On slow decay of Peetre's K-functional

TL;DR

This paper investigates the conditions under which Peetre's K-functional decays slowly, providing a characterization that reveals strict inclusions between certain real interpolation spaces.

Contribution

It offers a new characterization of slow decay of Peetre's K-functional and applies it to establish strict inclusions between real interpolation spaces.

Findings

Characterization of slow decay of Peetre's K-functional

Demonstration of strict inclusions between real interpolation spaces

Insights into the structure of interpolation spaces

Abstract

We characterize when Peetre's K-functional decays to zero slowly and we use this characterization to demonstrate certain strict inclusions between real interpolation spaces.

Equations

$$K(x,t,X,Y)=\underset{y\in Y}{\inf }\Vert x-y{\Vert }_X+t\Vert y{\Vert }_Y$$

$$K(x,0^{+},X,Y)=\underset{t\to 0^{+}}{\lim }K(x,t,X,Y)=0$$

$$K(x,t,X,Y)\ge c$$

$$\text{If }y\in Y\text{ satisfies }\Vert x-y{\Vert }_X<c,\text{ then }\Vert y{\Vert }_Y\ge \frac{c-\Vert x-y{\Vert }_X}{t}$$

$$K(A,t,X,Y)=\underset{x\in A}{\sup }K(x,t,X,Y).$$

$$K(S(X),t,X,Y)>c\text{ for all }t>0$$

$$E(y,A_n)\le C{n}^{-r}\Vert y{\Vert }_Y\text{ for all }y\in Y\text{ and all }n\ge 0$$

$$\Vert a{\Vert }_Y\le C{n}^r\Vert a{\Vert }_X\text{ for all }a\in A_n,$$

$$A_q^{\alpha }(X,\{A_n\})=(X,Y{)}_{\alpha /r,q}\text{ for all }0<\alpha <r\text{ and all }0<q\le \infty .$$

$$(X,Y{)}_{\theta ,q}=\{x\in X:{\rho }_{\theta ,q}(x)=\Vert t^{-\left(\theta +\frac{1}{q}\right)}K(x,t,X,Y){\Vert }_{L^q(0,\infty )}<\infty \},$$

$$\Vert x{\Vert }_{\theta ,q}=\Vert \{2^{\theta k}K(x,\frac{1}{2^k},X,Y){\}}_{k=0}^{\infty }{\Vert }_{{\ell }_q}.$$

$$K(x,t_n,X,Y)\ne \mathbf{O}({\epsilon }_n)$$

$$X=\bigcup _{m=1}^{\infty }{\Gamma }_m,$$

$${\Gamma }_m=\{x\in X:K(x,t_n,X,Y)\le m{\epsilon }_n\text{ for all }n\in \mathbb{N}\}.$$

$$\text{conv}({\Gamma }_m)\subseteq {\Gamma }_{Cm},$$

$$K(\lambda x+(1-\lambda )y,t_n,X,Y)$$

$$\frac{1}{2}(B_X(x_0,r)+B_X(-x_0,r))\subseteq {\Gamma }_{C{m}_0}.$$

$$B_X(0,r)\subseteq \frac{1}{2}(B_X(x_0,r)+B_X(-x_0,r)),$$

$$B_X(0,r)\subseteq {\Gamma }_{C{m}_0}.$$

$$K(\frac{rx}{\Vert x\Vert },t_n,X,Y)\le C{m}_0{\epsilon }_n\text{ for all }n=1,2,\cdots$$

$$K(x,t_n,X,Y)\le \frac{\Vert x\Vert }{r}C{m}_0{\epsilon }_n\text{ for all }n=1,2,\cdots$$

$$c<K(x_n,t_n,X,Y)\le \frac{1}{r}C{m}_0{\epsilon }_n\text{ for all }n=1,2,\cdots ,$$

$$K(S(X),t_n,X,Y)\le c_n\text{ for all }n\in \mathbb{N}$$

$$K(\frac{x}{\Vert x{\Vert }_X},t_n,X,Y)\le K(S(X),t_n,X,Y)\le c_n\text{ for all }n\in \mathbb{N},$$

$$K(x,t_n,X,Y)=\Vert x{\Vert }_{X}K(\frac{x}{\Vert x{\Vert }_X},t_n,X,Y)\le \Vert x{\Vert }_X{c}_n\text{ for all }n\in \mathbb{N},$$

$$K(x,t_n,X_1,Y_1)\ne \mathbf{O}({\epsilon }_n).$$

$$K^{\ast }(x,t,X,Y)=\underset{y\in Y}{\inf }\Vert x-y{\Vert }_{\ast ,X}+t\Vert y{\Vert }_{\ast ,Y}$$

$$M\cdot K^{\ast }(x,t,X,Y)\le K(x,t,X,Y)\le N\cdot K^{\ast }(x,t,X,Y)$$

$$\Vert x_n-y{\Vert }_X<\delta \text{ and }y\in Y\text{ imply that }\Vert y{\Vert }_Y\ge b_n$$

$$1>\delta >\Vert x_n-y_n{\Vert }_X+\frac{1}{b_n}\Vert y_n{\Vert }_Y>1.$$

Full text

∗ Corresponding author

1. Introduction

In this paper we consider a couple $(X,Y)$, where $(X,\|\cdot\|_{X})$ is a quasi-Banach space and $Y\subset X$ is a quasi-semi-normed space, $(Y,\|\cdot\|_{Y})$ which is continuously embedded into $X$. Under these conditions, we also consider Peetre’s $K$-functional, which is defined by

[TABLE]

This functional is obviously connected to the approximation properties of the elements of $X$ by elements of $Y$. Concretely, if $Y$ is dense in $X$, then

[TABLE]

and the claim

[TABLE]

is equivalent to the following property:

[TABLE]

In particular, if $\|x-y\|_{X}<\frac{c}{2}$ with $y\in Y$, then $\|y\|_{Y}\geq\frac{c-\|x-y\|_{X}}{t}\geq\frac{c}{2t}$, which diverges to infinity when $t$ tends to [math]. Let $A$ be a subset of $X$, we define

[TABLE]

Thus, if $S(X)$ denotes the unit sphere of $X$, the condition

[TABLE]

implies the existence of elements in $X$ of small norm (in particular, of norm $\|x\|_{X}=1$) which are badly approximable by elements of $Y$ with small norm in $Y$, which is a property that holds true in many cases. For example, if we set $X=C[a,b]$ and $Y=C^{(m)}[a,b]$ with $m\geq 1$, then there are oscillating functions of small uniform norm that cannot be uniformly approximated by functions of small $C^{(m)}$-semi-norm.

The connection between the $K$-functional and approximation theory is, in fact, a strong one, and there are many results which connect the behavior of this functional to the properties of best approximation errors with respect to an approximation scheme $(X,\{A_{n}\})$. In particular, the so called Central Theorems in Approximation Theory lead to a proof that, when the approximation scheme $(X,\{A_{n}\})$ satisfies Jackson’s and Bernstein’s inequalities with respect to the space $Y\hookrightarrow X$, which are given by

[TABLE]

and

[TABLE]

respectively, then the approximation spaces $A_{q}^{\alpha}(X,\{A_{n}\})=\{x\in X:\{2^{\alpha k}E(x,A_{2^{k}})\}\in\ell_{q}\}$ are completely characterized as interpolation spaces by the formula (see [2])

[TABLE]

Here $(X,Y)_{\theta,q}$ denotes the real interpolation space

[TABLE]

which, in case that $Y\hookrightarrow X$, can be renormalized with the following equivalent quasi-norm:

[TABLE]

It follows that, when $(X,\{A_{n}\})$ satisfies Jackson’s and Bernstein’s inequalities with respect to a subspace $Y$, the rates of convergence to zero of the sequence of best approximation errors $E(x,A_{n})$ and the sequence of evaluations of Peetre’s $K$-functional $K(x,t,X,Y)$ at points $1/n$ seem to have similar roles and, in particular, serve to describe the very same subspaces of $X$. Thus, it is an interesting question to know under which conditions on the couple $(X,Y)$, with $Y\hookrightarrow X$, the $K$-functional $K(\cdot,\cdot,X,Y)$ approaches to zero slowly, not only in presence of an approximation scheme but also for the general case. In this paper we characterize couples whose associated $K$-functional decays to zero slowly. As an application, we use this characterization to demonstrate that some natural embeddings between real interpolation spaces are strict, a question that was posed by Lions in the decade of 1960 and was solved by Janson, Nilsson, Peetre and Zafran in 1984 [4] with some sophisticated tools [6]. A complete solution, for the complex method of interpolation, was previously shown by Stafney in 1970 [7]. Bergh and Löfström presented a partial solution for the real interpolation method in their famous book [1]. Their approach was inspired by Stafney’s work. In this paper we study the very same cases by means of less sophisticated tools.

In section 2 of this paper we characterize $K$-functionals which slowly decay to zero as those satisfying (1) and we also characterize them in terms of the approximation properties of $Y$ as a subspace of $X$. In section 3 we demonstrate that for any couple $(X,Y)$ of quasi-Banach spaces such that $Y\hookrightarrow X$, $Y\neq X$, the $K$-functional $K(\cdot,\cdot,X,Y)$ decays to zero slowly and, as a consequence, we prove that, under the very same hypotheses, all interpolation spaces $(X,Y)_{\theta,p}$ are strictly embedded between $X$ and $Y$ and that, if $\theta_{1}\neq\theta_{2}$, then $(X,Y)_{\theta_{1},p}\neq(X,Y)_{\theta_{2},q}$ for all $0<p,q\leq\infty$. Moreover, if $0<\theta<1$ and $0<p,q\leq\infty$ are such that $(X,Y)_{\theta,p}\neq(X,Y)_{\theta,q}$, then we show that the spaces $(X,Y)_{\theta,r}$ with $r\in[p,q]$ are pairwise distinct. The proofs we introduce for these cases are new, elementary, and very natural ones. Moreover, the results we get about $K$-functionals with slow decay are also new and, in our opinion, interesting by themselves. In the last section of the paper, we show that, if $Y$ is densely embedded into $X$ and $Z\hookrightarrow X$ is a subspace compactly embedded into $X$, then there is a function $\varphi(t)$ satisfying $\lim_{t\to 0^{+}}\varphi(t)=0$ such that $K(z,t,X,Y)\leq\varphi(t)\|z\|_{Z}$ for all $z\in Z$. Thus, all elements of $Z$ share a common decay to zero in their $K$-functionals.

2. Characterizations of slowly decaying $K$-functionals

The importance of condition (1) comes from the following result:

Theorem 1**.**

The following are equivalent claims:

• $(a)$

$K(S(X),t,X,Y)>c$* for all $t>0$ and a certain constant $c>0$.*

• $(b)$

For every non-increasing sequences $\{\varepsilon_{n}\},\{t_{n}\}\in c_{0}$, there are elements $x\in X$ such that

[TABLE]

Proof.

$(a)\Rightarrow(b)$. Let us assume, on the contrary, that $K(x,t_{n},X,Y)=\mathbf{O}(\varepsilon_{n})$ for all $x\in X$ and certain sequences $\{\varepsilon_{n}\},\{t_{n}\}\in c_{0}$. This can be reformulated as

[TABLE]

where

[TABLE]

Now, $\Gamma_{m}$ is a closed subset of $X$ for all $m$ and Baire category theorem implies that $\Gamma_{m_{0}}$ has nonempty interior for some $m_{0}\in\mathbb{N}$. On the other hand, $\Gamma_{m}=-\Gamma_{m}$ since $K(x,t,X,Y)=K(-x,t,X,Y)$ for all $x\in X$ and $t\geq 0$. Furthermore, if $C>1$ is the quasi-norm constant of $X$ and $Y$, then

[TABLE]

since, if $x,y\in\Gamma_{m}$ and $\lambda\in[0,1]$, then

[TABLE]

Thus, if $B_{X}(x_{0},r)=\{x\in X:\|x_{0}-x\|_{X}<r\}\subseteq\Gamma_{m_{0}}$, then $\textbf{conv}(B_{X}(x_{0},r)\cup B_{X}(-x_{0},r))\subseteq\Gamma_{Cm_{0}}$. In particular,

[TABLE]

Now, it is clear that

[TABLE]

since, if $x\in B_{X}(0,r)$, then $\|-x_{0}-(x+x_{0})\|=\|x_{0}-(x-x_{0})\|=\|x\|_{X}=\leq r$, so that $x-x_{0}\in B_{X}(x_{0},r)$, $x+x_{0}\in B_{X}(x_{0},r)$, and $x=\frac{1}{2}((x-x_{0})+(x+x_{0}))$. Thus,

[TABLE]

This means that, if $x\in X\setminus\{0\}$, then

[TABLE]

Hence

[TABLE]

On the other hand, $(a)$ implies that, for each $n$, there is $x_{n}\in S(X)$ such that $K(x_{n},t_{n},X,Y)>c$, so that

[TABLE]

which is impossible, since $\varepsilon_{n}$ converges to [math] and $c>0$. This proves $(a)\Rightarrow(b)$.

Let us demonstrate the other implication. Assume that $(a)$ does not hold. Then there are non-increasing sequences $\{t_{n}\},\{c_{n}\}\in c_{0}$ such that

[TABLE]

In particular, if $x\in X$ is not the null vector, then

[TABLE]

so that

[TABLE]

and $K(x,t_{n},X,Y)=\mathbf{O}(c_{n})$ for all $x\in X$. This proves $(b)\Rightarrow(a)$. ∎

Definition 2**.**

We say that the $K$-functional $K(\cdot,\cdot,X,Y)$ slowly decays to zero if either condition $(a)$ or $(b)$ of Theorem 1 hold true.

Remark 3*.*

Note that, if $Y\subseteq X$ and the $K$-functional $K(\cdot,\cdot,X,Y)$ decays to zero slowly, then $(X,Y)_{\theta,q}$ is properly contained into $X$ for all $0<\theta<1$ and all $0<q\leq\infty$.

Next proposition, whose proof is omitted, states that if the elements of $S(X)$ are not easily approximated by elements of $Y$, they cannot be easily approximated by elements of any other subspace $Y_{1}\hookrightarrow Y$.

Proposition 4**.**

Assume that $Y$ is densely embedded into $X$ and $K(S(X),t,X,Y)>c$ for all $t>0$ and a certain constant $c>0$. Then:

• (a)

If $Y_{1}\hookrightarrow Y$ is another quasi-semi-normed space continuously embedded into $Y$, then $K(S(X),t,X,Y_{1})>c$ for all $t>0$.

• (b)

If $Y_{1}\hookrightarrow Y$ is another quasi-semi-normed space continuously embedded into $Y$, then for every every non-increasing sequences $\{\varepsilon_{n}\},\{t_{n}\}\in c_{0}$, there are elements $x\in X$ such that

[TABLE]

Remark 5*.*

Assume that $\|\cdot\|_{*,X}$ is an equivalent quasi-norm of $X$ and $\|\cdot\|_{*,Y}$ is an equivalent quasi-semi-norm of $Y$, and let us denote by

[TABLE]

the $K$-functional associated to the couple $(X,Y)$ when doted with the norms $\|\cdot\|_{*,X}$ and $\|\cdot\|_{*,Y}$, respectively. Then there are two constants $M,N>0$ such that

[TABLE]

In particular, if $S^{*}(X)=\{x\in X:\|x\|_{*,X}=1\}$, the following are equivalent claims:

• $(a)$

$K^{*}(S^{*}(X),t,X,Y)>c^{*}$ for all $t>0$ and a certain constant $c^{*}>0$.

• $(b)$

$K(S(X),t,X,Y)>c$ for all $t>0$ and a certain constant $c>0$.

Let us now characterize, in terms of the approximation properties of $Y$ as a subspace of $X$, the couples $(X,Y)$ whose associated $K$-functional slowly decays to zero:

Theorem 6**.**

The following are equivalent claims:

• $(a)$

$K(S(X),t,X,Y)>c$* for all $t>0$ and a certain constant $c>0$.*

• $(b)$

For a certain constant $\delta>0$ there exist sequences $\{x_{n}\}\subset S(X)$ and $\{b_{n}\}\subset]0,\infty[$, $\lim b_{n}=\infty$, such that, for every $n\in\mathbb{N}$,

[TABLE]

Proof.

Note that we can assume $0<c,\delta<1$ without loss of generality. Assume $(a)$ and take, for each $t>0$, an element $x_{t}\in S(X)$ such that $K(x_{t},t,X,Y)>c$ then we have already shown that if $\|x_{t}-y_{t}\|_{X}<\frac{c}{2}$ with $y_{t}\in Y$, then $\|y_{t}\|_{Y}\geq\frac{c-\|x_{t}-y_{t}\|_{X}}{t}\geq\frac{c}{2t}$. Thus, $(b)$ holds for $\delta=c/2$ and $b_{n}=\frac{c}{2t_{n}}$ for every decreasing sequence $\{t_{n}\}\in c_{0}$. This proves $(a)\Rightarrow(b)$.

Let us now assume that $K(x_{n},\frac{1}{b_{n}},X,Y)<\delta$ with $\{x_{n}\}$, $\delta$ and $\{b_{n}\}$ satisfying $(b)$. Then there is $y_{n}\in Y$ such that $\|x_{n}-y_{n}\|_{X}+\frac{1}{b_{n}}\|y_{n}\|_{Y}<\delta$, which leads to a contradiction since $\|x_{n}-y_{n}\|_{X}<\delta$ and $y_{n}\in Y$ imply $\|y_{n}\|_{Y}\geq b_{n}$, so that

[TABLE]

It follows that $K(S(X),\frac{1}{b_{n}},X,Y)\geq\delta$ for all $n$ and $(a)$ follows with $c=\delta$ from the monotonicity of $K$ and the fact that $\lim_{n\to\infty}\frac{1}{b_{n}}=0$. ∎

Theorem 6 reveals an easy way to confirm that Peetre’s $K$-functional decays to zero slowly, since condition $(b)$ of this theorem is, in many cases, easy to verify. We include here just a few examples to show the way this theorem can be used.

Example 7*.*

Set $X=C[a,b]$ with the uniform norm $\|f\|_{C[a,b]}=\sup_{t\in[a,b]}|f(t)|$ and $Y=C^{(1)}[a,b]$ with the seminorm $\|g\|_{C^{(1)}[a,b]}=\|g^{\prime}\|_{C[a,b]}$. Take $\alpha_{n}=\frac{a+b}{2}-\frac{1}{2n},\beta_{n}=\frac{a+b}{2}+\frac{1}{2n}$ and set

[TABLE]

Then $\|f_{n}\|_{C[a,b]}=1$ for all $n$ and, if $\|f_{n}-g\|_{C[a,b]}<\frac{1}{2}$ with $g\in C^{(1)}[a,b]$ then $g(\alpha_{n})<-1/2$ and $g(\beta_{n})>1/2$ so that, there is a point $\rho_{n}\in[\alpha_{n},\beta_{n}]\subseteq[a,b]$ such that

[TABLE]

and condition $(b)$ of Theorem 6 holds true with $\delta=\frac{1}{2}$ and $b_{n}=n$. Note that this example also works when we consider on $Y=C^{(1)}[a,b]$ the norm $\|g\|_{1}=\|g\|_{C[a,b]}+\|g^{\prime}\|_{C[a,b]}$.

Example 8*.*

Given $0<p<q\leq\infty$ we set $X=\ell_{q}$ with the standard norm (or quasi-norm) $\|\{a_{n}\}\|_{q}=\left(\sum_{n=0}^{\infty}|a_{n}|^{q}\right)^{\frac{1}{q}}$, for $q<\infty$, and $X=c_{0}$ with the supremum norm $\|\{a_{n}\}\|_{\infty}=\sup_{n}|a_{n}|$, if $q=\infty$. We also set $Y=\ell_{p}$ with the standard norm (or quasi-norm). Take $\mathbf{a}_{n}=\{a_{k}^{n}\}_{k=0}^{\infty}$, where $a_{k}^{n}=1$ for $k=0,1,\cdots,2n-1$ and $a_{k}^{n}=0$ for $k\geq 2n$. If $\|\mathbf{a}_{n}-\mathbf{b}\|_{\infty}<\frac{1}{2}$ (analogously if $\|\mathbf{a}_{n}-\mathbf{b}\|_{q}<\frac{1}{2}$), with $\mathbf{b}=\{b_{k}\}\in\ell_{p}$, then $b_{k}>\frac{1}{2}$ for all $k\in\{0,1,\cdots,2n-1\}$, which implies that $\|\mathbf{b}\|_{p}\geq 2^{-1+\frac{1}{p}}n^{\frac{1}{p}}$. Thus, condition $(b)$ of Theorem 6 holds true in all these cases with $\delta=\frac{1}{2}$ and $b_{n}=2^{-1+\frac{1}{p}}n^{\frac{1}{p}}$

Example 9*.*

If we do not impose $Y\hookrightarrow X$ but we maintain $Y\subseteq X$ (i.e., we do not require continuity of the inclusion) then it is easy to find a couple $(X,Y)$ whose $K$-functional does not decay to zero slowly. Concretely, we can take $X=\ell_{1}$ with its natural norm, $\|\cdot\|_{1}$, and $Y=c_{00}=\{\text{ finite sequences }\}$ with the norm of supremum , $\|\cdot\|_{\infty}$. Obviously $Y\subset X$. Take $(a_{n})\in S(\ell_{1})$ and take $\delta>0$ arbitrary. Then there is a number $N_{0}\in\mathbb{N}$ such that $\sum_{n=N_{0}+1}^{\infty}|a_{n}|<\delta$. Take $(b_{n})\in c_{00}$ given by $b_{k}=a_{k}$ for $k\leq N_{0}$ and $b_{k}=0$ otherwise. Then

[TABLE]

and this holds for every $\delta>0$, so that

[TABLE]

and this $K$-functional does not decay to zero slowly.

3. The Quasi-Banach setting

The following result shows that, under very mild conditions on the couple $(X,Y)$, the $K$-functional $K(\cdot,\cdot,X,Y)$ slowly decays to zero.

Theorem 10**.**

Assume that $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ are quasi-Banach spaces. Then

• $(a)$

If $Y\hookrightarrow X$, $Y\neq X$ (i.e., $Y$ is properly embedded into $X$). Then $K(S(X),t,X,Y)>c$ for all $t>0$ and a certain constant $c>0$.

• $(b)$

If $X$ and $Y$ are both $p$-normed spaces, $Y\hookrightarrow X$, then either $K(S(X),t,X,Y)=1$ for all $t>0$, or $X=Y$.

Proof.

It follows from Aoki-Rolewicz Theorem (see [2, Theorem 2.1.1] or [5, pages 7-8]) that $X$ and $Y$ can be renormed with equivalent quasi-norms in such a way that they become $p$-normed spaces for a certain $p\in]0,1]$ (indeed, Aoki-Rolewicz gives two equivalent quasi-norms $\|\cdot\|_{*,X}$ and $\|\cdot\|_{*,Y}$, respectively and two numbers $p_{X},p_{Y}\in]0,1]$ such that $\|x_{1}+x_{2}\|_{*,X}^{p_{X}}\leq\|x_{1}\|_{*,X}^{p_{X}}+\|x_{2}\|_{*,X}^{p_{X}}$ and $\|y_{1}+y_{2}\|_{*,Y}^{p_{Y}}\leq\|y_{1}\|_{*,Y}^{p_{Y}}+\|y_{2}\|_{*,Y}^{p_{Y}}$ for all $x_{1},x_{2}\in X$ and all $y_{1},y_{2}\in Y$. Thus, if we set $p=\min\{p_{X},p_{Y}\}$ then $\|x_{1}+x_{2}\|_{*,X}^{p}\leq\|x_{1}\|_{*,X}^{p}+\|x_{2}\|_{*,X}^{p}$ and $\|y_{1}+y_{2}\|_{*,Y}^{p}\leq\|y_{1}\|_{*,Y}^{p}+\|y_{2}\|_{*,Y}^{p}$ for all $x_{1},x_{2}\in X$ and all $y_{1},y_{2}\in Y$ and both spaces are $p$-normed with the very same $p$. It follows from this and from Proposition 5 that we only need to demonstrate part $(b)$ of the Theorem.

Thus, we assume that $X$, $Y$ are quasi-Banach $p$-normed spaces and $Y\hookrightarrow X$ and we want to demonstrate that, if $K(S(X),t,X,Y)<1$ for a certain $t>0$, then $X=Y$. Assume, on the contrary, that $K(S(X),t_{0},X,Y)=c<1$ and $X\neq Y$. Take $\rho\in]0,1[$ such that $c<\rho^{1/p}<1$. Then $K(\frac{x}{\|x\|_{X}},t_{0},X,Y)<\rho^{1/p}$ for every $x\in X$, $x\neq 0$. Hence every element $x$ of $X$ which is different from zero satisfies

[TABLE]

which implies that $\|x-y_{0}\|_{X}+t_{0}\|y_{0}\|_{Y}<\rho^{1/p}\|x\|_{X}$ for certain $y_{0}\in Y$. Thus, if we set $x_{0}=x-y_{0}$, we have that

[TABLE]

Let us take $x\in X\setminus Y$ and apply (2) to this concrete element. We can repeat the argument just applying it to $x_{0}$ (if $x_{0}=0$ then $x=y_{0}\in Y$, which contradicts our assumption). Hence, there are elements $x_{1}\in X$ and $y_{1}\in Y$ such that

[TABLE]

Moreover, $x=x_{0}+y_{0}=x_{1}+y_{1}+y_{0}$. Again $x_{1}\neq 0$ since $x\not\in Y$. We can repeat the argument $m$ times to get a decomposition $x=x_{m}+y_{m}+\cdots+y_{0}$ with $x_{m}\in X$, $x_{m}\neq 0$, $y_{k}\in Y$ for all $0\leq k\leq m$ and

[TABLE]

Let us set $z_{m}=x-x_{m}=y_{0}+\cdots+y_{m}$. Then

[TABLE]

and $x$ is the limit of $z_{m}$ in the norm of $X$. On the other hand, if $n>m$, then

[TABLE]

which converges to [math] for $n,m\to\infty$. Hence $\{z_{m}\}$ is a Cauchy sequence in $Y$ and its limit belongs to $Y$ since $Y$ is topologically complete. This implies $x\in Y$, which contradicts our assumptions. Thus, we have demonstrated, for $p$-normed quasi-Banach spaces $X$ and $Y$ satisfying $Y\hookrightarrow X$, that if $K(S(X),t_{0},X,Y)<1$ for a certain $t_{0}>0$, then $X=Y$. In particular, if $X\neq Y$ then $K(S(X),t,X,Y)=1$ for all $t>0$. ∎

Corollary 11**.**

Let $(X,Y)$ be an ordered couple of quasi-Banach spaces, $Y\hookrightarrow X$, $Y\neq X$. Then there exist a constant $\delta>0$; a sequence $\{b_{n}\}\subset[0,\infty)$ with $(b_{n})\to\infty$ as $n\to\infty$ and a sequence $\{x_{n}\}\subseteq S(X)$ such that for all $n\in\mathbb{N}$ and $y\in Y$

[TABLE]

Proof.

The result is a direct application of Theorems 10 and 6. ∎

Theorem 10 has some interesting consequences on the theory of interpolation spaces. Concretely, if $X$, $Y$ are quasi-Banach spaces and $Y\hookrightarrow X$, $Y\neq X$, then all interpolation spaces $(X,Y)_{\theta,q}$ are strictly embedded into $X$ and, if $Y$ is not closed in $X$, they strictly contain $Y$. Moreover, the inclusions we get by applying the reiteration theorem are also strict. This is stated in the following theorems:

Theorem 12**.**

Let $(X,Y)$ be a couple of quasi-Banach spaces, $Y\hookrightarrow X$, $Y\neq X$ and $Y$ not closed in $X$. Then

[TABLE]

Proof.

The fact that the inclusion $(X,Y)_{\theta,q}\hookrightarrow X$ is strict follows directly from Theorems 1 and 10, since they allow us to claim that there are elements $x\in X$ such that $K(x,\frac{1}{2^{k}},X,Y)$ goes to zero as slow as we want, which implies that we can find $x\in X$ such that $\|x\|_{(X,Y)_{\theta,q}}=\infty$. The inclusion $Y\hookrightarrow(X,Y)_{\theta,q}$ is also strict. This should be well known, but we include here a proof for the sake of completeness. We assume $q<\infty$, since the case $q=\infty$ admits a similar proof. The fact that $Y\hookrightarrow(X,Y)_{\theta,q}$ as soon as $\theta<1$ is a direct computation, based on the fact that, for $y\in Y$, $K(y,t,X,Y)\leq t\|y\|_{Y}$ for all $t>0$. As $Y\hookrightarrow X$ with strict inclusion, we know that $\|y\|_{X}\leq M\|y\|_{Y}$ for a certain $M>0$ and all $y\in Y$ and, moreover, there exists a sequence $\{y_{n}\}_{n=0}^{\infty}\subseteq S(Y)$ such that $\|y_{n}\|_{X}$ converges to [math] when $n$ goes to infinity. In particular, for every $N_{0}\in\mathbb{N}$ there exists $y_{N_{0}}\in S(Y)$ such that $\|y_{N_{0}}\|_{X}\leq 2^{-N_{0}}$. This obviously implies that

[TABLE]

Hence

[TABLE]

which converges to [math] when $N_{0}$ goes to infinity. This demonstrates that the norms $\|\cdot\|_{Y}$ and $\|\cdot\|_{\theta,q}$ are not equivalent on $Y$, which implies that $Y\neq(X,Y)_{\theta,q}$. ∎

Theorem 13**.**

Let $(X,Y)$ be a couple of quasi-Banach spaces, $Y\hookrightarrow X$, $Y\neq X$. Then

• $(a)$

Assume that $0<\theta_{0},\theta_{1}<1$, $\theta_{0}\neq\theta_{1}$, and $0<p,q\leq\infty$. Then

[TABLE]

• $(b)$

Let $\theta\in]0,1[$ and assume that $0<p,q\leq\infty$ are such that $(X,Y)_{\theta,p}\neq(X,Y)_{\theta,q}$ and that $r_{1},r_{2}\in[p,q]$, $r_{1}\neq r_{2}$. Then

[TABLE]

• $(c)$

If $0<\theta<1$ and $0<p\leq\infty$, then $(X,Y)_{\theta,p}$ is an infinite-codimensional subspace of $X$ and $Y$ is an infinite-codimensional subspace of $(X,Y)_{\theta,p}$.

Proof.

Part $(a)$ follows from Theorem 10 and the fact that $X$ is a space of class $\mathcal{C}(0,(X,Y))$, which implies that we can use the following reiteration formula:

[TABLE]

Indeed, if $0<\theta_{1}<\theta_{0}<1$, then $\theta_{1}=\alpha\theta_{0}$ for a certain $\alpha\in]0,1[$, which implies that

[TABLE]

Thus, Theorem 10 guarantees that $(X,Y)_{\theta_{0},p}$ is (a quasi-Banach space) strictly embedded into $X$, so that applying the very same result to the couple $(X,(X,Y)_{\theta_{0},p})$, and formula (5), we get that $(X,Y)_{\theta_{1},q}$ is strictly embedded between $(X,Y)_{\theta_{0},p}$ and $X$. In particular,

[TABLE]

Let us assume $p>q>0$. To demonstrate part $(b)$ we use the reiteration formula

[TABLE]

which, in conjunction with Theorem 10, implies that, if $(X,Y)_{\theta,p}\neq(X,Y)_{\theta,q}$ then all interpolation spaces $(X,Y)_{\theta,r}$ with $r\in]p,q[$ are strictly embedded between the spaces $(X,Y)_{\theta,p}$ and $(X,Y)_{\theta,q}$ (note that, if $p>q$, then $(X,Y)_{\theta,q}\hookrightarrow(X,Y)_{\theta,p}$ since $\ell_{q}\hookrightarrow\ell_{p}$). This is so because $\frac{1}{r}=\frac{1-\eta}{p}+\frac{\eta}{q}$ with $0\leq\eta\leq 1$ is another way to claim that $\frac{1}{r}\in[\frac{1}{p},\frac{1}{q}]$ which is equivalent to $r\in[q,p]$. Thus, if $r_{1},r_{2}\in]q,p[$, $r_{1}<r_{2}$, then $(X,Y)_{\theta,r_{1}}$ is strictly embedded in $(X,Y)_{\theta,p}$ and henceforth $(X,Y)_{\theta,r_{2}}$ is strictly embedded between the spaces $(X,Y)_{\theta,p}$ and $(X,Y)_{\theta,r_{1}}$, since $r_{1}<r_{2}<p$. In particular, $(X,Y)_{\theta,r_{1}}\neq(X,Y)_{\theta,r_{2}}$.

Part $(c)$ follows directly from $(a)$ and Theorem 12.

∎

Remark 14*.*

Observe that statement $(b)$ of Theorem 13 also follows from part $(a)$ and formula (6), since choosing two distinct $r_{1},r_{2}$ in $]p,q[$ is the same as choosing two distinct $\eta_{1},\eta_{2}$ in the lefthand side of formula (6).

In the case we consider a couple $(X,Y)$ where $(X,\|\cdot\|_{X})$ is a quasi-Banach space and $Y\subset X$ is a quasi-semi-normed space $(Y,\|\cdot\|_{Y})$ which is continuously included into $X$, we can no longer guarantee the slow decay of the associated $K$-functional. For example, in [3] it was proved that, if $0<p<1$ and $r\in\mathbb{N}$ is a positive natural number, then $K(f,t,L_{p},W_{p}^{r})=0$ for all $f\in L_{p}[0,1]$, when we deal the Sobolev space $W_{p}^{r}$ with the quasi-seminorm $\|g\|_{p,r}=\|g^{(r)}\|_{L_{p}[0,1]}$, so that

[TABLE]

In spite of this, we can use Theorem 10 to prove the following:

Theorem 15**.**

Assume that $(X,Y)$ is a couple, where $(X,\|\cdot\|_{X})$ is a quasi-Banach space and $Y\subset X$ is a quasi-semi-normed space $(Y,\|\cdot\|_{Y})$ which is continuously embedded into $X$. If the interpolation space $(X,Y)_{\theta_{0},p_{0}}$ is strictly embedded into $X$ for some choice of $0<\theta_{0}<1$ and $0<p_{0}\leq\infty$, then

[TABLE]

In particular, the strict inclusion of one interpolation space $(X,Y)_{\theta_{0},p_{0}}$ into $X$ implies that all interpolation spaces $(X,Y)_{\theta,p}$ are strictly embedded into $X$, where $0<\theta<1$ and $0<p\leq\infty$.

Proof.

A direct application of Theorem 10 to the couple of quasi-Banach spaces $(X,(X,Y)_{\theta_{0},p_{0}})$ leads to

[TABLE]

Thus, for each $t,\varepsilon>0$ there is $x_{t}\in S(X)$ such that $K(x_{t},t,X,(X,Y)_{\theta_{0},p_{0}})>(1-\varepsilon)c$. On the other hand, we have that $Y\hookrightarrow(X,Y)_{\theta_{0},p_{0}}$, which means that $\|y\|_{\theta_{0},p_{0}}\leq M\|y\|_{Y}$ for all $y\in Y$ and a certain constant $M=M(\theta_{0},p_{0})$. Hence

[TABLE]

Hence

[TABLE]

which implies that $K(S(X),t,X,Y)\geq c>0$ for all $t>0$, since $\varepsilon,t>0$ are arbitrary. This ends the proof. ∎

4. Spaces whose elements share a common decay in their $K$-functionals respect to a given couple

Another interesting question is to know under which conditions on a subspace $Z$ of $X$ we have that, for each decreasing sequence $\{t_{n}\}$ all elements $z\in Z$ share a common rate of convergence to zero for the sequence $\{K(z,t_{n},X,Y)\}$. We give a sufficient condition in the following result:

Theorem 16**.**

Assume that $Y$ is a continuously embedded dense subspace of $X$. If $Z$ is a compactly embedded subspace of $X$, and $\{t_{n}\}\in c_{0}$ is non-increasing, then there exists a non increasing sequence $\{\varepsilon_{n}\}\in c_{0}$ such that

[TABLE]

In fact, if we set $\varphi(t)=K(S(Z),t,X,Y)$, then $\lim_{t\to 0^{+}}\varphi(t)=0$ and $K(z,t,X,Y)\leq\|z\|_{Z}\varphi(t)$ for all $z\in Z$.

Proof.

Take $\varepsilon_{n}=K(S(Z),t_{n},X,Y)$, where $S(Z)$ denotes the unit sphere in $Z$. Then

[TABLE]

so that the result is proved as soon as we demonstrate that $\{\varepsilon_{n}\}\in c_{0}$. Now, the sequence $\{\varepsilon_{n}\}$ is non-increasing because of the monotony of $K$. Thus, if this sequence does not converge to [math], is uniformly bounded by a positive constant $c>0$. In particular, there is a sequence $\{z_{n}\}\subseteq S(Z)$ such that $K(z_{n},t_{n}X,Y)\geq c$ for all $n$. The compactness of the imbedding $Z\to X$ implies that there is a subsequence $\{z_{n_{k}}\}_{k=1}^{\infty}$ and a point $z\in X$ such that $\lim_{k\to\infty}\|z_{n_{k}}-z\|_{X}=0$. Hence

[TABLE]

which is impossible. Thus, we have demonstrated that, if $\varphi(t)=K(S(Z),t,X,Y)$, then $\lim_{t\to 0^{+}}\varphi(t)=0$ and $K(z,t,X,Y)\leq\|z\|_{Z}\varphi(t)$ for all $z\in Z$. ∎

Example 17*.*

If $(X,Y)$ is a couple and $Y\hookrightarrow Z\hookrightarrow X$ with $Z$ compactly embedded into $X$, then we can define the space $(Z^{*},\|\cdot\|_{*})=(Z,\|\cdot\|_{X})$ which results from considering on $Z$ the norm of $X$ and, thanks to Theorem 16, the associated $K$-functional will satisfy:

[TABLE]

where $\varphi(t)=K(S(Z),t,X,Y)$. Thus, $K(z,t,Z^{*},Y)$ does not decay to zero slowly.