )^{*} w \right) \simeq \rho _{2} \left( U f^{*} w \right) \end{aligned}$$

from [15]. The proofs in [20] are more complicated. The conditions for the weighted Lorentz inequalities are similar to (6.1).

A brief survey of papers on these inequalities, from the pioneering work of Benedetto and Heinig [5] through that of G. Sinnamon [26] and Rastegari and Sinnamon [24], is given in the paper [22] of Nursultanov and Tikhonov.

In this paper we have seen the behaviour of ${\mathscr {F}}$ on r.i. spaces depends on its action on radially decreasing functions. But what about the size of f if ${\hat{f}}$ is radially decreasing? This question is taken up in [11] in the context of Fourier series where functions with a cosine series having decreasing coefficients as $|n| \rightarrow \infty$ are studied.