ISSN 0001-4346, Mathematical Notes, 2010, Vol. 87, No. 1, pp. 23–30. © Pleiades Publishing, Ltd., 2010. Original Russian Text © V. F. Babenko, S. A. Pichugov, 2010, published in Matematicheskie Zametki, 2010, Vol. 87, No. 1, pp. 26–34.
Sharp Estimates of the Norms of Fractional Derivatives ¨ of Functions of Several Variables Satisfying Holder Conditions V. F. Babenko1* and S. A. Pichugov2 1
2
Dnepropetrovsk National University Dnepropetrovsk National University of Railroad Engineering Received June 5, 2008; in final form, May 26, 2009
Abstract—We prove a new sharp Kolmogorov-type inequality that estimates the uniform norm of a mixed derivative of fractional order (in the sense of Marchaud) of a function of several variables via ¨ the uniform norm of the function and its norm on Holder spaces. DOI: 10.1134/S0001434610010049 Key words: Marchaud fractional derivative, essentially bounded function, Kolmogorov-type inequality, Holder ¨ condition, Holder ¨ space.
Many questions of analysis involve derivatives and integrals of fractional order (see, for example, [1]). α x of fractional order α ∈ (0, 1) of a The classical definition (due to Liouville) of the derivatives D± function x(u), u ∈ R given on the whole real axis, is as follows [1, p. 43]: u 1 d x(t) α dt, (D+ x)(u) := Γ(1 − α) du −∞ (u − t)α ∞ 1 d x(t) α dt. (D− x)(u) := − Γ(1 − α) du u (u − t)α Also important and sometimes more convenient are the derivatives Dα± x in the sense of Marchaud [2] (see also [1, pp. 95–97]); they are defined as follows: ∞ α x(u) − x(u ∓ t) dt. (Dα± x)(u) := Γ(1 − α) 0 t1+α In what follows, we shall put Aα = α/Γ(1 − α) for brevity. It is well known (see, for example, [1, p. 96]) that, for sufficiently well-behaved functions, the values of these derivatives coincide: α x)(u). (Dα± x)(u) = (D±
Let C(R) denote the space of bounded continuous functions x : R → R with norm xC := sup{|x(t)| : t ∈ R}. For the exponent β ∈ (0, 1], we let H β denote the space of functions x ∈ C(R) such that |x(t1 ) − x(t2 )| < ∞. |t1 − t2 |β t1 ,t2 ∈R
xH β := sup
t1 =t2
The following sharp Kolmogorov-type inequality for the derivatives of fractional order α < β ≤ 1 (in the sense of Marchaud) of functions of one variable was proved in [3]: Dα± xC ≤ *
Aα 21−α/β 1−α/β α/β xC xH β ; α 1 − α/β
E-mail:
[email protected]
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BABENKO, PICHUGOV
this inequality becomes an equality for the function x(u) defined as follows: |u|β − 1/2 if |u| ≤ 1; x(u) = 1/2 if |u| ≥ 1. For other known sharp inequalities of Kolmogorov type for the fractional derivatives of functions of a single variable, see, for example, [4]–[8]. There are also not many known sharp Kolmogorov-type inequalities for the derivatives of integral or fractional order of functions of two or more variables (see [9]–[17]). In the present paper, we obtain a generalization of inequality (1) to the case of functions of several variables. Let L∞ (Rm ) denote the space of essentially bounded functions x : Rm → R with norm x∞ = xL∞ (Rm ) := vrai sup{|x(t)| : t ∈ Rm }. For a vector t ∈ Rm , denote by Δtj ej x(u) the difference of the function x(u) with respect to the variable uj with step-width tj : Δtj ej x(u) := x(u) − x(u + tj ej ), where {e1 , . . . , em } is the standard basis in Rm . ¨ For a given Holder smoothness vector β = (β1 , . . . , βm ), βj ∈ (0, 1], j = 1, . . . , m, we introduce the ¨ condition with respect to the variable uj classes H βj of functions x(u) ∈ L∞ (Rm ) that satisfy the Holder with exponent βj i.e., Δtj ej x( · )∞ βj m 0,
dj > 0.
m 1 αj = dj Dαε x∞ , b
Dαε y∞
j=1
yH βj = = it follows that Dαε x∞
Δtj ej x(d1 u1 , . . . , dm um )∞ 1 sup b tj =0 |tj |βj Δtj ej x(d1 u1 , . . . , dm um )∞ 1 βj 1 β dj sup = dj j xH βj , β b b |dj tj | j tj =0
m m 1 1 βj βj 2 αj −αj −1 m−1 d t xH βj , x∞ dt. dj tj min min ≤ Aα 2 j b b j j b Rm + j=1
j=1
Let β
dj j =
b = 2x∞ , Then we obtain Dαε x∞
b 2x∞ = . xH βj xH βj
m m 1 2x∞ αj /βj −α −1 β m−1 ≤ Aα 2 tj j min min tj j , 1 dt, j 2x∞ xH βj Rm + j=1
j=1
i.e., 1−
Dαε x∞ ≤ x∞
m j=1
αj /βj
m j=1
where
I=
m Rm + j=1
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−αj −1
tj
2010
αj /βj
x
H
βj
Aα 2m−1 21−
m
β min min tj j , 1 dt. j
j=1
αj /βj
· I,
(4)
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BABENKO, PICHUGOV
To calculate I, let us split the domain of integration Rm + into the sets βj A0 = t ∈ Rm : min min t , 1 = 1 , + j j βj βi , i = 1, . . . , m. Ai = t ∈ Rm + : min min tj , 1 = ti Since Rm + =
j
m
i=0 Ai
and the pairwise intersections of the sets Ai have zero measure, we have I=
m
Ii ,
i=0
where
m
Ii =
Ai j=1
−αj −1
tj
β min min tj j , 1 dt. j
Obviously, I0 =
m A0 j=1
−α −1 tj j dt
=
m
∞
−αj −1
tj
j=1 1
dtj =
m 1 . αj
j=1
All the integrals Ij , j = 1, . . . , m, can be calculated identically. Let us calculate, for example, the integral I1 . We have
+∞ 1 +∞ m m −α −1 −αj −1 1 −1+β1 tj j tβ1 1 dt = t−α . . . t dt . . . dt I1 = m 2 dt1 1 j β /β β /β
t1 1
0
A1 j=1
2
t1 1
m
j=2
1 m m
1 1 1 −α1 −1+β1 − m −α1 −1+β1 j=2 β1 αj /βj = t1 dt = t dt1 1 1 α β /β 1 j α j α 0 j=2 j t1 j=2 j 0
m 1 α /β
1 1 = . · αj 1− m j=1 αj /βj 1
j=1
Similarly, Ii =
m j=1
Therefore, I=
m i=0
1 αj
·
1−
αi /βi
m , j=1 αj /βj
i = 1, . . . , m.
m m m m 1 1 αi /βi j=1 (1/αj )
m
Ii = + = . · αj αj 1 − j=1 αj /βj 1− m j=1 αj /βj j=1
i=1
(5)
j=1
Combining this with (4), we obtain (2). Now let us construct the function f (u), u ∈ Rm , that turns (2) in an equality. First, we define f (u) for u ∈ Rm + , and then extend f as an even function with respect to each variable to the whole space Rm . For u = (u1 , . . . , um ) ∈ Rm + , let u = (u1 , . . . , um ), where ui := min{ui , 1}. β1 βm β Further, consider the vector u := (u1 , . . . , um ) and put v = v(u) = (v1 (u), . . . , vm (u)) := (u β )∗ , βm arranged in nonincreasing where (u β )∗ is the vector whose coordinates are the numbers u1β1 , . . . , um m order. Now, for u ∈ R+ , we define the function f by setting
f (u) = v1 (u) − v2 (u) + · · · + (−1)m−1 vm (u) −
1 . 2
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Since 0≤
m
(−1)j−1 vj (u) ≤ 1,
j=1
we have f ∞ ≤
1 , 2
f ∞ =
1 2
while, actually, (6)
(the norm is attained, for example at the point u = e1 ). βj and estimate f , j = 1, . . . , m. To do this, consider the difference Let us verify that f ∈ m βj j=1 H (h > 0) f (u + hei ) − f (u) = v1 (u + hei ) − v2 (u + hei ) + · · · + (−1)m−1 vm (u + hei ) − (v1 (u) − v2 (u) + · · · + (−1)m−1 vm (u)). The vector u + hei differs from the vector u by only the ith coordinate, which is greater than the ith coordinate of the vector u. Therefore, the vector v(u + hei ) will differ from the vector v(u) as follows. Suppose that the number uiβi is the νth coordinate of the vector v(u). Then there exists a μ ≤ ν such that (u + hei )βi i is the μth coordinate of the vector v(u + hei ). In addition, the coordinates of the vectors v(u + hei ) and v(u) whose numbers are either less than μ or greater than ν coincide. Thus, the difference f (u + hei ) − f (u) will have the form f (u + hei ) − f (u) = (−1)μ−1 (ui + h)βi + (−1)μ vμ (u) + · · · + (−1)ν−1 vν−1 (u) − (−1)μ−1 vμ (u) − · · · − (−1)ν−2 vν−1 (u) − (−1)ν−1 uβi i = (−1)μ−1 (ui + h)βi + 2(−1)μ vμ (u) + · · · + 2(−1)ν−1 vν−1 (u) − (−1)ν−1 uβi i . To be definite, suppose that μ is odd (the case of an even μ is treated in a similar way). Then we can show that the following system of inequalities holds: uiβi − (ui + h)βi ≤ (ui + h)βi − 2vμ (u) + · · · + 2(−1)ν−1 vν−1 (u) − (−1)ν−1 uiβi ≤ (ui + h)βi − uiβi . The first inequality of this system is equivalent to the inequality 0 ≤ 2(ui + h)βi − 2vμ (u) + · · · + 2(−1)ν−1 vν−1 (u) − (1 + (−1)ν−1 )uiβi , which is obvious, because the numbers (ui + h)βi ,
vμ (u),
...,
vν−1 (u),
uiβi
are monotone nonincreasing. The second inequality of this system is equivalent to the inequality −2vμ (u) + · · · + 2(−1)ν−1 vν−1 (u) + (1 − (−1)ν−1 )uiβi ≤ 0, which is also obvious, because the numbers (ui + h)βi ,
vμ (u),
...,
vν−1 (u),
uiβi
are monotone nonincreasing. The system of inequalities proved above implies that f ∈ H βi . It is also clear that, for any index i = 1, 2, . . . , m, f βi ≤ 1. MATHEMATICAL NOTES
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Now let us calculate (Dαε )f (0) for ε = (+, . . . , +). To do this, let us first prove that β β Δt1 e1 . . . Δtm em f (0, . . . , 0) = −2m−1 min tβ1 1 , . . . , tβmm , 1 = −2m−1 min t1 1 , . . . , tmm . The proof will be by induction on m. For m = 2, this is verified immediately. This is the induction basis. In the calculation of Δt1 e1 . . . Δtm em f (0, . . . , 0), we can take Δt1 e1 , . . . , Δtm em in any convenient order, because the operators Δti ei and Δtj ej commute. β
β
To be definite, let us assume that, among the numbers t1 1 , . . . , tmm , the first one is the largest and the difference with respect to t1 will be calculated last. We express the difference Δt1 e1 . . . Δtm em f (0, . . . , 0) as 1 1 1 ··· (−1)j1 +···+jm f (j1 t1 , j2 t2 , . . . , jm tm ) Δt1 e1 . . . Δtm em f (0, . . . , 0) = j1 =0 j2 =0 1
=
···
j2 =0
−
jm =0
1
(−1)j2 +···+jm f (0, j2 t2 , . . . , jm tm )
jm =0 1
1
···
j2 =0
(−1)j2 +···+jm f (t1 , j2 t2 , . . . , jm tm ).
jm =0
By the induction assumption, 1 j2 =0
···
1
β β (−1)j2 +···+jm f (0, j2 t2 , . . . , jm tm ) = −2m−2 min t2 2 , . . . , tmm .
(8)
jm =0 β
Using the definition of the function f and the assumption that the number t1 1 is the largest among the β β numbers t1 1 , . . . , tmm , we obtain β
f (t1 , j2 t2 , . . . , jm tm ) = t1 1 − f (0, j2 t2 , . . . , jm tm ), and hence −
1
···
j2 =0
1
β β β (−1)j2 +···+jm (t1 1 − f (0, jm tm , . . . , jm tm )) = −2m−2 min t2 2 , . . . , tmm
jm =0
(here we have again used the induction assumption (8)). Finally, we obtain β β β β Δt1 e1 . . . Δtm em f (0, . . . , 0) = −2m−2 min t2 2 , . . . , tmm − 2m−2 min t2 2 , . . . , tmm β β β β β = −2m−1 min t2 2 , . . . , tmm = −2m−1 min t1 1 , t2 2 , . . . , tmm . Now, for ε = (+, +, . . . , +), taking (5) into account, we estimate Dαε f ∞ from below: m −α −1 β α α m−1 tj j min min tj j , 1 dt Dε f ∞ ≥ |(Dε )f (0, 0, . . . , 0)| = Aα 2 Rm + j=1
m = 2m−1 Aα
1−
j=1
m
1/αj
j=1 αj /βj
j
(9)
.
Using (6), (7), and (9), we obtain
m
Dαε f ∞ j=1 (1/αj ) 1− m m−1 j=1 αj /βj .
m
≥ 2 A 2 α m 1− j=1 αj /βj m αj /βj 1 − α /β j j j=1 f ∞ j=1 f βj H
The right-hand side of this inequality coincides with the constant in inequality (2). The sharpness of inequality (2) is proved. MATHEMATICAL NOTES
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m
Corollary 1. Suppose that αj > 0 for j = 1, . . . , m, with j=1 αj < 1. Then, for any function x(u) ∈ L∞ (Rm ) having, for each j = 1, . . . , m, an essentially bounded derivative ∂x/∂tj in the sense of Sobolev, the following sharp inequality holds:
m m
∂x αj 2m−1 21− j=1 αj 1− m j=1 αj α
m · · x∞ · . Dε x∞ ≤ m ∂tj Γ(1 − α ) 1 − α j j j=1 j=1 L∞ (Rm ) j=1
Proof. It is well known (see, for example, [18]) that, under the assumptions of the corollary, m ∂x βj H for β1 = · · · = βm = 1 and xβj = for all j = 1, . . . , m. x∈ ∂tj ∞ j=1
In conclusion, note that the condition m αj j=1
βj
