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9T. 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$;h,ii) mixed fractional integral with a Volterra convolution type of weight integral constructions from a mixed modulus of continuity w(p^;h,ij) its density <p(x,y) with weight p(x,y). These Zygmund type bounds and action theorems directly affect the nature of the improvement of the modulus of continuity by mixed fractional integration of the Volterra convolution type:
(K^)(x,y) = J* J*k(x — t)k(y - s)$(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) £ Va,A > 0, k(x) > 0, if k(x) £ [0, Z] and satisfies the conditions:
1)k(x) ± 0,xxk(x) - almost increases and xxk(x)lx = 0 = 0;
2)3e >0, 0 < e < A, that xx-£k(x) - almost decreases;
3)3C, that lk(x^-k(x2n < c^, if X *= max(x1,x2). (2)
I x±-x2 I x* 1 2
Definition 2. Let given bounded on [a, b] function <p(x). Under modulus of continuity <p(x) understood expression
sup l$(x + h) — $(x)l = 5), 0 < S < b — a.
h£[0,S]
Definition 3. We denote by &1 function class w(S) £ (0,b — a] and satisfying the conditions
1) m(S) >0 in (0, b-a], limM(S) = 0;
s^o
2) m(S) t in (0, b-a];
3) M(S1 + S2) < M(S1) + M(S2).
Below in the estimates we need inequalities:
1) if &(<$; h) is modulus continuity, then we have
I f(x + h)-f(x)I<C
х2ш(ф;х1) < Сх1ш(ф;х2), x2 < x1; (3)
2) if k(x) E Vx, then x^k(x1) < Cx%k(x2) and 30 < e < A,
x£-£k(x2) < Cxt£k(xi), xi<x2; (4)
Lemma 1. Let k(x) EVX, Я> 0 and "(x) > 0 and almost increases, then for any 0 < x < 1/2, rightly
"(x)k(x)<Cft-1"(t)k(t)dt. (5)
Consider a one-dimensional integral operator of the Voltero type convolutions (Кф)(x) = f* k(x - t^(t)dt, 0 < x < b. (6)
Theorem 1. Let k(x) EVл.0<Л<1и ф(x) E С([0,Ь]),ф(0) = 0. Then for integral (4), the following estimate is valid
"(Кф, h) <c(hik(h)M(ф, h) + h f* dt) (7)
Proof. The integral (4) will be presented in the form
rX rX
(Кф)(x) = ф(0) I k(x - t)dt + I k(x- t)ty(t) - ф(0)]dt.
'о '0
Since ф(0) = 0, then we have f(x) = f* g(t)k(t - a)dt, где ф(г) - ф(0) = g(t). Let h > 0, x,x + h E [0, b]. Consider the difference
I k(h + t)[g(x - t) - g(x)]dt +
J-h
+ fX (k(h + t)- k(t))[g(x -t)- g(x)]dt + g(x) f*X+h k(t)dtl = ^ + ¿2 + A3. (8) We estimate A1. Using inequality (3) and by definition 1, we have
A1 < t)k(h - t)dt < Ссо(ф,h)k(h) f* (-h) dt < Chk(h)"(<p,h). (9)
We estimate A2 . Using inequality (2), we have
A2<Cf0xk-^^;t)dt. Here we distinguish two cases: 1) h> x and 2) h < x. In the first case
X
,,] r ^ fx "(ф; t) „ „ [h dt A2 < Chl+X~Em I (h + t)1+Asdt = Chk(h) l "(ф. ht) {1 + t)1+xs <
< Chk(h) fX dt < Chk(h)"M,h). (10)
In the second case A2, it can be represented as the sum of two terms, i.e.A2 < f^ + f* a = A'2 + A",
для справедлива оценка (10). А для, имеем (for A'2 is rightly estimate (10). And for A"2, we have
(x k(t) Гь ш(ф. t)
A2 < Ch I ш(ф. t)——dt < Ch I k(t)dt.
Jh t Jh t
Finally, we estimateA3. By x < h:
, fx+h dt
A3 < Cш(ф.x)k(x + h)(x + h)A I —< Cш(ф.h)k(h)h.
Jx t
If x > h, then using Lemma 1 and we have
Jrb k(t) Cb k(t)
-"(ф; t)dt < Ch I -"(ф; t)dt.
x t Jx t
Collecting all estimates for A1. A2. A3, we get (7).
Let a continuous function ф^.у) be defined inR2. We introduce the necessary notation
/1,0 \ /0,1
' h,t
and
(A нф) (x,y) = ф^ + h,y) - ф^^^),^ цф) (x,y) = ф^,у + r) - ф^^), (AКлф) (x,y) = ф^ + h.y + rj)- ф^,у + г) - ф(x + h,y) + ф^,у),
p(x + h,y + V) = (Ahiricp) (x,y) + (x,y) + (Ahcp) (x,y) + p(x,y). (11)
Now we introduce the following characteristics: 1) Private modules of continuity
1,0
5,0) = sup sup
y 0<h<S
2) Mixed modulus continuity of order 1.1
1,1
w(cp; 5, a) = sup sup
x,y 0<h<S
0<1J<(T
1,0 0,1 0,1 ( A ^¡(x^) и ш(ф;0,а) = sup sup lA^i(x,y)
\ ' x 0<tJ<(T \ '
где 0 < S <b,0 < a <d.
1,1
to (Kcp; h, rj) < C3
1,1
It follows from the definition u> (cp; S, a) that this function belongs in each variable 01. In addition, we note that there is an inequality
1,1 r1,0 0,1
to (cp; 8,0) <2 mini to (cp; S, 0), to (cp; 0, a)}. (12)
Main result
In this section, we generalize Theorem 1 to the case k(x,y) = k1(x)k2(y).
Theorem 2. Let k1(x), k(y) £ VA, 0 < A < 1, <p(x,y) £ C(Q) and cp(x,y)lx=oy=o = 0. Then Zygmund type estimates are valid
to(Kcp; h,0) < C1[hk(h)toj(^; h,d)+ h J* ^p-1co(cp;t,d)dt], (13)
°co(Kcp;0,r]) <C2\pk(ri)1a>(<p;b,ri) + v J^ ^)1<io(cp;b, s)dsj, (14)
1,1 fb k(t) 1,1
hrjk(h)k(ri)to(cp; h,r\) + hrjk(ri) I -to (cp;t,r))dt +
h t
+ hk(h)vJ* ^a)(<p; h,s)ds + hV J* J* ^^-to^t, s) dt ds~\. (15)
Proof. Using (11), (1) we will present in the form
~ rx ry r ry 11,1 \
(Kcp)(x,y) = cp(0,0) I I k(x — t)k(y — s)dtds + I I k(x — t)k(y — s)(A tscp)(0,0)dtds 0 0 0 0 rx ry rx ry
+ + I I k(x — t)k(y — s)[cp(t, 0) — cp(0,0)]dtds + | | k(x — t)k(y 0 0 0 0 — s)[cp(0,s) — cp(0,0)]dtds. Since cp(x,y)lx=oy=o = 0, then we have
rx ry ¡1,1 \ rx ry
f(x,y)=I I k(x — t)k(y — s)(Ats0)(0,0)dtds=I | g(t,s)k(x — t)k(y — s)dtds. 0 0 0 0 Let h > 0, x,x + h £ [0, b]. Consider the difference
/•0 ry
[f(x + h,y) — f(x,y)]= I I k(h + t)k(s)[g(x — t,y — s) — g(x,y — s)]dtds +
J-h Jo
+ I I [k(h + t) — k(t)]k(s)[g(x — t,y — s)—g(x,y — s)]dtds +
00
+ (J.h k(h + t)dt — J* k(t)dt)J0 g(x,y — s)ds. (16)
Fair inequality
(fh fy 1,1
lf(x + h,y)— f(x,y)l<C(\ I k(h — t)k(s)to(cp;t,y — s)dtds +
00
ry 11 rx+h ry
)to(cb;t,v — s) + I I k(t)k(s)to(cb;t,v — s)dtds
rx ry 1,1 rx+n ry 1,1 \
+ I I lk(h + t) — k(t)lk(s)to(cp;t,y — s) + I I k(t)k(s)to (cp;t,y — s) dtds ) < Jo Jo Jx Jo '
( rh 1,1 rx 1,1
< C1 ( I k(h — t)to(cp;t,d)dt + I lk(x + h) — k(t)lto (cp;t,d)dt +
\Jo Jo
+ J*+hlk(x + h) — k(x)lto(cp; t, d)dt).
Using the estimates for A1, A2,A3 in the proof of Theorem 1, one can easily verify the inequality (13). Having made a symmetric permutation (16), we can similarly get (14). Let h,-q> 0, Vx,x + h £ [0,b], Vy.y + q £ [0,d]. Consider the difference
1,1 o o 1,1
A hnf(x.y) ) = I I M -t.-*a ) (x,y)k(t + h)k(s + v)dtds +
rx ry /1,1
+ I I ( A -t_-sa ) (x,v)[k(h + t) — k(t)][k(n + s) — k(s)]dtds +
(Ah,vf(x,y)) = J (a -t,-sg) (x,y)k(t + h)k(s + V)dtds I I (A- t,-sg)(x,y)[k(h + t) — k(t)][k(V + s) — k(s)]dtds
JrX+H ry + rj ro ry .1,1 .
I k(t)k(s)dtds + I I (A-t,-sg)(x,y)k(h + t)[k(r] + s) — k(s)]dtds +
x Jy J-h Jo ^ '
I I (j- t,-sg)(x,y)[k(h + t) — k(t)]k(s + V)dtds
Jy J-h
rx ro ,1,1
+ I I (A -tj-sg) (x, y)[k(h + t) — k(t)]k(s + v)dtds +
JrX+h ro
I [g(x,y — s) — g(x,y)]k(t)k(i] + s)dtds +
X J-r
ro ry+r
+ I I [g(x,y — s)—g(x,y)]k(t + h)k(s)dtds +
J-h Jy
Jrx+h ry
I [g(*,y — s) — g(x,y)]k(t)[k(T + s) — k(s)]dtds +
X Jo
rX ry + tj
+ j j [g(x-t,y)-g(x,y)][k(t + h) - k(t)]k(s)dtds. Jo Jy
Fairness inequality
1,1 \ (rh rj 1,1 rX ry 1,1
:Cjj I M«;t,s)k(h-t)k(r]-s)dtds + + I I w(<;t,s)\k(h + t) 0 0 0 0 -k(t)\\k(i] + s) -k(s)\dtds +
11 rx+h ry+t rh ry 11
+ M(<;x,y) j j k(t)k(s)dtds + j j w(<p;t,s)k(h - t)\k(r] + s) - k(s)\dtds +
X y 0 0
X t 1,1 X+h t 1,1
+ j j u(<p;t,s)\k(h + t) - k(t)\k(tj - s)dtds + j j m (<p; x,s)k(t)k(r] - s)dtd +
0 0 X 0
h y+ t 1,1 X+h y 1,1 + j j w(<p;t,y)k(h - t)k(s)dtds + j j m (<; x,r)k(t)\k(rj + s) - k(s)\dtd +
0 y X 0
y+ t 1,1
+ j m (<; t,y)\k(t + h) - k(t)\k(s)dtds}. Estimating the obtained terms in the standard way, we arrive at inequality (15).
References
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