1 Introduction

Let \(L_{\mathrm{loc}}\) be the space of all locally integrable functions f on \({\mathbf {R}}^{n}\) with lebesgue measure. The Riesz potential operator \(R^{s}\), \(0< s< n\), \(n\geq1\) is defined by

$$R^{s} f (x)= \int_{{\mathbf {R}}^{n}} f(y) |x-y|^{s-n}\,dy, $$

where \(f\in L_{\mathrm{loc}}\).

It is well known that in the supercritical case \(s>n/p\),

$$ R^{s} : L^{p} \mapsto{\mathcal {C}}^{s-n/p},\quad s>n/p, $$
(1.1)

where \({\mathcal {C}}^{\gamma}\); \(\gamma>0\) is Hölder-Zygmund space [1], but in the critical case \(s=n/p\) the function \(R^{s}f\) may not be even continuous. We prove the optimal one is obtained if in above \(L^{p}\) is replaced by Marcinkiewicz space \(L^{p,\infty}\). In this paper we prove similar optimal results, when \(L^{p,\infty}\) is replaced by more general rearrangement invariant spaces E. More precisely, we consider quasi-norm rearrangment invariant space E, consisting of functions \(f\in L^{1}+L^{\infty}\), such that the quasi-norm \(\|f\|_{E}=\rho (f^{\ast})<\infty\), where \(\rho_{E}\) a monotone quasi-norm, defined on \(M^{+}\) with values in \([0,\infty]\). Here \(M^{+}\) is the cone of all locally integrable functions \(g\ge0\) on \((0,\infty)\) with Lebesgue measure.

Monotonicity means that \(g_{1}\leq g_{2}\) implies \(\rho_{E}(g_{1})\leq\rho_{E}(g_{2})\). We suppose that \(L^{1}\cap L^{\infty}\hookrightarrow E\hookrightarrow L^{1}+L^{\infty}\), which means continuous embeddings. Here \(f^{\ast}\) is the decreasing rearrangement of f, given by

$$f^{\ast}(t)=\inf\bigl\{ \lambda>0:\mu_{f}(\lambda)\leq t\bigr\} ,\quad t>0, $$

and \(\mu_{f}\) is the distribution function of f, defined by

$$\mu_{f}(\lambda)=\bigl\vert \bigl\{ x\in{\mathbf {R}}^{n}: \bigl\vert f(x)\bigr\vert >\lambda\bigr\} \bigr\vert _{n}, $$

\(\vert \cdot \vert _{n}\) denoting the Lebesgue n-measure.

Finally,

$$f^{\ast\ast}(t):=\frac{1}{t} \int_{0}^{t} f^{\ast}(s)\,ds. $$

Let \(\alpha_{E}\), \(\beta_{E}\) be the Boyd indices of E (see [2–4]). For example, if \(E=L^{p}\), then \(\alpha_{E}=\beta_{E}=1/p\) and the condition \(1>s/n\geq1/p\) means \(p>1\), \(\beta_{E} <1\). For these reasons we suppose that for the general E, \(0<\alpha_{E}=\beta_{E}\leq1\), and the case \(s/n>\alpha_{E}\) is called supercritical, while the case \(s/n=\alpha_{E}\) is called critical. In the supercritical case the function \(R^{s}f\); \(f\in E\) is always continuous [5], while the spaces in the critical case \(\alpha_{E}=s/n\), can be divided into two subclasses: in the first subclass the functions \(R^{s}f\) may not be continuous; then the target space is rearrangement invariant, while these functions in the second subclass are continuous and the target space is the generalized Hölder-Zygmund space \({\mathcal {C}}H\) [6, 7]. The separating space for these two subclasses is given by the Lorentz space \(L^{n/s,1}\). The continuity of fractional maximal operator and Bessel potential operator is discussed in [8] and [9]. Gogatishvili and Ovchinnikov in [10] discussed the optimal Sobolev’s embeddings. The problem of the optimal target rearrangement invariant spaces for potential type operators is considered in [11] by using \(L_{p}\)-capacities. The problem of mapping properties of the Riesz potential in optimal couples of rearrangement invariant spaces is treated in [12–15]. The characterization of the continuous embedding of the generalized Bessel potential spaces into Hölder-Zygmund spaces \(\mathcal{CH}\), when H is a weighted Lebesgue space, is given in [7]. For further literature and reviews, we refer the reader to [16–20].

The main goal of this paper is to prove continuity of the Riesz potential operator \(R^{S}:E\mapsto{\mathcal {C}}H\) in an optimal couple E, \({\mathcal {C}}H\), for the supercritical case on unbounded domain. The same problem was considered in [5] for bounded domain. The critical and subcritical case for the continuity of Riesz potential operator was considered in [12] and [14].

The plane of this paper is as follows. In Section 2 we provide some basic definitions and known results. In Section 3 we characterize the continuity of the Riesz potential operator \(R^{S}:E\mapsto{\mathcal {C}}H\). The optimal quasi-norms are constructed in Section 4.

2 Preliminaries

We use the notations \(a_{1}\lesssim a_{2}\) or \(a_{2}\gtrsim a_{1}\) for nonnegative functions or functionals to mean that the quotient \(a_{1}/a_{2}\) is bounded; also, \(a_{1}\approx a_{2}\) means that \(a_{1}\lesssim a_{2}\) and \(a_{1}\gtrsim a_{2}\). We say that \(a_{1}\) is equivalent to \(a_{2}\) if \(a_{1}\approx a_{2}\).

There is an equivalent quasi-norm \(\rho_{p}\approx\rho_{E}\) that satisfies the triangle inequality \(\rho_{p}^{p}(g_{1}+g_{2})\leq\rho_{p}^{p}(g_{1})+\rho _{p}^{p}(g_{2})\) for some \(p\in(0,1]\) that depends only on the space E (see [21]). We say that the quasi-norm \(\rho_{E}\) satisfies Minkowski’s inequality if for the equivalent quasi-norm \(\rho_{p}\),

$$ \rho_{p}^{p} \Bigl(\sum g_{j} \Bigr) \lesssim\sum\rho_{p}^{p}(g_{j}),\quad g_{j}\in M^{+}. $$

Usually we apply this inequality to functions \(g\in M^{+}\) with some kind of monotonicity.

Recall the definition of the lower and upper Boyd indices \(\alpha_{E}\) and \(\beta_{E}\). Let \(g_{u}(t)=g(t/u)\) where \(g\in M^{+}\), and let

$$h_{E}(u)=\sup \biggl\{ \frac{\rho_{E}(g^{\ast}_{u})}{\rho_{E}(g^{\ast})}: g\in M^{+} \biggr\} , \quad u>0, $$

be the dilation function generated by \(\rho_{E}\). Suppose that it is finite. Then

$$\alpha_{E}:=\sup_{0< t< 1} \frac{\log h_{E}(t)}{\log t} \quad\mbox{and}\quad \beta _{E}:=\inf_{1< t< \infty} \frac{\log h_{E}(t)}{\log t}. $$

The function \(h_{E}\) is sub-multiplicative, increasing, \(h_{E}(1)=1\), \(h_{E}(u) h_{E}(1/u)\geq1\) hence \(0\leq\alpha _{E}\leq\beta_{E}\). We suppose that \(0<\alpha_{E}=\beta_{E}\leq1\) and \(g^{\ast \ast}(\infty)=0\).

If \(\beta_{E}<1\) we have by using Minkowski’s inequality that \(\rho_{E}(f^{\ast})\approx\rho_{E}(f^{\ast\ast})\).

Recall that \(w\in M^{+}\) is slowly varying function, if for every \(\epsilon>0\), the function \(t^{\epsilon}w(t)\) is equivalent to increasing function and \(t^{-\epsilon}w(t)\) is equivalent to a decreasing function.

In order to introduce the Hölder-Zygmund class of spaces, we denote the modulus of continuity of order k by

$$\omega^{k}(t,f)=\sup_{|h|\leq t} \sup_{x\in{\mathbf {R}}^{n}}\bigl|\Delta^{k}_{h} f(x)\bigr|, $$

where \(\Delta^{k}_{h} f\) are the usual iterated differences of f. When \(k=1\) we simply write \(\omega(t,f)\). Let H be a quasi-normed space of locally integrable functions on the interval \((0,1)\) with the Lebesgue measure, continuously embedded in \(L^{\infty}(0,1)\) and \(\|g\|_{H}=\rho_{H}(|g|)\), where \(\rho_{H}\) is a monotone quasi-norm on \(M^{+}\) which satisfies Minkowski’s inequality. The dilation function \(h_{H}\), generated by \(\rho_{H}\), is defined as follows:

$$h_{H}(u)=\sup \biggl\{ \frac{\rho_{H}( {\chi_{(0,1)}\tilde{g}}_{u})}{\rho_{H}(\chi _{(0,1)}g)}: g\in M_{a} \biggr\} , $$

where \(( {\tilde{g}}_{u})(t)=g(ut)\) if \(ut<1\), \(( {\tilde{g}}_{u})(t)=g(1)\) if \(ut\geq1\), and

$$M_{a}:=\bigl\{ g \in M^{+}:t^{-a/n}g(t)\mbox{ is decreasing } g \mbox{ is increasing and } g(+0)=0 \bigr\} . $$

The choice of the space \(M_{a}\) is motivated by the fact that \(\omega ^{n}(t^{1/n},f)\), is equivalent to a function \(g\in M_{a}\).

The function \(h_{H}(u)\) is sub-multiplicative and \(u^{-1} h_{H}(u)\) is decreasing and

$$h_{H}(1)=1,\qquad h_{H}(u) h_{H}(1/u)\geq1. $$

Suppose that \(h_{H}\) is finite. Then the Boyd indices of H are well defined,

$$\alpha_{H}=\sup_{0< t< 1} \frac{\log h_{H}(t)}{\log t} \quad \mbox{and} \quad\beta_{H}=\inf_{1< t< \infty} \frac{\log h_{H}(t)}{\log t}, $$

and they satisfy \(\alpha_{H}\leq\beta_{H}\leq1\). In the following, we suppose that \(0\leq\alpha_{H}=\beta_{H}< 1\).

For example, let \(H=L^{q}_{\ast}(b(t) t^{-\gamma/n})\). Here \(0\leq\gamma< a/n\) and b is a slowly varying function, and \(L^{q}_{\ast}(w)\), or simply \(L^{q}_{\ast}\) if \(w=1\), is the weighted Lebesgue space with a quasi-norm \(\|g\|_{L^{q}_{\ast}(w)}=\rho_{w,q}(|g|)\). It turns out that \(\alpha_{H}=\beta_{H}=\gamma/n\).

Definition 2.1

Let \(j=0,1,\dots \) and let \({C}^{j}\) stand for the space of all functions f, defined on \({\mathbf {R}}^{n} \), that have bounded and uniformly continuous derivatives up to the order j, normed by \(\|f\|_{{C}^{j}}=\sup\sum_{l=0} ^{j} |P^{l} f(x)| \), where \(P^{l} f(x) =\sum_{|\nu|=l} D^{\nu}f(x) \).

  • If \(j/n<\alpha_{H}<(j+1)/n\) for \(j\geq1\) or \(0\leq\alpha_{H}<1/n\) for \(j=0\), then \({\mathcal {C}} H\) is formed by all functions f in \({C}^{j}\) having a finite quasi-norm

    $$ \|f\|_{{\mathcal {C}} H}= \|f\|_{{ C}^{j}}+\rho_{H} \bigl(t^{j/n}\omega \bigl(t^{1/n},P^{j} f\bigr)\bigr). $$

  • If \(\alpha_{H}=(j+1)/n\), then \({\mathcal {C}} H\) consists of all functions f in \({C}^{j}\) having a finite quasi-norm

    $$ \|f\|_{{\mathcal {C}} H}= \|f\|_{{ C}^{j}}+\rho_{H} \bigl(t^{j/n}\omega ^{2}\bigl(t^{1/n},P^{j} f\bigr)\bigr). $$

In particular, if \(H=L^{\infty}(t^{-\gamma/n})\), \(\gamma>0\), then \({\mathcal {C}} H\) coincides with the usual Hölder-Zygmund space \({\mathcal {C}}^{\gamma}\) (see [1]). Also, if \(H=L^{\infty}\), then \({\mathcal {C}} H= C^{0}\). We need the following result about the equivalent quasi-norm in the generalized Hölder-Zygmund spaces.

Theorem 2.2

(equivalence) ([6])

Let \(\rho_{H}\) be a monotone quasi-norm, satisfying Minkowski’s inequality and let \(0\leq\alpha_{H}=\beta_{H}< m/n\). If \(\rho _{H}(t^{\alpha})<\infty\) for \(\alpha>\alpha_{H}\), then, for all such m,

$$ \|f\|_{{\mathcal {C}} H}\approx\|f\|_{{C}^{0}} + \rho_{H} \bigl(\omega ^{m}\bigl(t^{1/n},f\bigr) \bigr) . $$
(2.1)

Let N be the class of all admissible couples, it will be convenient to use the following definitions.

Definition 2.3

(admissible couple)

We say that the couple \((\rho_{E},\rho_{H})\in N\) is admissible for the Riesz potential if

$$ \bigl\| R^{s} f\bigr\| _{{\mathcal {C}} H}\lesssim\rho_{E}\bigl( f^{\ast}\bigr),\quad f\in E. $$
(2.2)

Then the couple E, H is called admissible. Moreover, \(\rho_{E}\) (E) is called domain quasi-norm (domain space), and \(\rho_{H}\) (H) is called the target quasi-norm (target space).

To prove our result we introduce the classes of the domain and target quasi-norms, where the optimality is investigated.

Let \(N_{d}\) consist of all domain quasi-norms \(\rho_{E}\) that are monotone, satisfy Minkowski’s inequality, \(0<\alpha_{E}=\beta_{E}<1\), and the condition

$$ \int_{0}^{\infty} t^{s/n-1}g(t)\,dt\lesssim \rho_{E}(g),\quad g\downarrow, $$
(2.3)

\(\int_{0}^{\infty} g^{\ast}(u)\,du\lesssim\rho_{E}(g^{\ast})\) and \(\rho_{E}(\chi _{(0,1)}t^{-\alpha})<\infty\) if \(\alpha<\alpha_{E}\).

Let \(N_{t}\) consist of all target quasi-norms \(\rho_{H}\) that are monotone, satisfy Minkowski’s inequality, \(0\leq\alpha_{H}=\beta_{H}< 1\), \(\rho _{H}(t^{\alpha})<\infty\) if \(\alpha>\alpha_{H}\) and \(\sup\chi _{(0,1)}g(t)\lesssim\rho_{G}(\chi_{(0,1)}g)\), \(g \in M_{n}\).

Finally

$$N:=\bigl\{ (\rho_{E},\rho_{H})\in N_{d}\times N_{t}: \rho_{H}\bigl(\chi _{(0,1)}t^{s/n}g(t) \bigr)\lesssim\rho_{E}(g), g\downarrow\bigr\} . $$

Definition 2.4

(optimal target quasi-norm)

Given the domain quasi-norm \(\rho_{E}\), the optimal target quasi-norm, denoted by \(\rho_{H(E)}\), is the strongest target quasi-norm, such that \((\rho_{E},\rho_{H(E)})\in N\) and

$$ \rho_{H}(\chi_{(0,1)}g)\lesssim \rho_{H(E)}(\chi_{(0,1)}g), \quad g\in M_{n}, $$
(2.4)

for any target quasi-norm \(\rho_{H}\) such that the couple \((\rho_{E}, \rho _{H})\in N\) is admissible. Since \({\mathcal {C}}H(E)\hookrightarrow {\mathcal {C}}H\), we call \({\mathcal {C}}H(E)\) the optimal Hölder-Zygmund space. For shortness, the space \(H(E)\) is also called an optimal target space.

Definition 2.5

(optimal domain quasi-norm)

Given the target quasi-norm \(\rho_{H}\in N_{t}\), the optimal domain quasi-norm, denoted by \(\rho_{E(H)}\), is the weakest domain quasi-norm, such that \((\rho_{E(H)},\rho_{H})\in N\) and

$$ \rho_{E(H)}\bigl(f^{\ast}\bigr)\lesssim\rho_{E} \bigl(f^{\ast}\bigr),\quad f\in E, $$

for any domain quasi-norm \(\rho_{E}\in N_{d}\) such that the couple \((\rho _{E}, \rho_{H})\in N\) is admissible. The space \(E(H)\) is called an optimal domain space.

Definition 2.6

(optimal couple)

The admissible couple \((\rho_{E},\rho_{H})\in N\) is said to be optimal if both \(\rho_{E}\) and \(\rho_{H}\) are optimal. Then the couple E, H is called optimal.

3 Admissible couples

Here we give a characterization of all admissible couples \((\rho_{E}, \rho _{H})\in N\). By using the following Hardy-Littlewood inequality [2], p.44, we get the well-known mapping property:

$$R^{s} : \Lambda^{1}\bigl(t^{s/n}\bigr)\mapsto L^{\infty}. $$

Now from (2.3) it follows that

$$ R^{s} : E \rightarrow L^{\infty}. $$
(3.1)

We have the following basic estimate.

Theorem 3.1

If \(f\in E\), then

$$ \chi_{(0,1)}\omega^{m}\bigl(t^{1/n}, R^{s} f\bigr)\lesssim S\bigl( f^{\ast}\bigr) (t), \quad s< m, $$
(3.2)

where

$$ Sg(t):= \int_{0}^{t} u^{s/n-1} g(u)\,du, \quad g\in M^{+}. $$
(3.3)

Proof

The proof of this result follows from Theorem 3.1 in [5]. □

Now we discuss the mapping property \(R^{s}:E\mapsto C^{0}\).

Theorem 3.2

A necessary and sufficient condition for the mapping

$$R^{s} : E\mapsto C^{0} $$

is the following:

$$ \int_{0}^{\infty} t^{s/n-1}g(t)\,dt\lesssim \rho_{E}(g),\quad g\downarrow. $$
(3.4)

Proof

We already know that

$$ R^{s} : E \rightarrow L^{\infty}. $$
(3.5)

To prove that \(R^{s}(E)\subset C^{0}\), it remains to show that \(R^{s} f \) is a uniformly continuous function. It is enough to show that

$$\lim_{t\rightarrow0} \omega\bigl(t^{\frac{1}{n}},R^{s}f \bigr)=0 \quad \mbox{if } f\in E. $$

By using Marchaud’s inequality,

$$\omega\bigl(t^{\frac{1}{n}},R^{s}f\bigr)\lesssim t^{\frac{1}{n}} \int_{t}^{\infty} u^{\frac{-1}{n}} \omega^{m}\bigl(u^{\frac{1}{n}},R^{s}f\bigr) \frac{du}{u}, $$

L’Hôpital’s rule, and (3.2), we get

$$\begin{aligned} \lim_{t\rightarrow0} \omega\bigl(t^{\frac{1}{n}},R^{s}f \bigr) \lesssim& \lim_{t\rightarrow0} \frac{ t^{\frac{-1}{n}}\omega^{m}(t^{\frac {1}{n}},R^{s}f)}{t^{\frac{-1}{n}}} \\ =&\lim_{t\rightarrow0}\omega^{m}\bigl(t^{\frac {1}{n}},R^{s}f \bigr) \\ \lesssim& \lim_{t\rightarrow0}S f^{\ast}(t)=0. \end{aligned}$$

Hence

$$R^{s}f \in C^{0}. $$

It remains to prove that if \(R^{s}:E \rightarrow C^{0}\), then (3.4) is true for \(\alpha_{E}\leq s/n\). To this end we choose a test function h as follows. Let \(g\in D_{n-s}\), \(\rho_{E}(g)<\infty\) and

$$ h(x)= \int_{0}^{\infty} g(u)\varphi\bigl(x u^{-1/n} \bigr) \frac{du}{u}, $$
(3.6)

where \(\varphi\geq0\) is a smooth function with compact support in \((-c^{-1/n},c^{-1/n})\) such that if \(\psi=R^{s} \varphi\), then \(\psi(0)>0\). To see that this is possible, we calculate \(\psi(0)\). Since

$$\psi(x)= \int_{R^{n}}\varphi(y)|x-y|^{s-n}\,dy, $$

we have for appropriate \(d>0\),

$$\psi(0)\geq \int _{|y|\leq d}\varphi(y)|y|^{s-n}\,dy\gtrsim \int _{|y|\leq d}\varphi(y)\,dy>0. $$

Note also that, for large \(c>0\),

$$ \psi(x)\lesssim|x|^{s-n},\quad u>c. $$
(3.7)

Indeed

$$\psi(x)= \int _{|y|\leq d}\varphi(y)|x-y|^{s-n}\,dy\lesssim |x|^{s-n} \int _{|y|\leq d}\varphi(y)\,dy $$

since

$$|x-y|\geq|x|-|y|\geq|x|-d\geq|x|/2, \quad\mbox{if } c>2d. $$

We also have

$$R^{s}\bigl(\varphi\bigl(xu^{-1/n}\bigr)\bigr)=u^{s/n} \psi\bigl(xu^{-1/n}\bigr). $$

Hence

$$f(x):=R^{s} h(x)= \int_{0}^{\infty} u^{s/n}g(u)\psi \bigl(xu^{-1/n}\bigr)\frac{du}{u}. $$

We may take

$$h(x)\lesssim \int_{c|x|^{n}}^{\infty}g(u)\,du/u, $$

hence, for appropriate \(c>0\),

$$h^{\ast}(t)\lesssim \int_{t}^{\infty}g(u)\,du/u. $$

Applying Minkowski’s inequality and using \(\alpha_{E}>0\), we have

$$ \rho_{E}\bigl(h^{\ast}\bigr)\lesssim \rho_{E}(g). $$
(3.8)

Given that

$$\sup\bigl|R^{s}h(x)\bigr|\lesssim\|h\|_{E}, $$

we have in particular

$$\bigl|R^{s}h(0)\bigr|\lesssim\|h\|_{E}, $$

whence

$$R^{s} h(0)=\psi(0) \int_{0}^{\infty} u^{s/n-1}g(u)\,du\lesssim\|h\| _{E}\lesssim\rho_{E}(g). $$

Thus (3.4) is proved. □

In the following theorem, we characterize the admission couple. Note that this result cannot obtained directly from Theorem 3.4 [5], because here we consider an unbounded domain.

Theorem 3.3

The couple \((\rho_{E},\rho_{H})\in N\) is admissible if and only if

$$ \rho_{H} (\chi_{(0,1)}S g)\lesssim\rho_{E} (g), \quad g\downarrow. $$
(3.9)

Proof

Let (3.9) be true. By using (3.2) and (3.9), we get

$$\rho_{H} \bigl(\chi_{(0,1)}\omega^{m} \bigl(t^{\frac{1}{n}},R^{s}f\bigr) \bigr)\lesssim\rho_{H} \bigl(\chi_{(0,1)}S\bigl(f^{*}\bigr) \bigr)\lesssim \rho_{E} \bigl(f^{*} \bigr),\quad m>s. $$

Therefore

$$\begin{aligned} \bigl\| R^{s}f\bigr\| _{{\mathcal {C}} H} \approx& \bigl\| R^{s}f \bigr\| _{C^{0}}+\rho_{H} \bigl(\omega^{m} \bigl(t^{\frac{1}{n}},R^{s}f\bigr) \bigr) \\ \lesssim& \rho_{E}\bigl(f^{*}\bigr)+\bigl\| R^{s}f \bigr\| _{C^{0}} \\ \lesssim& \rho_{E}\bigl(f^{*}\bigr)+\rho_{E} \bigl(f^{*}\bigr) \\ \lesssim& \rho_{E}\bigl(f^{*}\bigr). \end{aligned}$$

Thus \(\rho_{E}\), \(\rho_{H}\) is an admissible couple.

For the converse, we have to prove that (2.2) implies (3.9). To this end we choose a test function in the form \(f(x)=R^{s}h(x)\), where h is given by (3.6). We have

$$f(x)=R^{s}h(x)= \int _{0}^{\infty}u^{s/n}g(u)\psi\bigl(x u^{-\frac {1}{n}}\bigr)\frac{du}{u}. $$

To estimate the modulus of continuity of f from below, we split f as follows:

$$f=f_{1t}+f_{2t}, $$

where

$$f_{1t}(x)= \int _{0}^{t}u^{\frac{s}{n}}g(u)\psi\bigl(x u^{-\frac {1}{n}} \bigr)\frac{du}{u},\qquad f_{2t}(x)= \int _{t}^{\infty}u^{\frac {s}{n}}g(u)\psi\bigl(x u^{-\frac{1}{n}} \bigr)\frac{du}{u}. $$

First we prove that, for some large \(C>0\),

$$\omega^{m}\bigl(Ct^{\frac{1}{n}},f_{1t}\bigr)\geq \frac{\psi(0)}{2} Sg(t). $$

To this aim consider

$$\Delta_{h}^{m}f_{1t}(x)= \int _{0}^{t}u^{\frac{s}{n}}g(u)\Delta _{h}^{m}\psi\bigl(x u^{-\frac{1}{n}} \bigr) \frac{du}{u}. $$

Also consider

$$\begin{aligned}[b] \Delta_{h}^{m}\psi\bigl(x u^{-\frac{1}{n}}\bigr)&=\sum _{k=0}^{m}(-1)^{m-k}\dbinom{m}{k} \psi\bigl((x+hk) u^{-\frac{1}{n}}\bigr)\\ &=(-1)^{m}\psi(0)+\sum_{k=1}^{m}(-1)^{m-k} \dbinom{m}{k}\psi \bigl(hku^{-\frac{1}{n}}\bigr) \quad\mbox{at } x=0. \end{aligned} $$

If \(|h|=Ct^{\frac{1}{n}}\), then for \(u< t\), \(k\geq1\), \(|h|ku^{-\frac{1}{n}} \geq Ck \geq C\), hence by (3.7) and for large \(C>0\),

$$\psi\bigl(h ku^{-\frac{1}{n}}\bigr)\lesssim C^{s-n}, \quad u< t, k\geq1. $$

Therefore,

$$\Delta_{h}^{m}f_{1t}(0)= \int _{0}^{t}u^{\frac {s}{n}}g(u) \Biggl[(-1)^{m}\psi(0)+\sum_{k=1}^{m}(-1)^{m-k} \psi \bigl(hku^{-\frac{1}{n}}\bigr)\Biggr]\frac{du}{u} $$

and, for large \(C>0\),

$$\begin{aligned} \bigl|\Delta_{h}^{m}f_{1t}(0)\bigr| =& \Biggl|(-1)^{m}\psi(0) \int _{0}^{t}u^{\frac{s}{n}}g(u) \frac {du}{u}+\sum_{k=1}^{m}(-1)^{m-k} \int _{0}^{t}u^{\frac{s}{n}}g(u) \psi \bigl(hku^{-\frac{1}{n}}\bigr)\frac{du}{u} \Biggr| \\ \geq& \psi(0) \int _{0}^{t}u^{\frac{s}{n}}g(u)\frac{du}{u}- c_{m} \int _{0}^{t}u^{\frac{s}{n}}g(u)\psi \bigl(hku^{-\frac{1}{n}}\bigr)\frac {du}{u} \\ \geq& \psi(0) \int _{0}^{t}u^{\frac{s}{n}}g(u)\frac{du}{u}- C^{s-n}c_{m} \int _{0}^{t}u^{\frac{s}{n}}g(u)\frac{du}{u} \\ =&\frac{\psi(0)}{2} \int _{0}^{t}u^{\frac{s}{n}}g(u) \frac {du}{u}. \end{aligned}$$

Hence

$$\omega^{m}\bigl(Ct^{\frac{1}{n}},f_{1t}\bigr)\geq \frac{\psi(0)}{2} Sg(t) $$

or

$$ \omega^{m}\bigl(t^{\frac{1}{n}},f_{1t}\bigr)\approx \omega^{m}\bigl(Ct^{\frac {1}{n}},f_{1t}\bigr)\geq \frac{\psi(0)}{2} Sg(t). $$
(3.10)

Further,

$$\omega^{m}\bigl(t^{\frac{1}{n}},f\bigr)\geq\omega^{m} \bigl(t^{\frac {1}{n}},f_{1t}\bigr)-\omega^{m} \bigl(t^{\frac{1}{n}},f_{2t}\bigr). $$

Now we estimate the modulus of continuity of the second function from above. To this aim, by using the formula [2], p.336, we get

$$\begin{aligned} \bigl|\Delta_{h}^{m}f_{2t}(x)\bigr| =& \biggl| \int _{-\infty}^{\infty}M_{m}(u)\sum _{|v|=m}\frac{m!}{v!}D^{v}f_{2t}(x+uh)h^{v}\,du\biggr| \\ \lesssim& \int _{-\infty}^{\infty}M_{m}(u)\sum _{|v|=m}\frac{m!}{v!}\bigl|D^{v}f_{2t}(x+uh)\bigr||h|^{|v|}\,du. \end{aligned}$$

Hence

$$\begin{aligned} \sup_{x}\bigl|\Delta_{h}^{m}f_{2t}(x) \bigr|\lesssim|h|^{m} \int _{-\infty}^{\infty}M_{m}(u) \sup\bigl|P^{m}f_{2t}(x+uh)\bigr|\,du \lesssim& |h|^{m} \bigl\| P^{m}f_{2t}\bigr\| _{L_{\infty}}. \end{aligned}$$

Therefore

$$ \sup_{x}\bigl|\Delta_{h}^{m}f_{2t}(x) \bigr|\lesssim |h|^{m}\bigl\| P^{m}f_{2t}\bigr\| _{L_{\infty}}. $$
(3.11)

To simplify (3.11), consider

$$\begin{aligned} &\bigl|P^{m}f_{2t}\bigr|=\biggl| \int _{t}^{\infty}u^{\frac{s}{n}}g(u)P^{m} \bigl(\psi\bigl(x u^{-\frac{1}{n}}\bigr) \bigr)\frac{du}{u}\biggr|, \\ &\sup_{x}\bigl|P^{m}f_{2t}\bigr|\lesssim \int _{t}^{\infty}u^{\frac{s}{n}}g(u)u^{-\frac{m}{n}} \bigl\| P^{m}\psi\bigr\| _{L_{\infty }} \frac{du}{u}, \\ &\bigl\| P^{m}f_{2t}\bigr\| _{L_{\infty}}\lesssim \int _{t}^{\infty}u^{\frac{s-m}{n}}g(u)\frac{du}{u}. \end{aligned}$$
(3.12)

So (3.11) becomes

$$ \omega^{m}\bigl(t^{\frac{1}{n}},f_{2t}\bigr) \lesssim t^{\frac{m}{n}} \int _{t}^{\infty}u^{\frac{s-m}{n}}g(u)\frac{du}{u} $$
(3.13)

whence for \(m>s\), we have

$$\omega^{m}\bigl(t^{\frac{1}{n}},f_{2t}\bigr) \lesssim \int_{t}^{\infty}t^{\frac {s}{n}}g(u)\frac{du}{u}. $$

Hence

$$\begin{aligned} &\chi_{(0,1)}Sg(t) \lesssim\chi_{(0,1)} \omega^{m}\bigl(t^{\frac {1}{n}},f\bigr)+\chi_{(0,1)} \int_{t}^{\infty}t^{\frac{s}{n}}g(u)\frac {du}{u}, \\ &\rho_{H} (\chi_{(0,1)}Sg )\lesssim\rho_{H}\bigl( \chi_{(0,1)}\omega ^{m}\bigl(t^{1/n},f\bigr)\bigr)+ \rho_{H} \biggl(\chi_{(0,1)} \int_{t}^{\infty}t^{\frac {s}{n}}g(u)\frac{du}{u} \biggr). \end{aligned}$$
(3.14)

Now since \((\rho_{E},\rho_{H})\in N\), we get

$$\rho_{H} (\chi_{(0,1)}Sg )\lesssim\rho_{E}(g). $$

 □

4 Optimal quasi-norms

Here we give a characterization of the optimal domain and optimal target quasi-norms.

4.1 Optimal domain quasi-norms

We can construct an optimal domain quasi-norm \(\rho_{E(H)}\) by Theorem 3.3 as follows.

Definition 4.1

(construction of an optimal domain quasi-norm)

For a given target quasi-norm \(\rho_{H}\in N_{t}\) we set

$$ \rho_{E(H)}(g):=\rho_{H} (\chi_{(0,1)}S g), \quad g\in M^{+}. $$
(4.1)

Note that

$$\alpha_{E(H)}=\beta_{E(H)}= s/n -\alpha_{H}. $$

Theorem 4.2

The couple \(\rho_{E(H)}\), \(\rho_{H}\) is admissible and the domain quasi-norm \(\rho_{E(H)}\) is optimal. Moreover, the target quasi-norm \(\rho_{H}\) is also optimal and

$$ \rho_{E(H)}(g)\approx\rho_{H} \bigl(\chi_{(0,1)}t^{s/n} g\bigr),\quad g\downarrow \textit{ if } \alpha_{H}>0. $$
(4.2)

Proof

The couple \(\rho_{E(H)}\), \(\rho_{H}\) is admissible since

$$\rho_{H}(\chi_{(0,1)}Sg)=\rho_{E(H)}(g). $$

Moreover, \(\rho_{E(H)}\) is optimal, since for any admissible couple \((\rho_{E_{1}},\rho_{H})\in N\) we have

$$\rho_{H}(\chi_{(0,1)}Sg) \lesssim\rho_{E_{1}}(g). $$

Therefore,

$$\rho_{E(H)}\bigl(f^{\ast}\bigr)={\rho_{H}}\bigl( \chi_{(0,1)}S\bigl(f^{\ast}\bigr)\bigr)\lesssim\rho _{E_{1}}\bigl(f^{\ast}\bigr), \quad f\in E. $$

To prove that \(\rho_{H}\) is also optimal, let \((\rho_{E(H)},\rho _{H_{1}})\in N\) be an arbitrary admissible couple. Then

$$\rho_{H_{1}}(\chi_{(0,1)}Sg)\lesssim\rho_{E(H)}(g). $$

We have to show that

$$ \rho_{H_{1}}(\chi_{(0,1)}g)\lesssim\rho_{H} ( \chi_{(0,1)}g), \quad g\in M_{n}. $$
(4.3)

Since \(g\in M_{n}\) is a quasi-concave, it is equivalent to a concave one, hence

$$g(t)\approx \int_{0}^{t}h_{1}(u)\,du,\quad h_{1}\downarrow. $$

Let

$$h(t)=t^{1-s/n}h_{1}(t). $$

Therefore

$$\rho_{H_{1}}(\chi_{(0,1)}g)\lesssim\rho_{H_{1}}(\chi _{(0,1)}Sh)\lesssim\rho_{E(H)}(h)\lesssim\rho_{H}(\chi _{(0,1)}Sh)\lesssim\rho_{H}(\chi_{(0,1)}g). $$

Thus (4.3) is proved.

To prove the equivalence (4.2), first we prove that

$$\rho_{E(H)}(g)\lesssim\rho_{H}\bigl(\chi_{(0,1)}t^{\frac{s}{n}}g \bigr),\quad g\downarrow \mbox{ if } \alpha_{H}>0. $$

To this aim we consider

$$\begin{aligned} \rho_{H}(\chi_{(0,1)}Sg) =&\rho_{H} \biggl( \chi_{(0,1)} \int _{0}^{t}u^{\frac{s}{n}}g(u)\frac{du}{u} \biggr) \\ =&\rho_{H} \biggl(\chi_{(0,1)} \int _{0}^{1}(tv)^{\frac {s}{n}}g(tv) \frac{dv}{v} \biggr). \end{aligned}$$

Applying Minkowski’s inequality and using \(\alpha_{H}>0\), we have

$$\rho_{E(H)}(g)=\rho_{H}(\chi_{(0,1)}Sg)\lesssim \rho_{H}\bigl(\chi _{(0,1)}t^{\frac{s}{n}}g(t)\bigr). $$

For the reverse we use

$$t^{\frac{s}{n}}g(t) \lesssim Sg(t), \quad g\downarrow, $$

whence

$$\rho_{H}\bigl(\chi_{(0,1)}t^{\frac{s}{n}}g(t)\bigr) \lesssim \rho_{H}\bigl(\chi _{(0,1)}Sg(t)\bigr)=\rho_{E(H)}(g). $$

 □

Example 4.3

Consider the space \(H=L^{1}_{\ast}(v)\), where v is slowly varying and \(v>1\). Then \(\rho_{H}\in N_{t}\) and by Theorem 4.2, we can construct an optimal domain \(E(H)\), where

$$\begin{aligned} \rho_{E(H)}(g)&=\rho_{H} (S g)= \int_{0}^{1} v(t)Sg(t)\,dt/t \\ &= \int_{0}^{1}v(t) \int_{0}^{t} u^{\frac{s}{n}}g(u)\frac{du}{u} \frac {dt}{t}= \int_{0}^{1} w(u)g(u)\frac{du}{u}, \end{aligned}$$

and \(w(u)=\int_{u}^{1} v(t)\frac{dt}{t}\). Hence \(E(H)=\Lambda^{1}(t^{s/n}w)\) and this couple is optimal. Also \(\alpha_{E}=\beta_{E}=s/n\).

Example 4.4

Let \(H=L^{\infty}(v)\), where v is slowly varying and \(v>1\). Then \(\rho _{H}\in N_{t}\). Let

$$\rho_{E}(g)=\sup v(t) \int_{0}^{t} u^{s/n} g^{\ast}(u)\,du/u. $$

Then by Theorem 4.2 this is an optimal domain quasi-norm and the couple \(\rho_{E}\), \(\rho_{H}\) is optimal. In particular, the couple \(\Lambda^{1}(t^{s/n})\), \(C^{0}\) is optimal.

4.2 Optimal target quasi-norms

Definition 4.5

(construction of an optimal target quasi-norm)

For a given domain quasi-norm \(\rho_{E}\in N_{d}\), we set

$$ \rho_{H(E)}(\chi_{(0,1)}g):=\inf\bigl\{ \rho_{E}(h): \chi_{(0,1)}g\leq S h, h\downarrow\bigr\} ,\quad g\in M^{+}. $$
(4.4)

Note that

$$\alpha_{H(E)}=\beta_{H(E)}=s/n- \alpha_{E}. $$

Theorem 4.6

The target quasi-norm \(\rho_{H(E)}\in N_{t}\), the couple \(\rho_{E}\), \(\rho_{H(E)}\) is admissible, and the target quasi-norm is optimal.

Proof

The couple \(\rho_{E}\), \(\rho_{H(E)}\) is admissible since

$$\rho_{H(E)}(\chi_{(0,1)}Sh)\leq\rho_{E}(h), \quad h \downarrow. $$

Now to prove that \(\rho_{H(E)}\) is optimal, we take any admissible couple \(\rho_{E}, \rho_{H_{1}}\in N_{t}\). Then

$$\rho_{H_{1}}(\chi_{(0,1)}Sh)\lesssim\rho_{E}(h),\quad h\downarrow. $$

Therefore, if \(g\leq Sh\), h↓, then

$$\rho_{H_{1}}(\chi_{(0,1)}g)\leq\rho_{H_{1}}( \chi_{(0,1)}Sh)\lesssim\rho_{E}(h), $$

whence, taking the infimum, we get

$$\rho_{H_{1}}(\chi_{(0,1)}g)\lesssim\rho_{H(E)}( \chi_{(0,1)}g). $$

Hence \(\rho_{H(E)}\) is optimal. □

Theorem 4.7

If \(\alpha_{E}< s/n\), then

$$\rho_{H(E)}(\chi_{(0,1)}g)\approx\rho_{E} \bigl(t^{-s/n}g(t)\bigr),\quad g\in M_{n}. $$

Moreover, the couple \(\rho_{E}\), \(\rho_{H(E)}\) is optimal.

Proof

Consider

$$\begin{aligned} \rho_{E}\bigl(t^{-s/n}Sh(t)\bigr) =& \rho_{E} \biggl(t^{-s/n} \int_{0}^{t} u^{s/n}h(u)\frac{du}{u} \biggr) \\ =& \rho_{E} \biggl( \int_{0}^{1} v^{s/n}h(tv)\frac{dv}{v} \biggr),\quad h\downarrow. \end{aligned}$$

Applying Minkowski’s inequality and using \(\beta_{E}< s/n\), we have

$$\rho_{E}\bigl(t^{-s/n}Sh(t)\bigr)\lesssim \rho_{E}(h), \quad h\downarrow. $$

If \(\chi_{(0,1)}g\leq Sh\), \(g\in M_{n}\), then

$$\rho_{E}\bigl(t^{-s/n}g(t)\bigr)\lesssim\rho_{E} \bigl(t^{s/n}Sh(t)\bigr)\lesssim\rho_{E}(h) $$

and, taking the infimum, we get

$$\rho_{E}\bigl(t^{-s/n}g(t)\bigr)\lesssim\rho_{H(E)}( \chi_{(0,1)}g). $$

On the other hand, for \(g\in M_{n}\), let \(h(t)=t^{-s/n}g(t)\chi _{(0,1)}(t)\). Then h↓ and

$$\begin{aligned} Sh(t) =& \int_{0}^{t}u^{s/n}h(u)\frac{du}{u} \\ =& \int_{0}^{t} u^{s/n}u^{-s/n}g(u) \frac{du}{u} \\ \geq& g(t). \end{aligned}$$

Therefore

$$\rho_{H(E)}(\chi_{(0,1)}g)\lesssim \rho_{E}(h)= \rho_{E}\bigl(t^{-s/n}g(t)\bigr). $$

Now we show that the domain quasi-norm \(\rho_{E}\) is also optimal. We have

$$\begin{aligned} \rho_{E(H(E))}\bigl(f^{\ast}\bigr) =& \rho_{H(E)}\bigl( \chi_{(0,1)}Sf^{\ast}\bigr) \\ \approx& \rho_{E}\bigl(t^{-s/n}Sf^{\ast}(t)\bigr) \\ =&\rho_{E} \biggl(t^{-s/n} \int_{0}^{t} u^{s/n}f^{\ast}(u) \frac{du}{u} \biggr) \\ \gtrsim& \rho_{E}\bigl(f^{\ast}\bigr), \quad f\in E. \end{aligned}$$

Therefore

$$\rho_{E(H(E))}\bigl(f^{\ast}\bigr)\gtrsim\rho_{E} \bigl(f^{\ast}\bigr);\quad f\in E. $$

 □

Example 4.8

Consider the space \(E=\Lambda^{q}(t^{\alpha}w(t))\), \(0< q\leq\infty\), where w is slowly varying and \(s/n>\alpha>0\). Then \(\beta_{E}=\alpha _{E}=\alpha\) and \(\rho_{E}\in N_{d}\). Hence by Theorem 4.7,

$$\rho_{H(E)}(g)\approx\rho_{E}\bigl(t^{-s/n}g(t) \bigr)= \biggl( \int _{0}^{1}\bigl(t^{-s/n}w(t)g^{\ast}(t) \bigr)^{q}\frac{dt}{t} \biggr)^{1/q}, $$

which implies that \(H(E)=L^{q}_{\ast}(t^{-s/n}w)\).

Moreover, the couple \(\rho_{E}\), \(\rho_{H}(E)\) is optimal. In particular, the couple

$$L^{p,\infty},{\mathcal {C}}^{s-n/p},\quad s>n/p,1< p< \infty, $$

is optimal.