2 Rearrangement-invariant spaces

Let $(\Omega ,\mu )$ be a $\sigma $ -finite measure space. Denote by $\mathfrak M(\Omega )$ the set of $\mu $ -measurable real-valued functions on $\Omega $ and by $\mathfrak M_+(\Omega )$ the nonnegative functions in $\mathfrak M(\Omega )$ . A Banach function norm is a functional $\rho : \mathfrak M_+(\Omega )\to [0,\infty ]$ satisfying

1. (A1) $\rho (f)=0$ if and only if $f=0\, \mu $ -a.e.;

2. (A2) $\rho (cf)=c\rho (f)$ , $c\ge 0$ ;

3. (A3) $\rho (f+g)\le \rho (f)+\rho (g)$ ;

4. (A4) $0\le f_n\uparrow f$ implies $\rho (f_n)\uparrow \rho (f)$ ;

5. (A5) $\mu (E)<\infty $ implies $\rho (\chi _E)<\infty $ ;

6. (A6) $\mu (E)<\infty $ implies $\int _E f\,d\mu \le c_E(\rho )\rho (f)$ , for some constant $c_E(\rho )$ depending on E and $\rho $ but not on $f\in \mathfrak M_+(\Omega )$ .

A Banach function norm is said to be r.i. if $\rho (f)=\rho (g)$ whenever $f,g\in \mathfrak M_+(\Omega )$ are equimeasurable in the sense that $f^*=g^*$ ; the nonincreasing rearrangement, $f^*$ , of $f\in \mathfrak M(\Omega )$ on $\mathbb {R}_+$ is defined as

$$\begin{align*}f^*(t):=\inf\{\lambda>0: \mu(\{ x\in \Omega: |f(x)|>\lambda\})\le t\}, \qquad t\in I_\mu:=(0,\mu(\Omega)). \end{align*}$$

It satisfies

$$\begin{align*}|\{ t\in I_\mu: f^*(t)>\tau\}|=\mu(\{ x\in \Omega: |f(x)|>\tau\}), \qquad f\in \mathfrak M(\Omega),\ \tau\in\mathbb{R}_+. \end{align*}$$

Although $f\mapsto f^*$ is not subadditive, the mapping $f\mapsto t^{-1}\int _0^t f^*(s)\,ds$ is, namely,

(2.1) $$ \begin{align} t^{-1}\int_0^t(f+g)^*(s)\,ds \le t^{-1}\int_0^t f^*(s)\,ds+t^{-1}\int_0^t g^*(s)\,ds \end{align} $$

for all $f,g\in \mathfrak M(\Omega )$ and $t\in \mathbb {R}_+$ .

A basic technique for working with r.i. norms is the Hardy–Littlewood–Pólya (HLP) principle, which asserts that

$$\begin{align*}\int_0^t f^*\le \int_0^t g^* \quad\text{for all } t>0 \quad\text{implies}\quad \rho(f)\le \rho(g) \end{align*}$$

(see [Reference Bennett and Sharpley1, Chapter 2, Theorem 4.6, p. 61].

The Köthe dual of a Banach function norm $\rho $ is another such norm, $\rho '$ , defined by

$$\begin{align*}\rho'(g):=\sup_{\rho(f)\le 1}\int_\Omega fg\,d\mu, \quad f,g\in \mathfrak M_+(\Omega). \end{align*}$$

It satisfies the Lorentz–Luxemburg theorem (sometimes referred to as the principle of duality):

$$\begin{align*}\rho":=(\rho')'=\rho \end{align*}$$

(see [Reference Bennett and Sharpley1, Chapter 1, Theorem 2.7, p. 10]).

The space $L_\rho (\Omega )$ is the vector space

$$\begin{align*}\{f\in \mathfrak M(\Omega): \rho(|f|)<\infty\}, \end{align*}$$

endowed with the norm $\|f\|_{L_\rho }:=\rho (|f|)$ . This Banach space is said to be an r.i. space, provided $\rho $ is an r.i. function norm.

The Luxemburg–Orlicz norm $\rho _{{}_A}$ defined in (1.1) is an r.i. norm; indeed,

$$\begin{align*}\rho_{{}_A}(f)=\inf\left\{\lambda>0:\int_{I_\mu}A\!\left(\frac{f^*(t)}{\lambda}\right)dt\le 1\right\},\qquad f\in \mathfrak M(\Omega). \end{align*}$$

Its Köthe dual, $\rho _{{}_A}'$ , satisfies

$$\begin{align*}\rho_{\widetilde A}(g)\le \rho_{{}_A}'(g)\le 2\,\rho_{\widetilde A}(g), \qquad g\in \mathfrak M(\Omega), \end{align*}$$

with

$$\begin{align*}\widetilde A(t):=\int_0^t a^{-1}(s)\,ds, \qquad t\in \mathbb{R}_+, \end{align*}$$

called the Young function complementary to A.

For further properties of r.i. spaces, see [Reference Bennett and Sharpley1]; for Orlicz spaces in particular, see [Reference Rao and Ren12].

3 Proofs of Theorems 1.1 and 1.3

We require the following inequalities, which are analogs of those for the Hardy–Littlewood maximal function $Mf$ in [Reference Stein13, Theorem 1(b), p. 5 and Theorem 5.2(b), p. 23]. Namely, for all $\tau>0$ ,

(3.1) $$ \begin{align} &\frac{1}{2\tau}\int_{\{t\in I_\mu:\, f^*(t)>\tau\}} f^*(t)\,dt \;\le\; \tfrac12\left|\{t\in I_\mu: t^{-1}\!\int_0^t f^*(s)\,ds>\tau\}\right|\\&\quad\le\; \frac{1}{\tau}\int_{\{t\in I_\mu:\, f^*(t)>\tau/2\}} f^*(t)\,dt. \nonumber\end{align} $$

(The factor $\tfrac 12$ at the start, which cancels the $2$ in the right-hand integrand, has been incorporated into the inequality so that no change of variable is needed in the conclusion.)

To prove (3.1), let $t_0$ be the least t for which $t^{-1}\int _0^t f^*(s)\,ds=\tau $ . (The inequalities are trivial if there is no such $t_0$ .) Then,

$$\begin{align*}\left|\{t\in I_\mu: t^{-1}\!\int_0^t f^*(s)\,ds>\tau\}\right|=t_0= \frac{1}{\tau}\int_0^{t_0} f^*(t)\,dt\ge \frac{1}{\tau}\int_{\{t\in I_\mu:\, f^*(t)>\tau\}} f^*(t)\,dt, \end{align*}$$

which yields the first inequality. For the second, define

$$\begin{align*}f_\tau(t):=\min\!\left\{f^*(t),\tfrac{\tau}{2}\right\}, \qquad f^\tau(t):=f^*(t)-f_\tau(t). \end{align*}$$

Then,

$$\begin{align*}\left|\{t\in I_\mu: t^{-1}\!\int_0^t f^*(s)\,ds>\tau\}\right|\le \left|\{t\in I_\mu: t^{-1}\!\int_0^t f^\tau(s)\,ds>\tfrac{\tau}{2}\}\right| \le \frac{2}{\tau}\int_{I_\mu} f^\tau(t)\,dt \end{align*}$$

$$\begin{align*}\le \frac{2}{\tau}\int_{\{t\in I_\mu: f^*(t)>\tau/2\}} f^\tau(t)\,dt \le \frac{2}{\tau}\int_{\{t\in I_\mu: f^*(t)>\tau/2\}} f^*(t)\,dt, \end{align*}$$

which is the second inequality after multiplication by $\tfrac 12$ .

We can now turn to the proof of Theorem 1.3. From the definition of c, we get

$$\begin{align*}A(t)=t\int_0^t c(s)\,\frac{ds}{s}. \end{align*}$$

The first inequality in (3.1) ensures, for all $\lambda>0$ ,

$$\begin{align*}\int_{\mathbb{R}_+}\int_{\{t\in I_\mu:\, f^*(t)/\lambda>\tau\}} \frac{f^*(t)}{\lambda}\,dt\,c(\tau)\frac{d\tau}{\tau} \;\le\; \int_{\mathbb{R}_+}\left|\{t\in I_\mu: \tfrac{t^{-1}\int_0^t f^*(s)\,ds}{\lambda}>\tau\}\right|c(\tau)\,d\tau, \end{align*}$$

that is,

$$ \begin{align*} \int_{I_\mu} A\!\left(\tfrac{f^*(t)}{\lambda}\right)dt&= \int_{I_\mu}\frac{f^*(t)}{\lambda}\int_0^{f^*(t)/\lambda} c(\tau)\frac{d\tau}{\tau}\,dt\\ &=\int_{\mathbb{R}_+}\int_{\{ t\in I_\mu:\, f^*(t)/\lambda>\tau\}}\frac{f^*(t)}{\lambda}\,dt\,c(\tau)\frac{d\tau}{\tau}\\ &\le \int_{\mathbb{R}_+}\left|\{t\in I_\mu: \tfrac{t^{-1}\int_0^t f^*(s)\,ds}{\lambda}>\tau\}\right|c(\tau)\,d\tau\\ &=\int_{I_\mu}C\!\left(\tfrac{t^{-1}\int_0^t f^*(s)\,ds}{\lambda}\right)dt. \end{align*} $$

Conversely, the second inequality in (3.1) yields

$$ \begin{align*} \int_{I_\mu}C\!\left(\tfrac{t^{-1}\int_0^t f^*(s)\,ds}{\lambda}\right)dt&= \int_{\mathbb{R}_+}\left|\{t\in I_\mu:\, \tfrac{t^{-1}\int_0^t f^*(s)\,ds}{\lambda}>\tau\}\right|c(\tau)\,d\tau\\ &\le 2\int_{\mathbb{R}_+}\int_{\{ t\in I_\mu:\, f^*(t)/\lambda>\tau/2\}}\frac{f^*(t)}{\lambda}\,dt\,c(\tau)\frac{d\tau}{\tau}\\ &=\int_{I_\mu}\frac{2 f^*(t)}{\lambda}\int_0^{2 f^*(t)/\lambda} c(\tau)\frac{d\tau}{\tau}\,dt\\ &=\int_{I_\mu} A\!\left(\tfrac{2 f^*(t)}{\lambda}\right)dt. \end{align*} $$

Together, these give

$$\begin{align*}\tfrac12\,\rho_{{}_{\Gamma_C}}(f)\le\rho_A(f^*)\le \rho_{{}_{\Gamma_C}}(f), \end{align*}$$

which, since $\rho _{{}_A}(f)=\rho _{{}_A}(f^*)$ , completes the proof of Theorem 1.3.

The result of Theorem 1.1 is now a consequence of the following corollary.

Corollary 3.1 Let $A(t)=\int _0^t a(s)\,ds$ , $t\in \mathbb {R}_+$ , be a Young function for which

$$\begin{align*}\int_{\mathbb{R}_+}A\!\left(\frac{k}{1+t}\right)dt<\infty \end{align*}$$

for some constant $k>0$ , so that, as shown before,

$$\begin{align*}\int_0^t a(s)\,\frac{ds}{s}<\infty, \quad t\in \mathbb{R}_+. \end{align*}$$

Set

(3.2) $$ \begin{align} \mathcal A(t):=t\int_0^t a(s)\,\frac{ds}{s}, \qquad t\in \mathbb{R}_+. \end{align} $$

Then, $\mathcal A(t)$ is a Young function such that

$$\begin{align*}a(t)= \mathcal A'(t)-\frac{\mathcal A(t)}{t}, \quad t\in \mathbb{R}_+, \end{align*}$$

whence, for any $\sigma $ -finite measure space $(\Omega ,\mu )$ ,

$$\begin{align*}\tfrac12\,\rho_{\Gamma_{\mathcal A}}(f)\le \rho_{\mathcal A}(f)\le \rho_{\Gamma_{\mathcal A}}(f), \qquad f\in \mathfrak M(\Omega). \end{align*}$$

Example 3.2 Consider the Young function

$$\begin{align*}A(t)=\int_0^t \ln^\beta(1+s)\,ds, \qquad 0<\beta<1,\ t\in \mathbb{R}_+, \end{align*}$$

for which $a(s)=\ln ^\beta (1+s)$ , $s\in \mathbb {R}_+$ . A short computation gives

$$\begin{align*}c(t)=\frac{\beta}{t}\int_0^t \frac{s}{1+s}\ln^{\beta-1}(1+s)\,ds\sim \beta\ln^{\beta-1}(1+t), \qquad t\to\infty, \end{align*}$$

so $c(t)$ essentially decreases rather than increases as $t\to \infty $ . That is, $C(t)=\int _0^t c(s)\,ds$ is not convex, and consequently, $\rho _{\Gamma _C}$ is genuinely a non-homogeneous sublinear functional rather than a norm (cf. Theorem 1.3). This example shows that the functionals $\rho _{\Gamma _C}$ produced by Theorem 1.3 are strictly more general than Luxemburg–Orlicz norms.

The complementary Young function of $\mathcal A$ will be needed in Section 4 (Theorem 4.6) for the proof of the CZ bound; we record its description now.

Remark 3.3 The complementary Young function $\widetilde {\mathcal A}$ of $\mathcal A$ satisfies

$$\begin{align*}\tfrac{3}{2}\int_0^{t/3} b^{-1}(s)\,ds\le \widetilde{\mathcal A}(t)\le \int_0^t b^{-1}(s)\,ds, \end{align*}$$

where $b(t):=\int _0^t a(s)\,\frac {ds}{s}$ , $t\in \mathbb {R}_+$ . To see this, observe that

$$ \begin{align*} b(t)\le {\mathcal A}'(t)=\int_0^t a(s)\,\frac{ds}{s}+a(t)&\le \frac{\ln 2+1}{\ln 2}\int_0^{2t} a(s)\,\frac{ds}{s}\le 3b(2t), \end{align*} $$

and so

$$\begin{align*}\tfrac12\,b^{-1}\!\left(\tfrac{t}{3}\right)\le (\mathcal A')^{-1}(t)\le b^{-1}(t), \qquad t\in \mathbb{R}_+. \end{align*}$$

The estimate then follows by integration, using that $\widetilde {\mathcal A}(t)=\int _0^t (\mathcal A')^{-1}(s)\,ds$ .

4 Applications of Theorem 1.1

The operators in the applications below are controlled in the same way by the Hardy–Littlewood maximal function, so we consider it first.

4.1 The Hardy–Littlewood maximal function

We begin with the following definition.

Definition 4.1 Given a function f locally integrable on $\mathbb {R}^n$ , we define its Hardy–Littlewood maximal function, $Mf$ , at $x\in \mathbb {R}^n$ by

$$\begin{align*}(Mf)(x)=\sup_{r>0}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)|\, dy, \end{align*}$$

where $B(x,r)=\{y\in \mathbb {R}^n: |x-y|<r\}$ .

The following result extends to Orlicz spaces, the well-known boundedness of M.

Theorem 4.2 Let A and $\mathcal A$ be as in Theorem 1.1. Then, there exists $C>0$ , independent of locally integrable f on $\mathbb {R}^n$ , such that

$$\begin{align*}\rho_A(Mf)\le C\rho_{\mathcal A}(f). \end{align*}$$

Moreover, $L_{\mathcal A}(\mathbb {R}^n)$ is the largest Orlicz space mapped by M into $L_A(\mathbb {R}^n)$ .

Proof It is proved, for $n=1$ , in [Reference Herz8] and for general n in [Reference Bennett and Sharpley1], that there exist positive constants c and $c'$ , depending only on n, with

$$\begin{align*}c(Mf)^*(t)\le (Pf^*)(t)\le c'(Mf)^*(t), \qquad t>0. \end{align*}$$

Thus, by Theorem 1.1,

$$\begin{align*}\rho_A(Mf)=\rho_A((Mf)^*)\le c^{-1}\rho_A(Pf^*)\le 2c^{-1}\rho_{\mathcal A}(f^*)= 2c^{-1}\rho_{\mathcal A}(f). \end{align*}$$

For the optimality, suppose $L_B$ is any Orlicz space with $M:L_B(\mathbb {R}^n)\to L_A(\mathbb {R}^n)$ . Testing against radial decreasing f and using $(Mf)^*\approx Pf^*$ , one obtains $P:L_B(\mathbb {R}_+)\to L_A(\mathbb {R}_+)$ , which by Theorem 1.1 forces $L_B\subset L_{\mathcal A}$ .

4.2 Approximate identities

Let $\phi $ be an integrable function on $\mathbb {R}^n$ and set

$$\begin{align*}\phi_{\epsilon}(x)=\epsilon^{-n}\phi(x/\epsilon), \qquad \epsilon>0,\ x\in \mathbb{R}^n. \end{align*}$$

Using the methods of [Reference Stein13, Theorem 2, pp. 62–65], one can show that

$$\begin{align*}(\phi_{\epsilon}\ast f)(x)=\int_{\mathbb{R}^n}\phi_{\epsilon}(x-y)f(y)\, dy \end{align*}$$

converges to f in $L_A(\mathbb {R}^n)$ as $\epsilon \to 0^+$ , provided

$$\begin{align*}\int_{\mathbb{R}^n}\mathcal A(\lambda|f(y)|)\, dy<\infty \end{align*}$$

for all $\lambda>0$ . (Here, A and $\mathcal A$ are as in Theorem 1.1.) For this reason, the family $\{\phi _{\epsilon }\}_{\epsilon>0}$ is called an approximate identity.

It is further proved in the same theorem in [Reference Stein13] that

$$\begin{align*}\sup_{\epsilon>0}|(\phi_{\epsilon}\ast f)(x)|\le \left[\int_{\mathbb{R}^n}|\psi(y)|\, dy\right](Mf)(x), \qquad x\in \mathbb{R}^n, \end{align*}$$

in which the radial majorant of $\phi $ ,

$$\begin{align*}\psi(x)=\sup_{|y|\ge |x|}|\phi(y)|, \end{align*}$$

is assumed to be integrable.

In view of Theorem 4.2, we obtain the following result.

Theorem 4.3 Let $\phi $ be an integrable function on $\mathbb {R}^n$ whose radial majorant

$$\begin{align*}\psi(x)=\sup_{|y|\ge |x|}|\phi(y)| \end{align*}$$

is also integrable on $\mathbb {R}^n$ . Then, with A and $\mathcal A$ as in Theorem 1.1,

$$\begin{align*}\rho_A\!\left(\sup_{\epsilon>0}|(\phi_{\epsilon}\ast f)(x)|\right)\le C \rho_{\mathcal A}(f), \end{align*}$$

whenever $\rho _{\mathcal A}(f)<\infty $ . Moreover, when $\phi $ is positive, radial, and decreasing, the domain $L_{\mathcal A}(\mathbb {R}^n)$ is the largest Orlicz space for which this bound holds.

Proof The first inequality is immediate from Theorem 4.2 since by hypothesis,

$$\begin{align*}\sup_{\epsilon>0}|(\phi_{\epsilon}\ast f)(x)|\le \left[\int_{\mathbb{R}^n}|\psi(y)|\, dy\right](Mf)(x), \qquad x\in \mathbb{R}^n; \end{align*}$$

applying $\rho _A$ to both sides and using $\rho _A(Mf)\le C\rho _{\mathcal A}(f)$ from Theorem 4.2 yields

$$\begin{align*}\rho_A\!\left(\sup_{\epsilon>0}|(\phi_{\epsilon}\ast f)(x)|\right)\le C\rho_{\mathcal A}(f). \end{align*}$$

For the optimality when $\phi $ is positive, radial, and decreasing, we show the reverse pointwise control $Mf\lesssim \sup _{\epsilon>0}|\phi _\epsilon \ast f|$ on nonnegative locally integrable f. Pick $r_0>0$ such that $\phi (r_0)>0$ (such an $r_0$ exists since $\phi $ is positive). Then, for any $r>0$ and any y with $|x-y|<r r_0$ , the monotonicity of $\phi $ gives $\phi (|x-y|/r)\ge \phi (r_0)$ . Hence,

$$ \begin{align*} (\phi_r\ast f)(x) &= r^{-n}\int_{\mathbb{R}^n}\phi\!\left(\frac{|x-y|}{r}\right)f(y)\,dy \ge r^{-n}\,\phi(r_0)\int_{|x-y|<r r_0}f(y)\,dy\\ &= \phi(r_0)\,r_0^{\,n}\,|B(0,1)|\cdot \frac{1}{|B(x,r r_0)|}\int_{B(x,r r_0)}f(y)\,dy. \end{align*} $$

Taking the supremum over $r>0$ gives $\sup _{\epsilon>0}|(\phi _\epsilon \ast f)(x)|\ge c\,(Mf)(x)$ for a constant $c=\phi (r_0)\,r_0^{\,n}\,|B(0,1)|>0$ . Combined with the upper bound, we have $\sup _{\epsilon>0}|(\phi _\epsilon \ast f)|\approx Mf$ pointwise (on nonnegative f), so the optimality of $L_{\mathcal A}$ follows from Theorem 4.2.

Corollary 4.4 Let $\phi $ , A, and $\mathcal A$ be as in Theorem 4.3 and assume in addition that $\int _{\mathbb {R}^n}\phi (y)\,dy=1$ . Let $E_A(\mathbb {R}^n)$ denote the closure in $L_A(\mathbb {R}^n)$ of the bounded functions of compact support (the standard “ $E_A$ ” subspace (see [Reference Rao and Ren12, Chapter 3, Section 4])). Then,

$$\begin{align*}\lim_{\epsilon \to 0^+} \rho_{A}(\phi_{\epsilon}\ast f -f)=0 \qquad \text{for every } f\in E_A(\mathbb{R}^n). \end{align*}$$

Proof The argument is the classical one (cf. [Reference Stein13, Theorem 2, p. 62]), adapted to the Luxemburg–Orlicz norm. By the density definition of $E_A(\mathbb {R}^n)$ and a $3\varepsilon $ -argument, it suffices to prove the conclusion for f bounded and of compact support.

Fix such an f, with $\|f\|_\infty \le M$ and $\mathrm {supp}\,f\subset B(0,R)$ . For $\epsilon \in (0,1]$ , $\phi _\epsilon \ast f$ is uniformly bounded by $M\|\phi \|_{L^1}$ and supported in $B(0,R+1)$ since $\mathrm {supp}\,\phi _\epsilon \subset \epsilon \,\mathrm {supp}\,\phi $ and one may approximate first by truncating $\phi $ to a compact set (with the radial-majorant control absorbing the error via Theorem 4.3). Hence, both f and $\phi _\epsilon \ast f$ lie in a common $L^\infty $ -ball supported in a fixed compact set, and pointwise convergence $\phi _\epsilon \ast f\to f$ a.e. is standard.

The Luxemburg–Orlicz norm on bounded sets of bounded compactly supported functions is dominated by a constant multiple of $\|\cdot \|_{L^\infty }\cdot \rho _A(\chi _{B(0,R+1)})$ , which in turn is finite by axiom (A5). Since pointwise convergence on a fixed bounded support of uniformly bounded functions implies $\rho _A$ -convergence (by the analog of bounded convergence for r.i. norms (see [Reference Bennett and Sharpley1, Chapter 1, Section 3])), we conclude $\rho _A(\phi _\epsilon \ast f - f)\to 0$ .

4.3 Calderón–Zygmund operators

A function K on $\mathbb {R}^n\setminus \{0\}$ , locally integrable away from the origin, is said to be a CZ kernel provided it satisfies the following four conditions:

1. (i) There exists a constant $C_1>0$ , independent of $\epsilon $ and N with $0< \epsilon <N$ , such that

   $$\begin{align*}\left|\int_{\epsilon<|x|<N}K(x)\, dx\right|\le C_1; \end{align*}$$
   moreover, for each $N>0$ , the limit $\lim _{\epsilon \to 0^+}\int _{\epsilon <|x|<N}K(x)\, dx$ exists.

2. (ii) There exists a constant $C_2>0$ , independent of $R>0$ , for which

   $$\begin{align*}\int_{|x|<R}|x||K(x)|\, dx\le C_2 R. \end{align*}$$

3. (iii) There exists a constant $C_3>0$ , independent of $y\in \mathbb {R}^n\setminus \{0\}$ , with

   $$\begin{align*}\int_{|x|>2|y|}|K(x-y)-K(x)|\, dx\le C_3. \end{align*}$$

The CZ singular integral operator, $T_K$ , associated with the kernel is

$$\begin{align*}(T_Kf)(x)=\lim_{\epsilon \to 0^+}\int_{|x-y|>\epsilon}K(x-y)f(y)\, dy, \end{align*}$$

the limit existing a.e. for all $f\in \mathfrak M(\mathbb {R}^n)$ for which

$$\begin{align*}\int_{\mathbb{R}^n}\frac{|f(y)|}{1+|y|^n}\, dy<\infty. \end{align*}$$

A theorem of CZ asserts that for each p, $1<p<\infty $ , there exists $C_p>0$ , independent of $f\in L_p(\mathbb {R}^n)$ , with

$$\begin{align*}\rho_p(T_Kf)\le C_p\rho_p(f) \end{align*}$$

(see [Reference Torchinsky14, pp. 286–290].

If, further, K satisfies

1. (iv) $|K(x_1-y)-K(x_2-y)|\le C_4\,\dfrac {|x_1-x_2|}{|x_3-y|^{n+1}}$ when $x_1, x_2, x_3, y\in \mathbb {R}^n$ are such that $|x_1-x_2|, |x_2-x_3|, |x_3-x_1|\le R/2$ , and $|x_i-y|\ge R$ , $i=1, 2, 3$ , where $C_4$ is independent of the points involved and of $R>0$ ,

then $T_K$ is of weak type $(1,1)$ , namely,

$$\begin{align*}\lambda\,|\{x\in \mathbb{R}^n: |(T_Kf)(x)|> \lambda\}|\le C_1'\,\|f\|_{L^1(\mathbb{R}^n)} \end{align*}$$

for some $C_1'>0$ independent of $\lambda>0$ and $f\in L^1(\mathbb {R}^n)$ (see [Reference Torchinsky14, p. 290]).

Remark 4.5 The operator, $T^{\prime }_K$ , associated with $T_K$ is given by $T^{\prime }_K=T_{K'}$ with $K'(x)=K(-x)$ . The kernel $K'$ satisfies (i)–(iv) whenever K does.

The next theorem provides the Orlicz analog of the CZ theorem.

Theorem 4.6 Let A and $\mathcal A$ be as in Theorem 1.1. Then,

$$\begin{align*}T_K: L_{\mathcal A}(\mathbb{R}^n)\to L_A(\mathbb{R}^n) \end{align*}$$

provided

(4.1) $$ \begin{align} t\int_0^t \widetilde{\mathcal A}(s)\,\frac{ds}{s^2}\le \widetilde{A}(Ct), \qquad t>0, \end{align} $$

for some $C>0$ ; here,

$$\begin{align*}\widetilde{\mathcal A}(t)=\int_0^t b^{-1}(s)\, ds, \qquad b(t)=\int_0^t a(s)\,\frac{ds}{s},\ a=A'. \end{align*}$$

Proof It is shown in [Reference Torchinsky14, Theorems 5.4 and 5.5] that both $T_K$ and its conjugate $T^{\prime }_K$ are of weak type $(1,1)$ and map $L^2(\mathbb {R}^n)$ into itself. According to the result in the Appendix of [Reference Calderón4], this implies

$$\begin{align*}\int_0^t(T_Kf)^*\le C \int_0^t (P+Q)f^*, \end{align*}$$

with $C>0$ independent of $f\in L_{\mathcal A}(\mathbb {R}^n)$ and $t>0$ . Here,

$$\begin{align*}(Qg)(t)=\int_t^{\infty} g(s)\,\frac{ds}{s}, \qquad g\in \mathfrak M_+(\mathbb{R}_+),\ t>0, \end{align*}$$

is the adjoint of P (with respect to the $L^2(\mathbb {R}_+;ds)$ pairing). Hence, by the HLP principle stated in Section 2,

$$ \begin{align*} \rho_A(T_Kf)=\rho_A((T_Kf)^*)&\le C\rho_A((P+Q)f^*)\\ &\le C\big[\rho_A(Pf^*)+\rho_A(Qf^*)\big]. \end{align*} $$

By Theorem 1.1,

$$\begin{align*}\rho_A(Pf^*)\le C'\rho_{\mathcal A}(f^*)=C'\rho_{\mathcal A}(f). \end{align*}$$

On the other hand, by Köthe duality (Section 2),

$$\begin{align*}\rho_A(Qf)\le C"\rho_{\mathcal A}(f) \quad\Longleftrightarrow\quad \rho_{\widetilde{\mathcal A}}(Pg)\le C"\rho_{\widetilde A}(g), \end{align*}$$

and the condition (4.1) guarantees the right-hand inequality via Theorem 1.1 applied to $\widetilde {\mathcal A}$ in place of $\mathcal A$ . Altogether,

$$\begin{align*}\rho_A(T_Kf)\le C[C'+C"]\rho_{\mathcal A}(f).\\[-35pt] \end{align*}$$

Remark 4.7 The condition (4.1) is the natural Boyd-type symmetric counterpart of (1.2) for the complementary function, and it is sharp in the sense that it is equivalent to the boundedness of $Q:L_{\mathcal A}\to L_A$ . We do not address here the question of whether $L_{\mathcal A}$ is optimal among Orlicz domains for $T_K$ itself (as opposed to $P+Q$ ): unlike M, a CZ operator $T_K$ does not in general dominate P pointwise on rearrangements. The optimal Orlicz domain question for $T_K$ has been studied by Cianchi [Reference Cianchi6] under reflexivity assumptions.

5 An application to the interpolation theory

Given Banach spaces $X_1$ and $X_2$ embedded in a common Hausdorff topological vector space $\mathcal H$ , the $\mathcal {K}$ -method of interpolation provides a concrete way to construct Banach spaces X lying between them, in the sense that any linear operator T with $T:X_i\to X_i$ , $i=1,2$ , satisfies $T:X\to X$ .

The key element of the method is the Peetre $\mathcal {K}$ -functional, defined at $x\in X_1+X_2$ and $t\in \mathbb {R}_+$ by

$$\begin{align*}\mathcal{K}(t,x;X_1,X_2):=\inf_{x=x_1+x_2}\left[\|x_1\|_{X_1}+t\|x_2\|_{X_2}\right]. \end{align*}$$

For our purposes, each interpolation space X corresponds to an r.i. norm $\rho $ on $\mathfrak M_+(\mathbb {R}_+)$ with $\rho (1/(1+t))<\infty $ ; more specifically, the norm of X is

$$\begin{align*}\|x\|_X:=\rho\!\left(\frac{ \mathcal{K}(t,x;X_1,X_2)}{t}\right), \quad x\in X_1+X_2. \end{align*}$$

The following is a special case of [Reference Kerman, Milman and Sinnamon9, Theorem 7.2] for $X_1$ and $X_2$ r.i. spaces and $\rho =\rho _{{}_A}$ an Orlicz norm. It elaborates, in a particular instance, the deep duality theorem of Brudnyi and Krugljak [Reference Brudnyi and Krugljak3].

Theorem 5.1 Let $(\Omega ,\mu )$ be a $\sigma $ -finite measure space and suppose $\rho _1$ and $\rho _2$ are r.i. norms on $\mathfrak M_+(\Omega )$ . Assume further that $L_{\rho _1'}(\Omega )\cap L_{\rho _2'}(\Omega )$ is dense in $L_{\rho _2'}(\Omega )$ and that

$$\begin{align*}\rho_2'(\chi_{E_k})\downarrow 0 \quad\text{as}\quad E_k\downarrow\emptyset,\ E_k \subset \Omega. \end{align*}$$

Consider a Young function $A(t)=\int _0^t a(s)\,ds$ , $t\in \mathbb {R}_+$ , satisfying $\int _{\mathbb {R}_+}A(k/(1+t))\,dt<\infty $ for some $k>0$ . Then, the functional

$$\begin{align*}\rho(f):=\rho_{{}_A}\!\left(\frac{\mathcal{K}(t,f; L_{\rho_1}(\Omega), L_{\rho_2}(\Omega))}{t}\right), \quad f\in L_{\rho_1}(\Omega)+L_{\rho_2}(\Omega) \end{align*}$$

is an r.i. norm on $\mathfrak M_+(\Omega )$ , and the r.i. space $L_\rho (\Omega )$ is an interpolation space between $L_{\rho _1}(\Omega )$ and $L_{\rho _2}(\Omega )$ .

Moreover, if in addition $\int _{\mathbb {R}_+}\widetilde A(k/(1+t))\,dt<\infty $ for some $k>0$ , and if $\mathcal A$ is the Young function defined in (3.2) and $\widetilde {\mathcal A}$ is its complementary function, then

$$\begin{align*}\rho'(g)\approx\rho_{\widetilde{\mathcal A}}\!\left(\frac{d}{dt}\mathcal{K}(t,g; L_{\rho_2'}(\Omega), L_{\rho_1'}(\Omega))\right),\quad g\in L_{\rho_2'}(\Omega)+L_{\rho_1'}(\Omega). \end{align*}$$

Now $\frac {d}{dt}\mathcal {K}(t,x;X_1,X_2)$ can be computed only when the $\mathcal {K}$ -functional is known exactly. More often, the latter is only known up to constant multiples. The motivation behind Theorem 1.3 is the following consequence of Theorem 5.1. A version of this result involving further assumptions on the Young function A is given in [Reference Kerman, Milman and Sinnamon9, Theorem 8.2].

Theorem 5.2 Let $\Omega $ , $\rho _1$ , $\rho _2$ , A, $\rho $ , and $\mathcal A$ be as in Theorem 5.1, with $a(t)$ absolutely continuous. Define the increasing function C by

$$\begin{align*}C(t):=\int_0^t c(s)\,ds, \end{align*}$$

where

$$\begin{align*}c(t):=\widetilde{\mathcal A}'(t) - \frac{\widetilde{\mathcal A}(t)}{t}=\frac{1}{t} \int_0^t s\,\widetilde{\mathcal A}"(s)\,ds, \quad t\in \mathbb{R}_+. \end{align*}$$

Then, provided $\int _{\mathbb {R}_+}C(k/(1+t))\,dt<\infty $ for some $k>0$ , one has $\rho _{\widetilde {\mathcal A}}\approx \rho _{\Gamma _C}$ , and so

$$\begin{align*}\rho'(g)\approx \inf\left\{\lambda>0: \int_{\mathbb{R}_+} C\!\left(\frac{\mathcal{K}(t,g; L_{\rho_2'}(\Omega), L_{\rho_1'}(\Omega))}{\lambda t}\right)dt\le 1\right\}, \end{align*}$$

for $g\in L_{\rho _2'}(\Omega )+ L_{\rho _1'}(\Omega )$ . In particular, $g\in L_{\rho _2'}(\Omega )+ L_{\rho _1'}(\Omega )$ belongs to $L_{\rho '}(\Omega )$ if and only if there exists $\lambda _g\in \mathbb {R}_+$ , such that

$$\begin{align*}\int_{\mathbb{R}_+} C\!\left(\frac{\mathcal{K}(t,g; L_{\rho_2'}(\Omega), L_{\rho_1'}(\Omega))}{\lambda_g t}\right)dt<\infty. \end{align*}$$