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6MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HOLDER SPACES DEFINED BY
USUAL HOLDER CONDITION
Mamatov Tulkin, Sabirova Rano, Barakaev Dilshod
Teachers of Department of Higher mathematics, Bukhara Technological Institute of Engineering 706030, Uzbekistan, Bukhara Region, Bukhara Street, K. Murtazayev, 15
Abstract. We study mixed fractional derivative in Marchaud form of function of two variables in Holder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Holder class defined by usual Holder condition.
Аннотация. Мы изучаем смешанную дробную производную в форме Маршо от двух переменных в пространствах Гельдера.
Keywords: functions of two variables, fractional derivative of Marchaud form, mixed fractional derivative, mixed fractional integral, Holder space.
Ключевые слова: функции двух переменных, дробный дифференциал форме Маршо, смешанный дробный производный, смешанный дробный интеграл, пространства Гельдера.
Introduction
The classical result of G.Hardi and D.Littlewood (1928, see [1, §3]) is known that the fractional integral (ICC+f)(x) = r-1(a)(t"-1 *f)(x),0 < a < lmaps isomorphically the space Hq([0,1]) of Holder order X £ (0,1) functions with a condition f(0) = 0on a similar space of a higher order X + a provided that X + a < 1. Further, this result was generalized in various directions: a space with a power weight, generalized Holder spaces, spaces of the Nikolsky type, etc. A detailed review of these and some other similar results can be found in [1].
In the multidimensional case, the statement about the properties of a map in Holder spaces for a mixed fractional Riemann - Liouville integral was studied in [2] - [7].
Mixed fractional derivatives form Marchaud
ina'P aV _^(x,y)_4.
(»a+,c+<P)(x,y) = F(1 - a)F(i - p){x - a)cc(y - C)P +
+---f f *(x:y2-*(t'TlBdtdT, x > a, y > c, (1)
were not studied in the Holder space. This paper is devoted to the study of the properties of a map in Holder spaces, defined by the usual Holder condition for functions of two variables.
Consider the operator (1) in a rectangle Q = {(x,y): 0 < x < b,0 < y < d}.
For a continuous function <(x, y) on R2 we introduce the notation
(Ah$)(x,y) = <(x + h,y)-<(x,y), (Av<)(x,y) = <(x,y + r) - <(x,y),
/i,i
(АКт,ф) (x,y) = ф(х + h,y + r)- ф(х + h,у) - ф(х,у + ц) + ф(х,у),
so that
<(x + h,y + r) = (AKr,4)(x,y) + (Ah4>)(x,y) + (Ar,4>)(x,y) + 4(x,y). (2)
Everywhere in the sequel by C, C1, C2 etc we denote positive constants which may different values in different occurences and even in the same line.
Definition 1. Let X,y £ (0,1]. We say that < £ HX'r(Q), if
№(xi,yi) - QfayX < Cilxi - x2lA + C2lyi - y2lr (3)
for all (xi, yi), (x2,y2) £ Q. Condition (3) is equivalent to the couple of the separate conditions
h <
uniform with respect to another variable. Note that
/i,i
A-0 \ z0'1 \
(A hф)(x,y) < Ci\h\x, (А^ф) (x,y)
<C2\rj\r (4)
<(x,y) £ Hx'r^ (a h,r<p) (x,y) < C9lhl0Xlrl(i-0)r < C min[lhlx,lrlr], where 9 £ [0,1]. (5)
By HqY(Q) we define a subspace of functions f £ HqY(Q), vanishing at the boundaries x = 0 and y = OofQ.
A one -dimensional statements
The following statements are known [1]. We use the schemes of the proofs to make the presentation easier for two-dimensional case.
Lemma 1. If f(x) £ Hx+a([0, b])and 0<X,0<a + X<l, then
g(x)=f^_m£Hl([0M)and\\g\\H, < C\\f\\H,+a, where C doesn't depend from f(x).
Proof. Let h > 0;x,x + h £ [0, b]. We consider the difference
\*(x+h) - ^ WI £+\m -
Since f 6 Hx+a, we have
I f(x + h)- f(x)I < C1hl+a, If(x) - f(0)\ < C2xA+a. Using these inqualities we obtain
Is(x + h)- g(x)\ <C^ + ^= Gi + g2.
For Gi we have
Gi = CihX(77h)a<ChX.
Let's estimate G2, here we shall consider two cases: x < h and x > ft. In the first case, we use inequality - o£l < \a1 - a2\(a1 ^ a2) and obtain
x^ha J
G2 < C2^-a <ChX.
2 2 (x+h)a
In second case, using (1 + t)a - 1 < at, t > 0 we have
G2 = C2£aa\(l+Xf -1\<Chx^<Ch\
which completes the proof.
The Marchaud fractional differentiation operator has a form:
(DZ+f)(x) + ^J'frN()dt, where 0 < a < 1. (6)
V °+J J Xar(1-a) r(1-a)J0 (x-t)1+a ' v '
Theorem 1. If f(x) 6 Hx+a([0, b]), 0 < a + A < 1, that
(Daa+f)(x) = X£^+^p(x) , (7)
where b(x) 6 Hx([0,b]) and b(0) = 0, thus \\ip\\Hi < C\\f\\Hi+a. Proof. We present (6) as
ma f\(v\ - f(°) I f(x)-f(°) . a rxf(x)-fit) (D0+J )(x) = xar(l-a) + xar(l-a) + r(1-a) J (x-t)i+a d, receive equality (7), where
<fV J r^w r^w xar(1-a) r(1-a)J0 (x-t)1+a
to sho
Let h > 0;x,x + h 6 [0, b]. Let's consider the difference
Here b1(x) 6 Hx([0,b]) by Lemme 1. Itisenoughto show b2(x) 6 Hx([0,b]).
fxf(x + h)-f(x) rx+h f(x + h)-f(t)
b2 (x + h) - b2 (x) = I —---—— dt + I —---—— i
r2V y Jn (x + h -t)1+a Jx (x + h -t)1+a
Since f 6 Hx+a([0,b]), then we have for/i
+ № - m] - dt=k + h + h.
ave for /i
\I1\ < Chx+a Jx(t + h)-1-adt < C1hx.
Let's estimate /2. We have
rx+h
(x + h - t)x-1dt = C2hx
x
For /3.
'' 31 < Ch Jo'1- \(1+t)i+a-~H+a
^ < ChX JotX \7nk+z -7i+a\dt< C3hX.
Finally, it remains to note that b2 (0) = 0, since
\b2(x)\ < c Joxtx-1dt.
Main result.
Lemma 2. Let f(x,y) 6 Hx,Y(Q), a<A<1,fi<y<1. Then for the mixed fractional differential operator (i) the representation
(r>a'P Afy - f(0,0)x-ay-P+y-PiPi(x)+x-aip2(y)+ip(x,y) (D°+,0+J)(x,y) = r(1-a)r(1-p) ,
(8)
and
where
\b1(x)\ < C1xx-a, \b2(y)\ < C2yy-P, (9)
\b(x,y)\ < Cx9x-ay(1-9)r-P (10)
b1(x) = x-a [f(x, 0) - f (0,0)] + a I [f(x, 0) - f(t, 0)](x - t)-a-1dt,
o
y
b2(y)=y-p[f(0,y)-f(0,0)]+p I [f(0,y)-f(0,T)](y-T)-1-?dT,
o
1 1,1 a x 1,1 d
b(x,y) = xayw(Ax,yf) (0,0) + yj I [Ax-t,yf)(t,+
ß fy f1'1 \ dx (* (y Z1,1 \ dtdT
Proof. Representation (8) itself is easily obtained by means of (2). Since f 6 Hx,r(Q), inequalities (9) are obvious. Estimate (10) is obtained by means of (5), i.e.
xb(x vXC \x^ev(1-e)y + (1-e) j*_&_+ ßxex jy_£_+ ßa j* jy (*-t)ex-1dtdT]
V(x,y)<C\x v + ay Jo (x-t)i-ex +ßx Jo (y-T)i-(i-e)Y +ßu co Jo (y-T)i-(i-e)Y]
It is easy to receive
rp(x,v) < CxSAy(1-S)r \1 + fo17^ + fo1?-gW + Jo1 fo1^Sf] < C3x°Y1-^. Theorem 2. Let f(x,y) 6 H^,r(Q),a < X< 1,ß < y < 1. Then the operatorD"/0+ continuously maps HX,r(Q) into HX-a,v-ß(Q). '
Proof. Since f(x,y) 6 HqY(Q), by (8) we have
^(x,y) = {Doffl+f)(x,v) = $(x,y).
Let h > 0;x,x + h 6 [0, b]. We consider the difference
^ (AKyf)(0,0) (1^Xyf)(0,0)
$(x + h,v) - $(x,y) = } <Pk: = —Rf , 7Vr +
k' yß(x + h)a vß
11
(x + h)a xa
+
(AKyf)(x,0) a ,*+h (A x+h-tyf) (t, 0) ß ry(^h,y-rf)(0,T)
(x + h -t)1+adt + V^L (x + h -t)1+a d+(x + h)a I (y - t)1+P
a C* l1,1 \ fy
^p J (A *-tyf) (t, 0)[(x + h - t)-1-a -(x- t)-1-a]dt + ß[(x + h)-a - x-a] J V (y_Ty+ß
+ (A*-t,yf)(t,0)[(x + h -t)-1-a-(x-t)-1-a]dt + ß[(x + h)-a-x-a] J (y_Ty+p dT +
r* ry (Ah,y-jf) (x,T^dtdr ,*+h ry(A*+h-ty-Tf)(x,T)dtdT +aßJo Jo (x + h -t)1+«(y-r)1+ß + aß J* Jo (x + h -t)1+a(y-r)1+ß +
x-t.y-rf )(*,*) ' (y-T)1+ß
Since f(x,y) 6 HqY(Q) we have
+aßf* % (v_TJß [(x + h - t)-1-a - (x - t)-1-a]dtdr. (11)
Zhx xÄr 1 11 ah" f* dt
№k\= C —¡T,-+ -T\---+—TT I 7-;-\7T--+
k yß(x + h)a yß (x + h)a xa_ yM„ (x + h - t)1+a
ah" r* dt
i=1
a r*+h hexß ry
+ —J (x + h -t)l-1-adt + --J (y-T)(1-s)r-1-ßdT
J * x (x + h)a Jo '
—ß J (x-t)
vß Jo
yß J* (x + h)a
* 1 1
a
+
d +
Y(x + h - t)1+a (x - t)1+
ry dT fl, f* fy(y-T)(1-d)r-1-ßdtdT
+ßx9Ä[(x + h)~a - x~a] Jo (y-r)d+ß-C1-»)r + aßh" Jo Jo (x + h -t)1+a +
ex
+aPCh ^t+Z^ + aPSX %yy_(;-tle)Y[(x + h - t)-i-c-(x - t)-i-c]dtdr, where
Jy(y-T)(i-9)r-i-PdT< W.
Using estimations G\, G2 of the proof of Lemma 1 and estimations /1, I2, I3 of the proof of the Theorem 1, it is easily possible to receive an estimation
lip(x + h,y) - xp(x,y)l < Chx-a. Rearranging symmetrically representation (11), we can similarly obtain that
lxp(x,y + r) - ^(x,y)l < Crr-^.
References:
S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach. Sci. Publ., N. York - London, 1993, 1012 pp. (Russian Ed. - Fractional Interals and Derivatives and Some of Their Applications, Nauka i Tekhnika, Minsk, 1987).
T.Mamatov Mixed fractional integration operators in Holder spaces. «Science and World ». Volgograd, № 1 (1). 2013, P. 30-38
T.Mamatov, S. Samko. Mixed Fractional Integration Operators in Mixed Weighted Holder Spaces. FC&AA. Vol.13, Num 3. 2010, p. 245-259
T. Mamatov. Weighted Zygmund estimates for mixed fractional integration. Case Studies Journal. Volume 7, Issue 5, 2018. p. 82-88
1o
T. Mamatov. Mixed Fractional Integration In Mixed Weighted Generalized Hölder Spaces. Case Studies Journal. Volume 7, Issue 6. 2018, P. 61-68
T. Mamatov. Mixed Fractional Integration Operators in Hölder spaces. Monograph. LAPLAMBERT Academic Publishing. P. 73
T. Mamatov. Mixed Fractional Integration Operators in Mixed Weighted Hölder Spaces. Monograph. LAPLAMBERT Academic Publishing. P. 73
ZIGMUND TYPE ESTIMATES FOR MIXED FRACTIONAL INTEGRALS OF THE VOLTERRA
CONVOLUTION TYPE
Mamatov Tulkin, Hamidov Farhod
Teachers of Department of Higher mathematics, Bukhara Technological Institute of Engineering 706030, Uzbekistan, Bukhara Region, Bukhara Street, K. Murtazayev, 15
Abstract. Non-weight Zygmund-type estimates are obtained for mixed fractional integrals of the Volterra convolution type for a function of two variables defined by a mixed modulus of continuity.
Аннотация. Получены оценки типа Зигмунда для смешанных дробных интегралов типа вольтеровской свертки в обобщенном Гельдервском пространстве для двух переменных определяемых смешанным модулем непрерывности.
Keywords: mixed fractional integral, mixed modulus of continuity, Zygmund type estimate, two variable function, Volterra convolution type.
Ключевые слова: функции двух переменных, смешанный дробный интеграл, смешанный модуль непрерывность, смешанный дробный интеграл типа вольтеровской свертки, оценки типа Зигмунда.
Introduction
An important stage in the study of fractional integro-differentiation of functions from generalized Holder spaces is the obtaining of Zygmund-type estimates, i.e. estimate of the modulus of continuity of a fractional integral through the modulus of continuity of the original function. A similar problem can be considered completely solved for the Holder space of functions of one variable and power weights (see [2], [6], see also [1]), as well as for the Holder space of functions of two variables and power weights (see [3]-[5]). Obtaining an estimate of the Zygmund type for mixed fractional integrals with arbitrary kernels and has not been studied.
The main focus of the work is to obtain an estimate of the type of Zygmund majorizing a mixed modulus of continuity w(pK<;h,r]) mixed fractional integral with a Volterra convolution type of weight integral constructions from a mixed modulus of continuity w(p<p;h,ij) its density <p(x,y) with weight p(x,y). These Zygmund type bounds and action theorems directly affect the nature of the improvement of the modulus of continuity by mixed fractional integration of the Volterra convolution type:
(K<)(x,y) = J0x J0yk(x - t)k(y - s)<(t,s)dtds, (1)
here we consider the degenerate kernels, as well as each k(x) and k(y) assumed to be close in some sense to a power function.
In this paper, we deal with arbitrary kernels, i.e. not necessarily power. We will consider the operator (1) in a rectangle Q = {(x,y): 0 < x < b,0 < y < d}.
Preliminary
1. One-dimensional statements. In this section, we present some well-known results and notation that we will need to present the issues under consideration (see [5]).
Everywhere in the results through C, C1, C2,... we denote absolute constants that can have different meanings in different cases.
Definition 1. We say that k(x) 6 Vx,A > 0, k(x) > 0, if k(x) 6 [0, Z] and satisfies the conditions:
1) k(x) ± 0,xxk(x) - almostincreasesand xxk(x)\x = 0 = 0;
2)3e >0, 0 < e < A, that xx-£k(x) - almost decreases;
3)3C, that \k(xi^-k(x2^\ < C^, if x *= max(x1,x2). (2)
I xi-x2 I x* 1 2
Definition 2. Let given bounded on [a, b] function <(x). Under modulus of continuity <(x) understood expression
sup \<(x + h) - <(x)\ = (¿(<p; 5), 0 < 5 < b - a.
h6[0,S]
Definition 3. We denote by (P1 function class w(S) 6 (0,b - a] and satisfying the conditions
1) m(S) >0 in (0, b-a], limu(S) = 0;
s^o
2) v(S) t in (0, b-a];
3) m(S± + S2) < M(S±) + M(S2).
Below in the estimates we need inequalities:
1) if m(<; h) is modulus continuity, then we have
