Georgian Mathematical Journal 1(1994), No. 5, 495-503

TWO-WEIGHTED ESTIMATES FOR SOME INTEGRAL

TRANSFORMS IN THE LEBESGUE SPACES WITH MIXED NORM AND IMBEDDING THEOREMS

V.KOKILASHVILI

Abstract. Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain the refinements of the well-known imbedding theorems in the scale of weighted Lebesgue spaces.

Two-weighted inequalities are obtained for anisotropic potentials in Lebesgue spaces with mixed norm. These estimates are used to prove imbedding theorems for different metrics and different dimensions for weighted spaces of anisotropic Bessel potentials.

Nonweighted cases were previously treated in [1–3]. One-weighted estimates for isotropic Bessel potentials can be found in [4].

1. A measurable almost everywhere positive function ϱ : ℝ^{1} → ℝ^{1} will be called a weight function. Let w = (w_{1}, w_{2}, …, w_{n}) be a vector-function where w_{i} (i = 1, 2, …, n) is a weight function. By definition a measurable function f(x) = f(x_{1}, x_{2}, …, x_{n}) given on the n-dimensional space ℝ^{n} belongs to L_{w}^{p}, p = (p_{1}, p_{2}, …, p_{n}), 1 < p_{i} < ∞ (1 = 1, 2, …, n), if the norm

∥f∥_{L_{w}^{p}(ℝ^{n})} =

=(∫_{−∞}^{∞} w_{1}^{p_{1}}(x_{1})dx_{1}(∫_{−∞}^{∞} w_{2}^{p_{2}}(x_{2})dx_{2}…(∫_{−∞}^{∞} f(x)w_{n}^{p_{n}}(x_{n})dx_{n})^{\frac{p_{n−1}}{pn}} )…)^{\frac{p_{1}}{p_{2}}})^{\frac{1}{p_{1}}}

is finite.

We shall introduce a class of pairs of weight functions.

For a given number r, 1 < r < ∞, we write r′ = \frac{r}{r−1}.

495

1991 Mathematics Subject Classification. 42B25, 46E30.

1072-947X/1994/0900-0495$07.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

Definition 1. A pair of weight functions (ϱ, σ) given on ℝ^{1} will be said to belong to the class G_{β}^{s,r}, 0 < β < 1, 1 < r < s < ∞, if the conditions

sup ( ∫_{I} ϱ^{s}(t)dt)^{\frac{1}{s}} ( ∫_{ℝ^{1}} \frac{σ^{−r′}(t)}{(|I| + |t_{I} − t|)^{βr′}} dt)^{\frac{1}{r′}} < ∞,  (1.1)

sup ( ∫_{I} σ^{−r′}(t)dt)^{\frac{1}{r′}} ( ∫_{ℝ^{1}} \frac{ϱ^{s}(t)}{(|I| + |t_{I} − t|)^{sβ}} dt)^{\frac{1}{s}} < ∞,  (1.2)

are fulfilled, where the supremum is taken over all bounded one-dimensional intervals I, with centre and length, t_{I} and |I| respectively.

In the sequel we shall proceed from

Theorem A [5–7]. The fractional integral

I_{γ}(f)(x) = ∫_{−∞}^{∞} \frac{f(τ)}{|t − τ|^{1−γ}} dτ, 0 < γ < 1,

generates a continuous operator from L_{σ}^{r}(ℝ^{1}) into L_{ϱ}^{s}(ℝ^{1}) if and only if (ϱ, σ) ∈ G_{1−γ}^{s,r}.

Let numbers a_{j} > 0 (j = 1, 2, …, n) be given. For x=(x_{1}, x_{2}, …, x_{n}) we set

|x|_{a} = ( ∑_{i=1}^{n} |x|^{\frac{2}{a_{j}}} )^{\frac{1}{2}} .

It is obvious that for a_{j} = 1 (j = 1, 2, …, n) we obtain an usual Euclidian distance.

Theorem 1. Let w = (w_{1}, w_{2}, …, w_{n}), v = (v_{1}, v_{2}, …, v_{n}), where w_{i} and v_{i} (i = 1, 2, …, n) are weight functions given on ℝ^{n}. We set

Kf(x) = ∫_{ℝ^{n}} \frac{f(y)}{|x − y|_{a}^{μ}} dy,

where

μ = ∑_{j=1}^{n} a_{j}(1 − γ_{j}), 0 < γ_{j} < 1 (j = 1, 2, …, n).

If 1 < p_{i} < q_{i} < ∞, (v_{i}, w_{i}) ∈ G_{1−γ_{i}}^{q_{i},p_{i}} (i = 1, 2, …, n), then there exists a positive number c such that the inequality ∥Kf∥_{L_{v}^{q}} ≤ c∥f∥_{L_{w}^{p}} holds for any f ∈ L_{w}^{p}.

The proof of Theorem 1 will be based on several lemmas. The first lemma is a weighted analogue of the well–known Hardy–Littlewood inequality (see [8], Theorem 382).

Lemma 1. Let (ϱ, σ) ∈ G_{1−γ}^{s,r}, 1 < r < s < ∞, 0 < γ < 1. Then there exists a constant c > 0, such that the inequality

| ∫_{−∞}^{∞} ∫_{−∞}^{∞} \frac{φ(x)ψ(y)}{|x − y|^{1−γ}} dx dy| ≤ c∥φ∥_{L_{σ}^{r}}∥ψ∥_{L_{1/ϱ}^{s′}}  (1.3)

holds for any arbitrary φ ∈ L_{σ}^{r}(ℝ^{1}) and ψ ∈ L_{1/ϱ}^{s′}(ℝ^{1}).

Proof. By virtue of the Hölder inequality we have

| ∫_{−∞}^{∞} ∫_{−∞}^{∞} \frac{φ(x)ψ(y)}{|x − y|^{1−γ}|}dx dy| ≤

≤ ∥φσ∥_{L^{r}}( ∫_{ℝ^{1}} ( ∫_{ℝ^{1}} \frac{|ψ(y)|dy}{|x − y|^{1−γ}} )^{r′} \frac{1}{σ^{r′}(x)}dx)^{\frac{1}{r′}} .

From the condition (ϱ, σ) ∈ G_{1−γ}^{s,r} readily follows that (\frac{1}{σ} , \frac{1}{ϱ}) ∈ G_{1−γ}^{r′,s′}. Using Theorem A we obtain the estimate

| ∫_{−∞}^{∞} ∫_{−∞}^{∞} \frac{φ(x)ψ(y)}{|x − y|^{1−γ}} dx dy| ≤ c∥φ∥_{L_{σ}^{r}} · ∥ψ∥_{L_{1/ϱ}^{s′}}. ■

Bellow we shall set ϱ = (\frac{1}{v_{1}} , \frac{1}{v_{2}} , …, \frac{1}{v_{n}}) for v = (v_{1}, v_{2}, …, v_{n}).

Lemma 2. Let 1 < p_{i} < q_{i} < ∞, (v_{i}, w_{i}) ∈ G_{1−γ_{i}}^{q_{i},p_{i}}, 0 < γ_{i} < 1. Then there exists a positive constant c such that

| ∫_{ℝ^{n}} ∫_{ℝ^{n}} \frac{φ(x)ψ(y)}{|x − y|_{a}^{μ}}dx dy| ≤ c∥φ∥_{L_{w}^{p}} · ∥ψ∥_{L_{ϱ}^{q′}}

for arbitrary φ ∈ L_{w}^{p}(ℝ^{n}) and ψ ∈ L_{ϱ}^{q′}(ℝ^{n}).

Proof. We shall apply the reduction technique to the one-dimensional case. Let x′ = (x_{1}, x_{2}, …, x_{n−1}), y′ = (y_{1}, y_{2}, …, y_{n−1}) and

μ′ = ∑_{j=1}^{n−1} a_{j}(1 − γ_{j}), a′ = (a_{1}, a_{2}, …, a_{n−1}).

It readily follows that

|x − y|_{a}^{μ} = |x − y|_{a}^{μ′}|x − y|_{a}^{a_{n}(1−γ_{n})} ≥ |x′ − y′|_{a′}^{μ′}|x_{n} − y_{n}|^{1−γ_{n}}.

Therefore by Lemma 1 we obtain

| ∫_{ℝ^{n}} ∫_{ℝ^{n}} \frac{φ(x)ψ(y)}{|x − y|_{a}^{μ}}dx dy|≤ ∫_{ℝ^{n−1}} ∫_{ℝ^{n−1}} \frac{dx′dy′}{|x′ − y′|_{a}^{μ′}} ∫_{−∞}^{∞} ∫_{−∞}^{∞} \frac{|φ(x)| |ψ(y)|}{|x_{n} − y_{n}|^{1−γ_{n}}} dx_{n}dy_{n} ≤

≤ c ∫_{ℝ^{n−1}} ∫_{ℝ^{n−1}} \frac{F (x′)H(y′)}{|x′ − y′|_{a′}^{μ′}}dx′dy′,

where

F (x′) = ( ∫_{ℝ^{1}} |φ(x)|^{p_{n}}w_{n}^{p_{n}}(x_{n})dx_{n})^{\frac{1}{pn}} ,

H(x′) = ( ∫_{ℝ^{1}} |ψ(y)|^{q′_{n}}v_{n}^{−q′_{n}}(y_{n})dy_{n})^{\frac{1}{q′_{n}}} .

Further reduction leads us to the proof of Lemma 2.

Proof of Theorem 1. By the property of the norm and also by Lemma 2 we have

∥Kf∥_{L_{v}^{q}} = sup | ∫_{ℝ^{n}} Kf(x)g(x)dx|,

where the least upper bound is taken over all functions g for which

∥g∥_{L_{ϱ}^{q′}} ≤ 1, ϱ = (\frac{1}{v_{1}} , \frac{1}{v_{2}} , …, \frac{1}{v_{n}} ).

Next, by Lemma 2 we have

| ∫_{ℝ^{n}} ∫_{ℝ^{n}} \frac{f(y)g(x)}{|x − y|_{a}^{μ}}dx dy| ≤ c∥f∥_{L_{w}^{p}} · ∥g∥_{L_{ϱ}^{q′}} ≤ c∥f∥_{L_{w}^{p}}. ■

Theorem 2. Let 1≤m≤n, v_{+}=(v_{1}, v_{2}, …, v_{m}), w_{+}=(w_{1}, w_{2}, …, w_{m}), w = (w_{1}, w_{2}, …, w_{m}, 1, …, 1), 1 < p_{i} < q_{i} < ∞ (i = 1, 2, …, m), q_{+} = (q_{1}, q_{2}, …, q_{m}), p_{+} = (p_{1}, p_{2}, …, p_{m}), 1 < p_{i} < ∞ (i = m + 1, …, n).

Next we set

μ = ∑_{i=1}^{m} a_{j}(1 − γ_{j}) + ∑_{j=m+1}^{n} \frac{a_{j}}{p′_{j}} ,  (1.4)

where a_{j} > 0, 0 < γ_{j} < 1 (j = 1, 2, …, m)

If (v_{i}, w_{i}) ∈ G_{1−γ_{i}}^{q_{i},p_{i}} (i = 1, 2, …, m), then there exists a constant c > 0 such that for any f ∈ L_{w}^{p}(ℝ^{n}) and arbitrary (x_{m+1}^{0}, …, x_{n}^{0}) the function

h(x) = ∫_{ℝ^{n}} \frac{f(y)}{|x − y|_{a}^{μ}} dy

TWO-WEIGHTED ESTIMATES FOR INTEGRAL TRANSFORMS 499

belongs to the space L_{v_{+}}^{q^{+}}(ℝ^{m}) and the inequality holds ∥h∥_{L_{v_{+}}^{q^{+}}} ≤ c∥f∥_{L_{w}^{p}}, where the constant c is independent of f.

The proof of Theorem 2 is based on the following Lemma 3. Let the conditions of Theorem 2 be fulfilled. If ϱ=(\frac{1}{v_{1}} , \frac{1}{v_{2}} , …,

\frac{1}{v_{n}}), q′_{+} = (q′_{1}, q′_{2}, …, q′_{m}), then there exists a constant c > 0 such that for all f : ℝ^{n} → ℝ^{1} and g : ℝ^{m} → ℝ^{1} we have

| ∫_{ℝ^{m}} ∫_{ℝ^{n}} \frac{g(x)f(y)}{|x − y|_{a}^{μ}} | ≤ c∥f∥_{L_{w}^{p}(ℝ^{n})} · ∥g∥_{L_{ϱ_{+}}^{q′_{+}} (ℝ^{m})}.  (1.5)

Proof. Let y′=(y_{1}, y_{2}, …, y_{m}), y′′=(y_{m+1}, …, y_{n}), p_{+} = (p_{1}, p_{2}, …, p_{m}), p_{−} = (p_{m+1}, …, p_{n}), p′_{−} = (p′_{m+1}, …, p′_{n}).

Obviously

| ∫_{ℝ^{m}} ∫_{ℝ^{n}} \frac{g(x)f(y)}{|x − y|_{a}^{μ}}dx dy| ≤ ∫_{ℝ^{m}} |g(x)|( ∫_{ℝ^{m}} ( ∫_{ℝ^{n−m}} \frac{|f(y)|dy′′}{|x − y|_{a}^{μ}} )dy′)dx. (1.6)

By virtue of the Hölder inequality

∫_{ℝ^{n−m}} \frac{|f(y)|dy′′}{|x − y|_{a}^{μ}} ≤ ∥f∥_{L^{p_{−}} (ℝ^{n−m})} · ∥ |x − y|_{a}^{−μ}∥_{L^{p′_{−}} (ℝ^{n−m})}.  (1.7)

We introduce some notation:

φ(y′) = ∥f(y′, ·)∥_{L^{p_{−}} (ℝ^{n−m})}, φ_{1}(x, y′) = ∥ |x − y|_{a}^{−μ}∥_{L^{p′_{−}} (ℝ^{n−m})},

T = ( ∑_{j=1}^{m} |x_{j} − y_{j}|^{\frac{2}{a_{j}}} )^{\frac{1}{2}} .

Let us prove that there exists a positive number c_{1} such that

φ_{1}(x, y′) ≤ c_{1}T^{( ∑_{j=m+1}^{n} \frac{a_{j}}{p′_{j}} −μ)}.  (1.8)

We have

φ_{1}(x, y′) = ∥(T^{2} + ∑_{j=m+1}^{n} |x_{j} − y_{j}|^{\frac{2}{a_{j}}} )^{− \frac{μ}{2}} ∥_{L^{p′_{−}} (ℝ^{n−m})} =

= ∥(T^{2} + ∑_{j=m+1}^{n} |y_{j}|^{\frac{2}{a_{j}}} )^{− \frac{μ}{2}} ∥_{L^{p′_{−}} (ℝ^{n−m})}.

The change of the variable y_{j} = T^{a_{j}}t_{j} in the latter expression leads to the equality

φ(x, y′) = T^{∑_{j=m+1}^{n} \frac{a_{j}}{p′_{j}} −μ}∥(1 + ∑_{j=m+1}^{n} |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{μ}{2}} ∥_{L^{p′_{−}} (ℝ^{n−m})}.

Now it is obvious that

(1 + ∑_{j=m+1}^{n} |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{μ}{2}} = (1 + ∑_{j=m+1}^{n} |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{1}{2} ∑_{k=m+1}^{n} _{\frac{a_{k}}{p{′ \atop k}}} +∑_{i=1}^{m} a_{i}(1−γ_{i})} =

= ∏_{j=m+1}^{n} (1 + ∑_{j=m+1}^{n} |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{1}{2} ( \frac{a_{j}}{p′_{j}} +ε)} ≤ ∏_{j=m+1}^{n} (1 + |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{1}{2} ( \frac{a_{j}}{p′_{j}} +ε)},

where ε > 0.

Therefore

∥(1 + ∑_{j=m+1}^{n} |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{μ}{2}} ∥_{L^{p′_{−}} (ℝ^{n−m})} ≤

≤ ∥ ∏_{j=m+1}^{n} (1 + |t_{j}|^{\frac{2}{a_{j}}} )^{− \frac{1}{2} ( \frac{a_{j}}{p′_{j}} +ε)}∥_{L^{p′_{−}} (ℝ^{n−m})}.

On the other hand,

∫_{ℝ^{1}} \frac{dt_{j}}{(1 + |t_{j}|^{\frac{2}{a_{j}}} )^{\frac{1}{2} ( \frac{a_{j}}{p′_{j}} +ε)p′_{j}}} ≤ ∫_{ℝ^{1}} \frac{dt_{j}}{(1 + |t_{j}|)^{1+ \frac{εp′_{j}}{a_{j}}}} < ∞.

Thus we have proved the estimate (1.8). It implies

φ(x, y′)≤( ∑_{j=1}^{m}|x_{j} − y_{j}|^{\frac{2}{a_{j}}} )^{− \frac{1}{2} ∑_{j=1}^{m} a_{j}(1−γ_{j})} ≤c_{1} ∏_{j=1}^{m} |x_{j} − y_{j}|^{(γ_{j}−1)}. (1.9)

Using the generalized Hölder inequality and Lemma 1 with (1.6), (1.7), (1.8) and (1.9) we obtain:

| ∫_{ℝ^{m}} dx ∫_{ℝ^{n}} \frac{g(x)f(y)}{|x − y|_{a}^{μ}}dy| ≤ c_{2} ∫_{ℝ^{m}} ∫_{ℝ^{m}} \frac{|g(x)|φ(y′)}{∏_{j=1}^{m} |x_{j} − y_{j}|^{1−γ_{j}}} dx dy ≤

≤ c_{3}∥φ(y′)∥_{L_{w_{+}}^{p_{+}} (ℝ^{m})}∥g∥_{L_{ϱ}^{q′} (ℝ^{m})}.

This implies that

| ∫_{ℝ^{m}} dx ∫_{ℝ^{n}} \frac{g(x)f(y)}{|x − y|_{a}^{μ}}dy| ≤ c_{3}∥f∥_{L_{w}^{p}(ℝ^{n})}∥g∥_{L_{ϱ}^{q_{+}} (ℝ^{m})}. ■

Proof of Theorem 2. Using the standard arguments, the validity of Theorem 2 readily follows from Lemma 3.

2. In this paragraph we shall prove the imbedding theorems for different metrics and different dimensions for weighted spaces of anisotropic Bessel potentials.

Definition 2 (see[2]). Let r = (r_{1}, r_{2}, …, r_{n}), p = (p_{1}, p_{2}, …, p_{n}), r_{j} > 0 (j = 1, 2, …, n). It will be said that f ∈ L_{w}^{p,r}(ℝ^{n}) if

f(x) = ∫_{ℝ^{n}} G_{r}(x − y)g(y)dy,

where G_{r} is the anisotropic Bessel–Macdonald kernel and g ∈ L_{w}^{p}(ℝ^{n}). By the definition, ∥f∥_{L_{w}^{p,r}} = ∥g∥_{L_{w}^{p}}.

The kernel G_{r} is characterized by its Fourier transform as follows (see[2]) (2π)^{\frac{n}{2}} Ĝ_{2}(λ) = [1 + σ^{2}(λ)]^{− \frac{r^{∗}}{2}} where the function σ(λ) is determined by the equation

∑_{j=1}^{n} \frac{λ_{j}^{2}}{σ^{2a_{j}}} = 1, a_{j} = \frac{r^{∗}}{r_{j}} , \frac{1}{r^{∗}} = \frac{1}{n} ∑_{j=1}^{n} \frac{1}{r_{j}} .

The kernel G_{r} obeys, along each j – the coordinate direction, the estimates

|G_{r}(x)| ≤ c|x_{j}|^{−r_{j}( ∑_{i=1}^{n} \frac{1}{r_{i}} −1)}.  (2.1)

Now we shall prove the imbedding theorem of different metrics.

Theorem 3. Let 1<p_{j} <q_{j} <∞, (v_{i}, w_{i})∈G_{1−γ_{i}}^{q_{i},p_{i}} (i = 1, 2, …, n). Put

κ = 1 − ∑_{j=1}^{n} \frac{γ_{j}}{r_{j}}

and ϱ = κr, r = (r_{1}, r_{2}, …, r_{n}).

Then each function f ∈ L_{w}^{p,r}(ℝ^{n}) belongs to the space L_{v}^{q,ϱ}(ℝ^{n}) and the inequality ∥f∥_{L_{v}^{q,ϱ}} ≤ c∥f∥_{L_{w}^{p,r}} holds, where the constant c is independant of f.

Proof. We have

f(x) = ∫_{ℝ^{n}} G_{ϱ}(x − y)h(y)dy,

where

h(x) = ∫_{ℝ^{n}} G_{r(1−κ)}(x − y)g(y)dy

and g ∈ L_{w}^{p}(ℝ^{n}).

Now it will be shown that h ∈ L_{v}^{q}(ℝ^{n}). Due to (2.1) we have

|G_{r(1−κ)}(x)| ≤ c|x_{i}|^{−r_{j}(1−κ)[ ∑_{i=1}^{n} \frac{1}{r_{i}(1−κ)} −1]},

or

|G_{r(1−κ)}(x)| ≤ c|x_{i}|^{−r_{j} ∑_{i=1}^{n} \frac{1−γ_{i}}{r_{i}}} .

Let a_{j} = \frac{1}{r_{j}} and

max_{1≤j≤n} |x_{j}|^{\frac{1}{a_{j}} ∑_{i=1}^{n} a_{i}(1−γ_{i})} = |x_{j_{0}}|^{_{\frac{1}{a_{j_{0}}}} ∑_{i=1}^{n} a_{i}(1−γ_{i})}.

As can be easily verified,

( ∑_{i=1}^{n} |x_{j}|^{\frac{2}{a_{j}}} )^{\frac{1}{2} ∑_{i=1}^{n} a_{i}(1−γ_{i})} ≤ c_{1}|x_{j_{0}}|^{_{\frac{1}{a_{j_{0}}}} ∑_{i=1}^{n} a_{i}(1−γ_{i})}.

Therefore

|G_{r(1−κ)}(x)| ≤ c_{2}|x|_{a}^{−μ},  (2.2)

where

μ = ∑_{j=1}^{n} a_{j}(1 − γ_{j}).

Hence

|h(x)| ≤ ∫_{ℝ^{n}} |g(y)| |x − y|_{a}^{−μ}dy.

Applying Theorem 1, we obtain ∥h∥_{L_{v}^{q}} ≤ c_{3}∥f∥_{L_{w}^{p}}, which implies ∥f∥_{L_{v}^{q,ϱ}} ≤ ∥f∥_{L_{w}^{p,r}}. ■

Using Theorem 2 one may prove an imbedding theorem of different dimensions in a similar manner.

Theorem 4. Let 1 < p_{i} < ∞ (i = 1, 2, …, n), 1 < p_{i} < q_{i} < ∞ (i = 1, 2, …, m), 1 ≤ m ≤ n. It is also assumed that (v_{i}, w_{i}) ∈ G_{1−γ_{i}}^{q_{i},p_{i}}, 0 < γ_{i} < 1 (i = 1, 2, …).

If

κ = 1 − ∑_{j=1}^{m} \frac{γ_{j}}{r_{j}} − ∑_{j=m+1}^{n} \frac{1}{r_{j}p_{j}} ,  (2.3)

then for an arbitrary function f from the space L_{w}^{p,r}(ℝ^{n}) the function F (x_{1}, x_{2}, …, x_{m}) = f(x_{1}, x_{2}, …, x_{m}, x_{m+1}^{0}, …, x_{n}^{0}) belongs to the space L_{v}^{q,ϱ}(ℝ^{m}) and the inequality ∥F ∥_{L_{v}^{q,ϱ}(ℝ^{m})} ≤ c∥f∥_{L_{w}^{p,r}(ℝ^{n})} holds where the constant c is independent of f.

Proof. In the case under consideration the kernel G_{(1−κ)r} admits the estimate |G_{(1−κ)r}(x)| ≤ c|x|_{a}^{−μ}, where a = (a_{1}, a_{2}, …, a_{n}), a_{i} = \frac{1}{r_{j}} and

μ = ∑_{i=1}^{m} \frac{1 − γ_{i}}{r_{i}} + ∑_{j=m+1}^{n} \frac{1}{r_{j}p′_{j}} .  (2.4)

Hence we can apply Theorem 2. The rest of the proof is as for the preceding theorem.

References

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(Received 02.08.1993)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, Z. Rukhadze St., Tbilisi 380093

Republic of Georgia