The Di ff erential Properties of Functions from Sobolev-Morrey Type Spaces of Fractional Order

where x = (x1, ..., xn) , α = (α1, ..., αn) , β = (β1, ..., βn) , and α j, β j ≥ 0 ( j = 1, ..., n) we introduce the new form of description of norm of the spaces W l p (G) and W l p,a,κ,τ (G) , when l = (l1, ..., ln) , l j > 0, j = 1, ..., n. Also, such approach were studied in (A. M. Najafov, 2010) and (A. M. Najafov, 2013). In other words the norms of the Sobolev and Sobolev-Morrey spaces of fractional order’s the generalized derivatives of fractional order Di i f = D [li] i D {li} +i f ([li] is the integer part, {li} is the non-integer part of the number li) expression by the ordinary Riemann-Liouville fractional derivatives of functions. But in the papers (T. I. Amanov, 1976; N. Aronszajn and K. Smith, 1961; A. Calderon and A. Zygmund, 1961; A. D. Jabrailov, 1972; P. I. Lizorkin, 1963; P. I. Lizorkin, 1972; A. M. Najafov, 2005a,b; A. M. Najafov and A. T. Orujova, 2012; Yu. V. Netrusov, 1984; L. N. Slobodetskiy, 1958a,b; H. Triebel, 1986) and etc the Sobolev and Sobolev-Morrey type spaces the generalized derivatives of fractional order expression by the differences of derivatives of functions. Also, we study the differential properties of functions from spaces W l p,a,κ,τ (G) (G ∈ Rn, l ∈ (0,∞)n , p ∈ [1,∞) , a ∈ [0, 1]n , τ ∈ [1,∞]) with parameters in terms of embedding theory and some properties of fractional order Sobolev-Morrey type spaces is proved. As application of obtained results we study a smoothness of solution of one class of higher order fractional quasielliptic equations (1.1).The fundamental difference of this work from earlier work is to obtain estimates for generalized derivatives of fractional order.


Introduction and preliminary notes
In this paper in connection with the investigation of differential equation of higher fractional order of type where x = (x 1 , ..., x n ) , α = (α 1 , ..., α n ) , β = (β 1 , ..., β n ) , and α j , β j ≥ 0 ( j = 1, ..., n) we introduce the new form of description of norm of the spaces W l p (G) and W l p,a,κ,τ (G) , when l = (l 1 , ..., l n ) , l j > 0, j = 1, ..., n.Also, such approach were studied in (A.M. Najafov, 2010) and (A.M. Najafov, 2013).In other words the norms of the Sobolev and Sobolev-Morrey spaces of fractional order's the generalized derivatives of fractional order is the integer part, {l i } is the non-integer part of the number l i ) expression by the ordinary Riemann-Liouville fractional derivatives of functions.But in the papers (T.I. Amanov, 1976; N. Aronszajn and K. Smith, 1961;A. Calderon and A. Zygmund, 1961;A. D. Jabrailov, 1972;P. I. Lizorkin, 1963;P. I. Lizorkin, 1972;A. M. Najafov, 2005a,b;A. M. Najafov and A. T. Orujova, 2012;Yu. V. Netrusov, 1984;L. N. Slobodetskiy, 1958a,b;H. Triebel, 1986) and etc the Sobolev and Sobolev-Morrey type spaces the generalized derivatives of fractional order expression by the differences of derivatives of functions.Also, we study the differential properties of functions from spaces with parameters in terms of embedding theory and some properties of fractional order Sobolev-Morrey type spaces is proved.As application of obtained results we study a smoothness of solution of one class of higher order fractional quasielliptic equations (1.1).The fundamental difference of this work from earlier work is to obtain estimates for generalized derivatives of fractional order.
The Hölder continuity of solutions of integer order quasielliptic equations with continuous or Hölder continuous coefficients of the leading derivatives was considered in (E. Guisti, 1967).In (L.Arkeryd, 1969), L p − estimates for solutions were studied, under the condition that the coefficients of leading derivatives are infinitely differentiable, and in (L. A. Bagirov, 1979;S. V. Uspenskii, G. V. Demidenko and V. G. Perepelkin, 1984) some other problems of the theory of quasielliptic equations were considered.In (R. V. Guseinov, 1992) and (A.M. Nadzhafov, 2005) the theorems were proved claiming that the solution belongs to the Hölder class inside the domain, and in (P. S. Filatov, 1997) local "interior" Hölder estimates were obtained for solutions to a quasielliptic type equation in the case when the right-hand side satisfies the anistropic Hölder condition.In this paper, as in (R. V. Guseinov, 1992) and (A.M. Nadzhafov, 2005), we study the Hölder continuity of a solution without any smoothness conditions on a αβ (x) .
Let G be a domain of R n , t > 0. Given x ∈ R n , we put Definition 1 Denote by W l p,a,κ,τ (G) the space of locally summable functions f on G having the weak derivatives D l i i f on G (i = 1, 2, ..., n) with the finite norm where is the integer part, {l i } is the non-integer part of the number l i .The partial generalized fractional derivatives D {l i } +i in S. L. Sobolev's sense are understood in the following sense: +i and D {l i } −i are the ordinary Riemann-Liouville fractional derivatives of order {l i } (0 < {l i } < 1) in the domain are understood as (A.M. Najafov, 2010) and(A. M. Najafov, 2013) where x is the inner point of the domain G. Γ (α) is a gamma function, the sets G (i) and G are determined as It should be noted that ordinary Riemann-Liouville fractional derivative on the segments and the real line are reminded in the monograph (S.G. Samko, A. A. Kilbas and O. N. Marichev, 1987).
Corollary 1 Putting r = ∞ for 0 < p ≤ 1or r = q for p > 1 in (1.17) , we infer Then the function A (i) T (x) defined by (1.7) satisfies the estimate where b = (b 1 , ..., b n ) and b j is an arbitrary number satisfying the inequalities (1.20)

Main Results
Now we reduce main result of this paper.
In particular, if µ i,0 > 0, i = 1, ..., n then D ν is continuous on G and where D ν f = D [ν] D {ν} + f, and Proof.First of all,observe that,since κ = cκ using the property 4, we can assume that f ∈ W l p,a, κ,τ 1 (G) and substitute κ for κ everywhere in (2.1)-(2.3)and for µ i in (1.6).We will prove these very inequalities (the greater κ, the greater µ i ).Existence of the generalized mixed derivatives of fractional order D ν f under the conditions of the theorem follows from (A. M. Najafov, 2010) and (A.M. Najafov, 2013).
, by Theorem 1 (A.M. Najafov, 2010) and (A.M. Najafov, 2013) the generalized mixed derivatives of fractional order exists on G andD ν f ∈ L p (G) .Then it is obtained integral representation for generalized mixed derivatives of fractional order of functions from Sobolev spaces of fractional order defined on the n-dimensional domains in R n and satisfying flexible horn conditions (The domains satisfying flexible horn condition introduced in (O.V. Besov, V. P. Ilyin and S. M. Nikolskii, 1996): where 0 < T ≤ min (1, T 0 ) the functions Ω (ν) (•, y) and L (ν) i (•, y, z) are of the class C ∞ 0 (R n ) with support in I 1 and the support of (2.4), (2.5) is contained in the flexible horn x + V (λ, x, 0) ⊂ G. Using Minkowski's inequality hence, we obtain that T q;G (2.6) From (1.17) for U = G, t = T as ρ → ∞ and r = q we derive f (ν) T λ q;G ≤ C 1 T µ 0 ∥ f ∥ p,a, κ.τ 1 ;G (2.7).
Assume now that µ i,0 > 0,i = 1, ..., n.Show that then D ν f is continuous on G. From (2.4),(2.5),and (2.8) for q = ∞ and µ i = µ i,0 > 0 we derive Hence, the left-hand side of the inequality tends to zero as T → 0. Since D ν f T λ is continuous on G, in this case the convergence of L ∞ (G) coincides with uniform convergence; consequently, the limit function D ν f is continuous on G.The theorem is proved.