Multiplicative Estimates for Norms of Derivatives on a Domain

Abstract

Multiplicative estimates for the L_p-norms of derivatives on a domain with flexible cone condition are established.

On a domain G of Euclidean space ${{\mathbb{R}}^{n}}$ with flexible cone condition (see the definition below), we establish multiplicative estimates for the L_p-norms of derivatives of a function in terms of the norms of its other derivatives.

Let $\mathbb{N}$ be the set of positive integers; ${{\mathbb{N}}_{0}} = \mathbb{N} \cup \{ 0\}$; $n \in \mathbb{N}$; ${{\mathbb{R}}^{n}}$ be n-dimensional Euclidean space of points $x = ({{x}_{1}},\; \ldots ,\;{{x}_{n}})$; $1 \leqslant p < \infty$; and ${{L}_{p}}(G)$ be the Lebesgue space of functions defined on an open set $G \subset {{\mathbb{R}}^{n}}$ and equipped with the norm

$$\left\| {f|{{L}_{p}}(G)} \right\| = {{\left( {\int\limits_{{{\mathbb{R}}^{n}}} {{{{\left| {f(x)} \right|}}^{p}}} dx} \right)}^{{1/p}}},\quad {{L}_{p}} = {{L}_{p}}({{\mathbb{R}}^{n}}).$$

All considered functions $f{\text{:}}\;G \to \mathbb{R}$ and their derivatives ${{D}^{\alpha }}f$ and $D_{i}^{s}f$ are assumed to be locally summable on their domains, and ${{G}_{\varepsilon }} = \{ x \in G{\text{:}}\;{\text{dist}}(x,\partial G)$ > ε} for ε > 0. For $\varphi {\text{:}}\;[0,T] \to \mathbb{R}$, we set

$${\text{||}}\varphi |L_{{2,s}}^{ * }{\text{||}} = {{\left\{ {\int\limits_0^T {{{{({{t}^{{ - s}}}\varphi (t))}}^{2}}} \tfrac{{dt}}{t}} \right\}}^{{1/2}}},\,\,L_{2}^{ * } = L_{{2,0}}^{ * }.$$

For comparison, we present the classical result of Gagliardo [3] and Nirenberg [4] for a domain $G \subset {{\mathbb{R}}^{n}}$ with a sufficiently smooth boundary:

$$\begin{gathered} {\text{||}}{{D}^{\beta }}f|{{L}_{q}}(G){\text{||}} \leqslant C{{\left( {\sum\limits_{|\alpha | = s} {{\text{||}}{{D}^{\alpha }}f|{{L}_{{{{p}_{1}}}}}(G){\text{||}}} } \right)}^{\theta }} \\ \times {{\left\| {f|{{L}_{{{{p}_{2}}}}}(G)} \right\|}^{{1 - \theta }}} + C\left\| {f|{{L}_{r}}(G)} \right\| \\ \end{gathered} $$

for $1 \leqslant {{p}_{1}},{{p}_{2}},q,r < \infty$, $s \in \mathbb{N}$, $\frac{{\left| \beta \right|}}{s} \leqslant \theta < 1$, and

$$\left| \beta \right| - \frac{n}{q} = \theta \left( {s - \frac{n}{{{{p}_{1}}}}} \right) + (1 - \theta )\left( { - \frac{n}{{{{p}_{2}}}}} \right).$$

Definition 1. A domain $G \subset {{\mathbb{R}}^{n}}$ is called a domainwith flexible cone condition if, for some $T \in (0,1]$ and κ > 0 and for any $x \in G$, there exists a piecewise smooth path

$$\gamma = {{\gamma }_{x}}{\text{:}}\;[0,T] \to G,\quad \gamma (0) = x,\quad \left| {\gamma {\text{'}}} \right| \leqslant 1\quad {\text{a}}.{\text{e}}.,$$

(1)

such that

$${\text{dist}}(\gamma (t),{{\mathbb{R}}^{n}}{\backslash }G) \geqslant \kappa t\quad {\text{for}}\quad 0 < t \leqslant T.$$

We prove the following result.

Theorem 1.Let G be a domain with flexible cone condition. Suppose that, for$J \geqslant 2$, $j = 1,\; \ldots ,\;J$, 1 < p_j < $\infty$, 1 < q < $\infty$, ${{s}_{j}} \in {{\mathbb{N}}_{0}}$, $0 < {{\theta }_{j}}$ < 1, and$\sum\limits_1^J {{{\theta }_{j}}} = 1$, $1 \leqslant r \leqslant q$,

$$\frac{1}{q} \leqslant \sum\limits_{j = 1}^J {\frac{{{{\theta }_{j}}}}{{{{p}_{j}}}}} ,\quad {\text{|}}\alpha {\text{|}} - \frac{n}{q} = \sum\limits_{j = 1}^J {{{\theta }_{j}}} \left( {{{s}_{j}} - \frac{n}{{{{p}_{j}}}}} \right).$$

Then, for some C > 0 and a sufficiently small ε > 0 independent of f,

$$\begin{gathered} {\text{||}}{{D}^{\alpha }}f|{{L}_{q}}(G){\text{||}} \\ \leqslant \;C\prod\limits_{j = 1}^J {{{{\left( {\sum\limits_{i = 1}^n {{\text{||}}D_{i}^{{{{s}_{j}}}}f|{{L}_{{{{p}_{j}}}}}(G){\text{||}}} } \right)}}^{{{{\theta }_{j}}}}}} + C\left\| {f|{{L}_{r}}({{G}_{\varepsilon }})} \right\| \\ \end{gathered} $$

(2)

for any function f such that the right-hand side of the inequality is finite.

Our consideration is based on integral representations of a function over a flexible cone in terms of its derivatives and on estimates of convolution operators.

We will use the following notation. The symbol $G$ stands for a domain in ${{\mathbb{R}}^{n}}$ with flexible cone condition that does not coincides with ${{\mathbb{R}}^{n}}$. For $t > 0$, $E \subset {{\mathbb{R}}^{n}}$, and $y \in {{\mathbb{R}}^{n}}$, we set

$$\begin{gathered} y + tE: = \left\{ {x{\text{:}}\,x = y + tz,\;z \in E} \right\}, \\ B(x,t): = \left\{ {y{\text{:}}\,{\text{|}}y - x{\text{|}} < t} \right\} = x + B(0,t). \\ \end{gathered} $$

Let χ be the indicator function of the ball $B(0,1)$ or the interval [–1, 1], and let $r(x) = {\text{dist}}(x,{{R}^{n}}{\backslash }G)$.

The following remark is important for the subsequent presentation.

Remark 1. Without loss of generality, we may assume that the paths in the definition of a domain with flexible cone condition have the following additional property: for some ${{\varepsilon }_{0}} > 0$, ${{\gamma }_{x}}(t) = x$ for $x \in G$ and $0 < t \leqslant min\left\{ {{{\varepsilon }_{0}}r(x),T} \right\} = :\rho (x)$.

Given below, the integral representation of a function in terms of its derivatives is an analogue of integral representations 7(75), 7(76) in [1].

Let $\vartheta \in {{C}^{\infty }}(\mathbb{R})$, $\vartheta (u) = 0$ for $u \leqslant - \delta$, $\vartheta (u) = 1$ for $u \geqslant \delta$, $\tau ,{{\tau }_{1}} \in \mathbb{R}$, $\delta \in (0,1)$ be sufficiently small, and

$$\omega (u,\tau ) = \frac{{{{d}^{k}}}}{{d{{u}^{k}}}}\left( {\frac{{{{u}^{{k - 1}}}}}{{(k - 1)!}}\vartheta (u - \tau )} \right).$$

(3)

For $f \in L(G,{\text{loc}})$ and $0 < t \leqslant T,$ we introduce the average

$$\begin{gathered} {{f}_{t}}(x) = {{t}^{{ - 2n}}}\iint {\Omega \left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t) - x}}{t}} \right)\Omega \left( {\frac{z}{t}} \right)} \\ \times f(x + y + z)dydz, \\ \end{gathered} $$

where

$$\begin{gathered} \Omega (y,z) = \prod\limits_{i = 1}^n \omega ({{y}_{i}},{{z}_{i}}), \\ \omega (u) = \omega (u,0),\quad \Omega (y) = \Omega (y,0), \\ \end{gathered} $$

$${\text{|}}D_{y}^{\alpha }\Omega (y,z){\text{|}} \leqslant {{C}_{\alpha }}\chi \left( {\frac{{y - z}}{{\delta \sqrt n }}} \right)\quad {\text{for}}\quad \left| z \right| \leqslant 1.$$

Let $f_{t}^{{(\alpha )}}(x)$ denote the derivative ${{D}^{\alpha }}$ of ${{f}_{t}}(x)$ calculated with fixed x in ${{\gamma }_{x}}(t) - x$ and in $\gamma _{x}^{'}(t)$. Applying the Newton–Leibniz formula with respect to t, for $0 < \varepsilon < T$, we obtain

$$f_{\varepsilon }^{{(\alpha )}}(x) = f_{T}^{{(\alpha )}}(x) + {{( - 1)}^{{|\alpha |}}}\int\limits_\varepsilon ^T {\int {\int {{{t}^{{ - 1 - 2n - |\alpha |}}}} } } $$

$$ \times \;\sum\limits_{i = 1}^n {\left[ {{{D}_{i}}{{\Phi }_{i}}\left( {\frac{y}{t}} \right.} \right.} ,\frac{{{{\gamma }_{x}}(t) - x}}{t},\gamma _{x}^{'}(t))D_{i}^{{k - 1}}{{D}^{\alpha }}\Omega \left( {\frac{z}{t}} \right)$$

$$\begin{gathered} + \,{{D}_{i}}\Omega \left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t)\, - \,x}}{t}} \right)D_{i}^{{k - 1}}{{D}^{\alpha }}{{\Phi }_{i}}\left. {\left( {\frac{z}{t}} \right)} \right] \\ \times \,f(x\, + \,y\, + \,z)dydzdt, \\ \end{gathered} $$

(4)

where ${{\Phi }_{i}}( \cdot ,z,w) \in C_{0}^{\infty }({{\mathbb{R}}^{n}})$, ${{\Phi }_{i}}(y) = {{\Phi }_{i}}(y,0,0)$, and, for some M > 0,

$$\begin{gathered} \left| {\frac{{{{\partial }^{j}}}}{{\partial y_{l}^{j}}}{{\Phi }_{i}}(y,z,w)} \right| \leqslant M\chi \left( {\frac{{y - z}}{{\delta \sqrt n }}} \right)\quad {\text{for}}\quad 1 \leqslant l \leqslant n, \\ j = 0,1,\; \ldots ,\;k,\quad \left| y \right| \leqslant 1,\quad \left| w \right| \leqslant 1. \\ \end{gathered} $$

Here, D_i and ${{D}^{\alpha }}$ denote the derivatives of functions with respect to the first argument.

To represent the integral operators on the right-hand side of (4) in the form of the product of two operators, on $\left\{ {(x,t){\text{:}}\;x \in G,0 < t \leqslant r(x)} \right\}$, we introduce the functions

$${{\varphi }_{i}}(x,t) = \int {{{t}^{{ - n}}}} D_{i}^{{k - 1}}{{D}^{\alpha }}\Omega \left( {\frac{z}{t}} \right)f(x + z)dz,$$

(5)

$${{\psi }_{i}}(x,t) = \int {{{t}^{{ - n}}}} D_{i}^{{k - 1}}{{D}^{\alpha }}{{\Phi }_{i}}\left\{ {\frac{z}{t}} \right\}f(x + z)dz.$$

(6)

Formula (4) can be rewritten as

$$\begin{gathered} f_{\varepsilon }^{{(\alpha )}}(x) = f_{T}^{{(\alpha )}}(x) + {{( - 1)}^{{|\alpha |}}}\int\limits_\varepsilon ^T {\int {{{t}^{{ - 1 - n - |\alpha |}}}} } \\ \times \;\sum\limits_{i = 1}^n {\left[ {{{D}_{i}}{{\Phi }_{i}}\left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t) - x}}{t},\gamma _{x}^{'}(t)} \right)} \right.} {{\varphi }_{i}}(x + y,t) \\ + \;{{D}_{i}}\Omega \left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t) - x}}{t}} \right){{\psi }_{i}}(x + y,t)]dydt,\quad 0 < \varepsilon < T. \\ \end{gathered} $$

(7)

Remark 2.Formula (7) holds for ε = 0 with $f_{0}^{{(\alpha )}}(x)$ = D^αf(x) if the derivative ${{D}^{\alpha }}f$ is locally summable on G. It is derived by passage to the limit, since $f_{\varepsilon }^{{(\alpha )}}(x)$ → D^αf(x) as ε → 0 at each Lebesgue point of the function ${{D}^{\alpha }}f$ and also in the sense of $L(G,{\text{loc}})$. Formula (7) with ε > 0 makes it possible to apply estimates of integral operators with locally summable kernels.

Theorem 2 (L. Hörmander, J.T. Schwartz, H. Tribel). Let B₁and B₂be two Banach spaces, G be a domain in${{\mathbb{R}}^{n}}$,$K(y)$for$y \in {{\mathbb{R}}^{n}}$be a linear bounded operator from${{B}_{1}}$to${{B}_{2}},$and$y \to K(y)$be a locally summable function on${{\mathbb{R}}^{n}}$.

Suppose that, on the set$^{ \circ }{{L}_{\infty }}({{\mathbb{R}}^{n}},{{B}_{1}})$consisting of${{B}_{1}}$-valued strongly measurable functions with a compact support,

$$(Af)(x) = \int K (x,y)f(y)dy.$$

For some${{p}_{0}} \in (1,\infty )$, $N > 0$, and$R > 0$, suppose that

$$\begin{gathered} ({\text{i}})\,\,\,{\text{||}}Af|{{L}_{{{{p}_{0}}}}}({{\mathbb{R}}^{n}},{{B}_{2}}){\text{||}} \leqslant N{\text{||}}\,f|{{L}_{{{{p}_{0}}}}}({{\mathbb{R}}^{n}},{{B}_{1}})\,{\text{||}}\,, \hfill \\ ({\text{ii}})\,\,\int\limits_{|x| > R|y|} {\left\| {K(x - y) - K(x)|{{B}_{1}} \to {{B}_{2}}} \right\|} \,dx \leqslant N\,\,\forall y \in {{\mathbb{R}}^{n}}. \hfill \\ \end{gathered} $$

Then, for all$p \in (1,\infty )$,

$$\left\| {Af|{{L}_{p}}({{\mathbb{R}}^{n}},{{B}_{2}})} \right\| \leqslant {{C}_{p}}N\left\| {f|{{L}_{p}}({{\mathbb{R}}^{n}},{{B}_{1}})} \right\|,$$

where C_pdoes not depend on f and N.

This theorem is given in [5, Section 2.2.2], where historical references can also be found. It is applied in the cases ${{B}_{i}} = \mathbb{R}$, $L_{2}^{ * }$ (i = 1, 2).

Lemma 1. Suppose that

$$\begin{gathered} \Omega \in C_{0}^{\infty }(B(0,1)),\quad \int \Omega (x)dx = 0, \\ 1 < p < \infty ,\quad 0 < T < \infty . \\ \end{gathered} $$

Then the operator

$$\begin{gathered} Sf(x,t) = \int {{{t}^{{ - n}}}} \Omega \left( {\frac{y}{t}} \right)f(x + y)dy, \\ S{\text{:}}\;{{L}_{p}}({{\mathbb{R}}^{n}}) \to {{L}_{p}}({{\mathbb{R}}^{n}},L_{2}^{ * }), \\ \end{gathered} $$

is a bounded operator with a norm estimate independent of T.

The proof of a version of this lemma for $T = \infty$ follows from [6]. For the case under consideration, the argument is similar and can be found in [2].

Lemma 2. Suppose that

$$\Omega \in C_{0}^{\infty }(B(0,1)),\quad \int \Omega (x)dx = 0,$$

$$\begin{gathered} 1 < p \leqslant q < \infty ,\quad 0 \leqslant \mu = \frac{n}{p} - \frac{n}{q}, \\ and\quad 0 < \varepsilon < T < \infty . \\ \end{gathered} $$

Then the operator

$$\begin{gathered} {{R}_{\mu }}f(x) = \int {\int\limits_\varepsilon ^T {{{t}^{{ - n + \mu }}}} } \Omega \left( {\frac{y}{t}} \right)f(x + y,t)\frac{{dt}}{t}dy, \\ {{R}_{\mu }}{\text{:}}\,\,{{L}_{p}}(L_{2}^{ * }) \to {{L}_{q}}, \\ \end{gathered} $$

(8)

is a bounded operator with a norm estimate independent of ε, T.

In the case q = p and μ = 0, the lemma is proved by applying Theorem 2 and the proof can be found in [2].

In the case $1 < p < q < \infty$ and $\mu = \frac{n}{p} - \frac{n}{q}$, the lemma is proved by applying the Hölder inequality with respect to t and then the Hardy–Littlewood inequality.

Lemma 3.Let$\tilde {\chi }(x, \cdot )$be the indicator function of the flexible cone$\bigcup\limits_{0 < t < T} B ({{\gamma }_{x}}(t),\kappa t)$, $1 < p \leqslant q < \infty$, and$0 \leqslant \mu$=$\frac{n}{p} - \frac{n}{q}$.

Then the operator

$$\begin{gathered} ({{A}_{\mu }}f)(x) = \int {\frac{{\tilde {\chi }(x,x + y)f(x + y)dy}}{{{{{\left( {r(x) + \left| y \right|} \right)}}^{{n - \mu }}}}}} , \\ {{A}_{\mu }}{\text{:}}\,\,{{L}_{p}}(G) \to {{L}_{q}}(G), \\ \end{gathered} $$

is bounded.

For the proof in the case p = q and μ = 0, see [1, Section 10.1] or [2]. To prove the lemma in the case $1 < p < q < \infty$, it suffices to majorize the kernel of the operator by the function $C{{\left| y \right|}^{{\mu - n}}}$ and to apply the Hardy–Littlewood inequality.

Lemma 4.Suppose that${{\varepsilon }_{1}} \in (0,1]$,

$$\Omega \in C_{0}^{\infty }(B(0,1)),\quad \int \Omega (x)dx = 0,$$

$1 < p \leqslant q < \infty$, $0 \leqslant \mu = \frac{n}{p} - \frac{n}{q}$, and$0 < \varepsilon < T < \infty$.

Then the operator

$$\begin{gathered} R_{\mu }^{{(1)}}f(x) = \int {\int\limits_\varepsilon ^T {\chi \left( {\frac{t}{{{{\varepsilon }_{1}}r(x)}}} \right)} } \,{{t}^{{ - n + \mu }}}\Omega \left( {\frac{y}{t}} \right)f(x + y,t)\frac{{dt}}{t}dy, \\ R_{\mu }^{{(1)}}{\text{:}}\,{{L}_{p}}(G,L_{2}^{ * }) \to {{L}_{q}}(G), \\ \end{gathered} $$

(9)

is a bounded operator with a norm estimate independent of ε, T.

As proof, we note that the operator

$$R_{\mu }^{{(2)}}f(x) = \int {\int\limits_\varepsilon ^T \chi } \left( {\frac{{{{\varepsilon }_{1}}r(x)}}{t}} \right){{t}^{{ - n + \mu }}}\Omega \left( {\frac{y}{t}} \right)f(x + y,t)\frac{{dt}}{t}dy$$

is a bounded one from ${{L}_{p}}(G,L_{2}^{ * })$ to L_p(G) with a norm estimate independent of T. It remains to be noted that, for $f \in {{L}_{p}}(G,L_{2}^{ * })$, it is true that $R_{\mu }^{{(1)}}f = {{R}_{\mu }}f - R_{\mu }^{{(2)}}f$.

The proof of Theorem 1 can be sketched as follows. Define

$$\begin{gathered} s = \sum\limits_1^J {{{\theta }_{j}}} {{s}_{j}},\quad \frac{1}{p} = \sum\limits_1^J {\frac{{{{\theta }_{j}}}}{{{{p}_{j}}}}} , \\ \mu = \frac{n}{p} - \frac{n}{q},\quad {\text{so}}\quad {\text{that}}\quad s - \left| \alpha \right| = \mu . \\ \end{gathered} $$

Formula (7) is rewritten as

$$f_{\varepsilon }^{{(\alpha )}} = f_{T}^{{(\alpha )}} + {{( - 1)}^{{|\alpha |}}}\int\limits_\varepsilon ^T {\int {{{t}^{{ - 1 - n - \mu }}}} } $$

$$\begin{gathered} \times \;\sum\limits_{i = 1}^n {\left[ {{{D}_{i}}{{\Phi }_{i}}\left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t) - x}}{t},\gamma _{x}^{'}} \right)} \right.} \,{{t}^{{ - s}}}{{\varphi }_{i}}(x + y,t) \\ + \;{{D}_{i}}\Omega \left. {\left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t) - x}}{t}} \right){{t}^{{ - s}}}{{\psi }_{i}}(x + y,t)} \right]dydt. \\ \end{gathered} $$

Applying the Young inequality yields the estimate

$${\text{||}}f_{T}^{{(\alpha )}}|{{L}_{q}}(G){\text{||}} \leqslant C\left\| {f|{{L}_{r}}({{G}_{\varepsilon }})} \right\|.$$

(10)

We estimate only one term on the right-hand side of the last equality (the estimates for the other terms are similar):

$$I(x) = {{I}_{1}}(x) + {{I}_{2}}(x)$$

$$\begin{gathered} = \int {\int\limits_\varepsilon ^T {{{t}^{{ - 1\, - \,n\, - \,\mu }}}\left[ {\chi \left( {\frac{t}{{{{\varepsilon }_{1}}r(x)}}} \right)} \right.} } {{D}_{i}}{{\Phi }_{i}}\left( {\frac{y}{t}} \right) + \chi \left( {\frac{{{{\varepsilon }_{1}}r(x)}}{t}} \right) \times \\ \times \;{{D}_{i}}{{\Phi }_{i}}\left. {\left( {\frac{y}{t},\frac{{{{\gamma }_{x}}(t) - x}}{t},\gamma {{{\text{'}}}_{x}}} \right)} \right]{{t}^{{ - s}}}{{\varphi }_{i}}(x + y,t)dtdy. \\ \end{gathered} $$

(11)

Here, ${{\varepsilon }_{1}} = {{\varepsilon }_{1}}(\kappa ) > 0$ is sufficiently small.

Next, I₁ is estimated with the help of Lemma 4, while I₂ is estimated by applying the Hölder inequality with respect to t and Lemma 3. As a result,

$$\left\| {I|{{L}_{p}}(G)} \right\| \leqslant C{\text{||}}{{\varphi }_{i}}|{{L}_{p}}(G,L_{{2,s}}^{ * }){\text{||}}.$$

(12)

Applying the Hölder inequality twice $\left( {\tfrac{{}}{{p{{\theta }_{j}}}}} \right.$first, with exponents $\tfrac{1}{{{{\theta }_{j}}}}$ and, then, with exponents $\left. {\tfrac{{{{p}_{j}}}}{{p{{\theta }_{j}}}}} \right)$ yields

$${\text{||}}{{\varphi }_{i}}|{{L}_{p}}(G,L_{{2,s}}^{ * }){\text{|}}{{{\text{|}}}^{p}} \leqslant \int\limits_G {\prod\limits_{j = 1}^J {{{{\left( {\int\limits_0^T {{{{[{{t}^{{ - {{s}_{j}}}}}{{\varphi }_{i}}(x,t)]}}^{2}}} \frac{{dt}}{t}} \right)}}^{{\tfrac{1}{2}{{\theta }_{j}}p}}}} } dx$$

$$\begin{gathered} \leqslant \;{{\prod\limits_{j = 1}^J {\left\{ {\int\limits_G {{{{\left( {\int\limits_0^T {{{{[{{t}^{{ - {{s}_{j}}}}}{{\varphi }_{i}}(x,t)]}}^{2}}} \frac{{dt}}{t}} \right)}}^{{\tfrac{1}{2}{{p}_{j}}}}}} } \right\}} }^{{\tfrac{{{{\theta }_{j}}}}{{{{p}_{j}}}}p}}} \\ = \;\prod\limits_{j = 1}^J {{{{\left\| {{{{\left( {\int\limits_0^T {{{{[{{t}^{{ - {{s}_{j}}}}}{{\varphi }_{i}}(x,t)]}}^{2}}} \frac{{dt}}{t}} \right)}}^{{\tfrac{1}{2}}}}|{{L}_{{{{p}_{j}}}}}(G)} \right\|}}^{{{{\theta }_{j}}p}}}} . \\ \end{gathered} $$

(13)

Assuming that $k \geqslant 2 + ma{{x}_{j}}{{s}_{j}}$, we have

$$\begin{gathered} {{t}^{{ - {{s}_{j}}}}}{{\varphi }_{i}}(x,t) = {{( - 1)}^{{{{s}_{j}}}}}\chi \left( {\frac{t}{{r(x)}}} \right) \\ \times \;\int {{{t}^{{ - n}}}} D_{i}^{{k - 1 - {{s}_{j}}}}\Omega \left( {\frac{y}{t}} \right)D_{i}^{{{{s}_{j}}}}f(x + y)dy. \\ \end{gathered} $$

By Lemma 4,

$${\text{||}}{{\varphi }_{i}}|{{L}_{p}}(G,L_{{2,s}}^{ * }){\text{||}} \leqslant C{\text{||}}D_{i}^{{{{s}_{j}}}}f|{{L}_{{{{p}_{j}}}}}(G){\text{||}}.$$

(14)

An estimate similar to (14) can also be derived for ψ_i in place of φ_i.

Combining these results with (10)–(14), we conclude that the set $\{ f_{\varepsilon }^{{(\alpha )}}{\text{:}}\,\,0 < \varepsilon < \varepsilon {\text{'}}\}$ with some $\varepsilon {\text{'}} > 0$ is bounded in $L(G,{\text{loc}})$ and, hence, is weakly compact (see, for example, [7]). Since the generalized differentiation operator is weakly closed, there exists ${{D}^{\alpha }}f \in L(G$, loc) and (11) holds for ε = 0 ($f_{0}^{{(\alpha )}}$ = D^αf). Therefore, the assertion of the theorem is proved.

Estimates of the form (2) in the case $G = {{\mathbb{R}}^{n}}$ can be found in [1]. Typically, they lack the last term on the right-hand side of (2). The results of Ven-tuan and M. Troisi concerning multiplicative estimates for norms of derivatives of functions from $C_{0}^{\infty }({{( - 1,1)}^{n}})$ are also given in [1].