V. A. SOLONNIKOV

ON SOME PROPERTIES OF SPACES \(\mathfrak{W}_p^l\) OF FRACTIONAL ORDER

(Presented by Academician V. I. Smirnov on 21 III 1960)

Let us define the space \(\mathfrak{W}_p^l(E_n)\) as the closure of smooth finite functions in the space \(E_n\) of functions in the norm

\[ \|f\|_{\mathfrak{W}_p^l(E_n)} = \sum_{i,j=1}^{n} \left( \int_{0}^{\infty} \frac{dh}{h^{1+p\lambda}} \int_{E_n} \left| D_i^{\bar l} f(x_1 \ldots x_j+h \ldots x_n) + D_i^{\bar l} f(x_1 \ldots x_j-h \ldots x_n) - 2D_i^{\bar l} f(x_1 \ldots x_n) \right|^p dx \right)^{1/p}, \]

where \(l=\bar l+\lambda,\ 0<\lambda\leqslant 1,\ D_i^{\bar l}=\dfrac{\partial^{\bar l}}{\partial x_i^{\bar l}}\). For \(\lambda<1\) this norm is equivalent to the following one:

\[ \sum_{i,j=1}^{n} \left( \int_{0}^{\infty} \frac{dh}{h^{1+p\lambda}} \int_{E_n} \left| D_i^{\bar l} f(x_1 \ldots x_j+h \ldots x_n) - D_i^{\bar l} f(x_1 \ldots x_n) \right|^p dx \right)^{1/p}. \]

This fact follows from the identity

\[ f(x+h)-f(x) = {}^{1}\!/_{2}[f(x+h)-f(x-h)] + {}^{1}\!/_{2}[f(x+h)+f(x-h)-2f(x)]. \]

Similar spaces were considered in the works \((^{1-7})\). The spaces \(\mathfrak{W}_p^l\) differ from the spaces \(W_p^l\), considered in \((^{4,7})\), only for integer \(l\); moreover, by means of Parseval’s equality it is easy to show that also for integer \(l\) the norm \(\mathfrak{W}_2^l\) is equivalent to the norm \(W_2^l\) on the set of functions finite in some bounded domain. As shown in \((^9)\), for \(p\ne 2\) the norms \(W_p^l\) and \(\mathfrak{W}_p^l\) are not equivalent.

It can be shown that on the set of functions finite in some bounded domain the norm \(\mathfrak{W}_p^l\) is equivalent to the norm \(B_{p,p}^l\), introduced in \((^6)\).

In the present paper we give a new, and, as it seems to us, simpler and more transparent method of proving a number of known facts, and also introduce several new theorems.

Theorem 1. If \(f\in\mathfrak{W}_p^l(E_n)\), \(m<n\),

\[ \sum_{i=1}^{n-m} r_i < l-\frac{n-m}{p}, \]

where the \(r_i\) are nonnegative integers, then

\[ D_{m+1}^{r_1}D_{m+2}^{r_2}\ldots D_n^{r_{n-m}} f \in \mathfrak{W}_p^{\,l-\Sigma r_i-\frac{n-m}{p}}(E_m) \]

for any fixed \(x_{m+1}, x_{m+2}, \ldots, x_n\), and

\[ \left\| D_{m+1}^{r_1}D_{m+2}^{r_2}\ldots D_n^{r_{n-m}} f \right\|_{\mathfrak{W}_p^{\,l-\Sigma r_i-\frac{n-m}{p}}(E_m)} \leqslant C\|f\|_{\mathfrak{W}_p^l(E_n)}. \]

Theorem 2. If functions
\(\varphi^{\,i}_{r_1^i r_2^i\ldots r_{n-m}^i}(x_1\ldots x_m)\in \mathfrak W_p^{\,l-s_i-\frac{n-m}{p}}(E_m)\) are given, where
\[ s_i=\sum_{k=1}^{n-m} r_k^i<l-\frac{n-m}{p}, \]
then one can construct a function \(f\in \mathfrak W_p^l(E_n)\) such that
\[ \varphi^{\,i}_{r_1^i r_2^i\ldots r_{n-m}^i}(x_1\ldots x_m) = D_{m+1}^{r_1^i}D_{m+2}^{r_2^i}\ldots D_n^{r_{n-m}^i} f \Big|_{x_{m+k}=0,\ k=1,\ldots,n-m}; \]
\[ \|f\|_{\mathfrak W_p^l(E_n)} \le C\sum_{s_i}\sum_{r_R} \left\|\varphi^{\,i}_{r_1^i r_2^i\ldots r_{n-m}^i}\right\|_ {\mathfrak W_p^{\,l-s_i-\frac{n-m}{p}}(E_m)}, \]
where \(C\) does not depend on \(\varphi^{\,i}_{r_1^i r_2^i\ldots r_{n-m}^i}\).

Theorem 3. If \(f\in\mathfrak W_p^l(E_n)\), \(l_1<l\), then \(f\in\mathfrak W_{p_1}^{l_1}(E_n)\), where
\[ l-\frac{n}{p}=l_1-\frac{n}{p_1},\qquad \|f\|_{\mathfrak W_{p_1}^{l_1}}\le C\|f\|_{\mathfrak W_p^l}. \]

Theorem 4. If \(f\in\mathfrak W_p^l(E_n)\), \(lp<n\), then \(f\in L_q(E_n)\), where
\[ q=\frac{np}{n-lp}, \]
and
\[ \|f\|_{L_q}\le C\|f\|_{\mathfrak W_p^l}. \]

Theorem 5. If \(f\in\mathfrak W_p^l(E_n)\), \(pl>n\), \(p(l-1)<n\), then \(f\in \mathrm{Lip}_\alpha\), i.e.
\[ \frac{|f(x+h)-f(x)|}{|h|^\alpha}\xrightarrow[|h|\to0]{}0, \qquad \alpha=l-\frac{n}{p}. \]

Theorem 6. If \(f\in\mathfrak W_p^l(E_n)\), \(pl>n\), \(p(l-1)=n\), then the function \(f\) satisfies Zygmund’s “condition \(\lambda^*\),” i.e.
\[ \frac{|f(x+h)+f(x-h)-2f(x)|}{|h|} \xrightarrow[|h|\to0]{}0. \]

The proofs of the theorems are comparatively simple. They are based on representations of the type of the equality
\[ f(x)=\frac{1}{h}\int_x^{x+h} f(\xi)\,d\xi -\int_x^{x+h}\frac{dt}{(t-x)^2}\int_x^t [f(t)-f(\xi)]\,d\xi, \tag{1} \]
obtained by V. P. Il’in \((^8)\) and valid for any \(h\).

Let us illustrate our method of proof on two characteristic examples. It is not difficult to show that the proof of Theorem 1 can be reduced to proving the fact that from \(f\in\mathfrak W_p^l(E_2)\) it follows that \(f\in\mathfrak W_p^{\,l-1/p}(E_1)\). Thus, let \(f\in\mathfrak W_p^l(E_2)\). We shall show that \(f\in\mathfrak W_p^{\,l-1/p}(E_1)\). We restrict ourselves to the case \(1/p<l<1\). Following Gagliardo \((^2)\), we estimate the \(\mathfrak W_p^{\,l-1/p}\)-norm of the function \(f(x)\) on the straight line \(x_1=x_2\), i.e.
\[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{|f(t,t)-f(\tau,\tau)|^p}{|t-\tau|^{lp}}\,d\tau\,dt. \]

We have
\[ f(t,t)-f(\tau,\tau) = [f(t,t)-f(t,\tau)]-[f(t,\tau)-f(\tau,\tau)]. \]

Let \(t\ge \tau\). Consider, for example, the second term. According to (1),
\[ f(t,\tau)-f(\tau,\tau) = \frac{1}{t-\tau}\int_\tau^t [f(t,\tau)-f(\xi,\tau)]\,d\xi + \int_\tau^t \frac{d\xi}{(\xi-\tau)^2} \int_\tau^\xi [f(\xi,\tau)-f(\eta,\tau)]\,d\eta \equiv F_1(t,\tau)+F_2(t,\tau). \]

Applying Hölder’s inequality, we obtain

\[ \begin{aligned} \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{|F_1|^p}{(t-\tau)^{lp}}\,d\tau &= \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{d\tau}{(t-\tau)^{p(l+1)}} \left| \int_{\tau}^{t} \frac{f(t,\tau)-f(\xi,\tau)}{(t-\xi)^{l+1/p}}(t-\xi)^{l+1/p}\,d\xi \right|^p \\ &\le \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{d\tau}{(t-\tau)^{p(l+1)}} \int_{\tau}^{t} \frac{|f(t,\tau)-f(\xi,\tau)|^p}{(t-\xi)^{1+pl}}\,d\xi \left[ \int_{\tau}^{t} (t-\xi)^{\frac{1+pl}{p-1}}\,d\xi \right]^{p-1} \\ &\le C\int_{-\infty}^{\infty} dt \int_{-\infty}^{t} d\tau \int_{\tau}^{t} \frac{|f(t,\tau)-f(\xi,\tau)|^p}{(t-\xi)^{1+pl}}\,d\xi \le C\|f\|_{\mathfrak W_p^l(E_2)}^p . \end{aligned} \]

We estimate the second term as follows:

\[ \begin{aligned} \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{|F_2|^p}{(t-\tau)^{lp}}\,d\tau &= \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{d\tau}{(t-\tau)^{lp}} \left| \int_{\tau}^{t}\frac{d\xi}{(\xi-\tau)^2} \int_{\tau}^{\xi} [f(\xi,\tau)-f(\eta,\tau)]\,d\eta \right|^p \\ &= \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{d\tau}{(t-\tau)^{lp}} \left| \int_{\tau}^{t}\frac{d\xi}{(\xi-\tau)^{2(1/p+1/p')}} \right. \\ &\qquad\qquad\left. \times \int_{\tau}^{\xi} \frac{f(\xi,\tau)-f(\eta,\tau)}{(\xi-\eta)^{l+1/p}} (\xi-\eta)^{(l+1/p)(1/p+1/p')} \left[ \frac{(\xi-\tau)^{2/p}}{(t-\tau)^{1/p}} \right]^{1/p'-1/p'} \,d\eta \right|^p \\ &\le \int_{-\infty}^{\infty} dt \int_{-\infty}^{t} \frac{d\tau}{(t-\tau)^{lp}} \int_{\tau}^{t}\frac{d\xi}{(\xi-\tau)^2} \int_{\tau}^{\xi} \frac{|f(\xi,\tau)-f(\eta,\tau)|^p}{(\xi-\eta)^{1+pl}} (\xi-\eta)^{l+1/p} \left[ \frac{(\xi-\tau)^{2/p}}{(t-\tau)^{1/p}} \right]^{p-1} \,d\eta \\ &\qquad\qquad\times \left\{ \int_{\tau}^{t}\frac{d\xi}{(\xi-\tau)^2} \int_{\tau}^{\xi} (\xi-\eta)^{l+1/p} \left[ \frac{(\xi-\tau)^{2/p}}{(t-\tau)^{1/p}} \right]^{-1} \,d\eta \right\}^{p-1} \\ &= C_1\int_{-\infty}^{\infty} dt \int_{\tau}^{\infty} \frac{dt}{(t-\tau)^{l+1/p'}} \int_{\tau}^{t}\frac{d\xi}{(\xi-\tau)^{2/p}} \int_{\tau}^{\xi} \frac{|f(\xi,\tau)-f(\eta,\tau)|^p}{(\xi-\eta)^{1+pl}} (\xi-\eta)^{l+1/p}\,d\eta \\ &= C_1\int_{-\infty}^{\infty} d\tau \int_{\tau}^{\infty} d\xi \int_{\tau}^{\xi} \frac{|f(\xi,\tau)-f(\eta,\tau)|^p}{(\xi-\eta)^{1+pl}} \frac{(\xi-\eta)^{l+1/p}}{(\xi-\tau)^{2/p}}\,d\eta \int_{\xi}^{\infty}\frac{dt}{(t-\tau)^{l+1/p'}} \\ &\le C_2\int_{-\infty}^{\infty} d\tau \int_{\tau}^{\infty} d\xi \int_{\tau}^{\xi} \frac{|f(\xi,\tau)-f(\eta,\tau)|^p}{(\xi-\eta)^{1+pl}}\,d\eta \le C\|f\|_{\mathfrak W_p^l(E_2)}^p . \end{aligned} \]

The remaining estimates are obtained analogously.

We now show how, with the aid of (1), one can prove Theorem 5 for the case \(n=1,\ l<1\).

On the basis of (1) one may write

\[ f(x+h)=\frac{1}{h}\int_x^{x+h} f(\xi)\,d\xi + \int_x^{x+h}\frac{dt}{(x+h-t)^2} \int_t^{x+h} [f(\xi)-f(t)]\,d\xi, \]

so that

\[ \frac{f(x+h)-f(x)}{h^{l-1/p}} \frac{1}{h^{l-1/p}} \int_x^{x+h}\frac{dt}{(x+h-t)^2} \int_t^{x+h} [f(\xi)-f(t)]\,d\xi - \frac{1}{h^{l-1/p}} \int_x^{x+h}\frac{dt}{(t-x)^2} \int_x^t [f(t)-f(\xi)]\,d\xi . \]

Since both terms are estimated in an analogous way, we shall restrict ourselves to estimating one of them:

\[ \begin{gathered} \frac{1}{h^{l-1/p}}\int_x^{x+h}\frac{dt}{(t-x)^2} \int_x^t \frac{f(t)-f(\xi)}{(t-\xi)^{l+1/p}}(t-\xi)^{l+1/p}\,d\xi \leq \\ \leq \frac{C_1}{h^{l-1/p}}\int_x^{x+h}\frac{dt}{(t-x)^{1-l}} \left|\int_x^t \frac{|f(t)-f(\xi)|^p}{|t-\xi|^{1+pl}}\,d\xi\right|^{1/p} \leq \\ \leq \frac{C_1}{h^{l-1/p}} \left|\int_x^{x+h}\frac{dt}{|t-x|^{(1-l)p'}}\right|^{1/p'} \left|\int_x^{x+h}dt\int_x^t \frac{|f(t)-f(\xi)|^p}{|t-\xi|^{1+pl}}\,d\xi\right|^{1/p} = \\ = C_2 \left|\int_x^{x+h}dt\int_x^t \frac{|f(t)-f(\xi)|^p}{|t-\xi|^{1+pl}}\,d\xi\right|^{1/p} \underset{h\to 0}{\longrightarrow}0, \end{gathered} \]

which was to be proved.

In conclusion I express my gratitude to V. P. Il’in, who gave many useful suggestions.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Received
18 III 1960

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