Non-tangential summability of power expansions of functions of Hardy classes Текст научной статьи по специальности «Математика»

Математика. Физика

УДК 517.518

DOI: 10.17277/vestnik.2016.03.pp.445-455

NON-TANGENTIAL SUMMABILITY OF POWER EXPANSIONS OF FUNCTIONS OF HARDY CLASSES

A. D. Nakhman

Department "Technical mechanics and Machine parts", TSTU, Tambov, Russia; [email protected]

Keywords: estimates of the weak and strong type; maximal operators; non-tangential summability.

Abstract: The class Нp of X -means of the power expansions of analytic

functions ф e Hp is constructed. The behavior of X -means for tending of arbitrary point (y, r) of the circle to a point (x, 1) along non-tangential paths are studied. The estimates for the corresponding maximal operators and theorems of summability almost everywhere are established. The results for Hardy classes of functions are linked to

performance Hp = Lp © iLp in which Lp and Lp, respectively, are classes of X -means of Fourier series and conjugate series of f = Re ф .

Introduction. Formulation of the problem

Denote Q = [-n, n]; let 9 = 9(z) be function of complex variable z = r exp(z'x), analytic in a circle {z : | z |< 1}, Im9(0) = 0, 0 < r < 1, x e Q and

Class Hp = Hp (Q), H = H 1(Q) of such functions is called the Hardy class [1, vol. 1, p. 431]. In addition to Hp consider the Lebesgue class Lp = Lp (Q) of 2n -periodic functions of a real variable, for which

(

N 1/p

iifiip = j I f (x)|p dx <« , p > 1;

le J

set L = L(0 = L1 (0 . The behavior of the power series

Ф(г exp(ix)) = к(ф)rk exp(ikx)

k=0

(1)

of functions 9 e Hp, p > 1 on the circle of convergence, when r ^ 1 - 0, is well studied. So [2, p. 541]

<(exp(ix)) = lim <(r exp(ix)) = f (x) + ig (x), (2)

r ^1-0

existsalmosteverywhere. Here f, g e Lp and the coefficients in the expansion (1) can be founded as

Mk(<) = — J<(exp(it))exp(-ikt) dt, k = 0,1,...; (3)

2n Q

it is natural to assume that Mk (<) = 0 for k < 0. If we put

ro

<(exp(ix)) ~ Y Mk (<) exp(ikx), (4)

k=0

then (1) can be considered as a family of Abel-Poisson means of power series (4) on the circle of convergence. A significant strengthening of the result (2) is as follows

[1, vol. 1, p. 438]. For every function < e Hp the limiting values

<(exp(ix)) = lim <(r exp(ix))

(y,r x,1)

exist for almost all x, if the point (y, r) is tending to (x, 1), staying in the "corner" area, characterized by the condition

I y - x I.

1 - r

■< d, d = const, d > 0

(tending along non-tangential paths). In the future, it will be convenient to change the

designation of taking h = ln1; so h ~1 - r (r ^ 1 - 0). Non-tangential paths for

r

(y, h) ^ (x,+0) will be now the paths within

rd (x) = -j (y,h) | y e [-n, n], 0 < h < 1, LL_Ll < d k d = const, d > 0 .

This approach to the behavior of the power expansions of function < e Hp on the boundary of circle of convergence can be extended as follows. Let

X = {Xk(h), k = 0,1,...; Mh) = 1} (5)

be an arbitrary sequence infinite, generally speaking, determined by values of parameter h > 0 . In this paper we study the behavior of X -means

®h (<) = ®(9, y; X, h) = Mk (<)X k (h) exp(iky) (6)

k=0

of series (4) for(y, h) ^ (x,0), (y, h) e rd (x). The main results will be linked to the so-called estimates of weak and strong type of maximal operators generated by X -means of (4), X -means of Fourier series and conjugate series of functions f e Lp (Q), p > 1, which are, respectively

ro

Uh (f) = U (f, y; X, h) = XX|k|(h)Ck (f)exp(iky) (7)

k=—ro

Uh(f) = U(f,y;P,h) = -i £(sgnk)Pw(h)ck(f)exp(iky). (8)

k=-œ

Here {ck ( f )} is a sequence of complex Fourier coefficients

1 n

ck (f) = f(t)exp(-ikt) dt, k = 0, ± 1, ± 2,... (9)

-n

Denote H p7 , Lp and Lp the classes of means (6), (7) and (8), respectively. The results for Hardy classes of functions will be linked to performance

Hp7 = Lp © iLp.

Specifically [1, vol. 1, p. 173], the relation

9(r exp(iy)) = CTr (f, y) + i5 r (f, y) (10)

holds for all 9 e Hp and Re 9 = f e Lp, p > 1 where CTr ( f, y), CT r ( f, y) are the Poisson-Abel means and the conjugate means, corresponding to the case of

P k (h) = exp(-hk ) (still, h = ln1 ). A more general (than (10)) statement

r

0(9, y; P, h) = U (f, y; P, h) + iU(f, y; P, h) (11)

was established in Theorem 3.1 of [3] by the using of (3) and arguments of type [2, p. 542-545].

Maximal operators

Introduce the following operators:

x+n

* * * 1 C

f » f , where f = f (x) = sup - \\f(t)\dt, (12)

n>o nx-n

(Hardy Littlewood maximal operator) and

f ^ f *, where f * = f*(x) = sup\ J f (x + ')dt \. (13)

n>0 n<|t|<n 2tg 2

Operators (12) and (13) are defined [1, vol. 1, p. 60, 401-402, 442, 443] for every f e L ; moreover, in this case there is almost everywhere a conjugate function

~ 1 f t f ( x) = — lim I f (x +1) ctg-dt.

n n^+0 n<|t|<n 2

For each p > 1 the following estimates of "weak type"

Mf ||

f W -f II \p ÎW -f II

ц{х e Q\/(x) > ç > o}< Cp I —, ц{х e Q\f * (x)> ç > 0} < Cp I- pJ

(14)

hold; here mis the Lebesgue measure of corresponding sets. Along with (14) the estimates of "strong type" [1, vol. 1, p. 58-59, 404]

|| f (x) || p + ||f *( x) || p < C p || f || p, p > 1 (the boundedness of operators (12) and (13) from Lp in Lp for all p > 1) and

||f (x) || + ||f *( x)||< C (1+||f (ln +|f |)||);

||f (x) || p +||f *( x)|| p < C p || f ||, 0 < p < 1

are valid too. Here and below C will represent constants, which depend only on clearly specified indexes.

In accordance with X -means (6) - (8), introduced above, we define the following maximal operators:

< ^ ©*(<), where ©*(<) = ©*(<, x; X) = sup | ©(<, x; X, h) |; (15)

(y,h)erd(x)

f ^ U*(f), where U*(f) = U*(f,x;X) = sup | U(f,x;X,h) |; (16)

(y,h)erd(x)

f ^ U (f), where U (f ) = U (f, x; X) = sup | U(f, x; X, h)|. (17)

(y,h)erd(x)

For each h > 0 denote m =

1

. The basis of the results of the behavior of means

_2dh _

(6) - (8) is the following statement.

Theorem 1. Let the sequence (5) decreases so rapidly that

N | X N (h)| + N2 | AXn (h)|= o(1), N ^ro, (18)

and there is a constant C = Cx such that

ro

Y A2Xk (h)| < C. (19)

k=1 m

Then, for all f e L(Q) the estimates

U *(f, x; X)< Cx f *( x), (20)

U*(f,x;X) < Cx(f *(x) + f *(x)) (21)

hold.

Auxiliary assertion

Consider [1, vol. 1, p. 86, 153] the conjugate Dirichlet kernel

k 1 cos(k + ^)t Dk (t) = Y sin v t =-J---^ (22)

v=l 2t^ t 2 sin— t

2 2

and the conjugate Fejer kernel

Fk(t) = T"~7¿DC) = - (t) where Fk(t)= cos(k + ^ ; (23)

k + v=0 2tg21 ' 2(k +1)sin2 21

k = 0,1,...; DD0(t) = F~_1(t) = 0.

Lemma. For all k = 0,1,... and (y,h) e r(x) the estimate

I j f (t) Fk (y _ t) dt| < + m^f \x) + f *(x)) (24)

holds.

Proof. Let's start with a few comments. At k = 0 the left side of (24) vanishes, so consider k = 1,2,...

If (y, h) e rd (x), then, obviously, | y _ 11 > | x _ 11 _dh. Hence, for x and t, such,

that

| x _ 11 > — > 2dh

m

the estimate

| y -11 > x -11 is valid. Indeed, (26) follows (cf. (25)) from inequality

| y-11 > | x-11 - dh >-2| x-11 for all (y, h) e rd (x). Then, by definitions (23), the estimates

(25)

(26)

F (t) < ck:

|t|<n ;

Fk (()

< 2, 0 <111 <n kt2

(27)

(28)

hold.

Assume firstly k < m and obtain the relation (24). By (27) and (28) we have

< C

j f (t)Fk (y _ t) dt

k j |f (t )| dt-

j f (t) Fk (y _ t) dt

j f (t) F~k (y _ t) dt

x _t|<n

|x _t|<-V k

j f (t )ctg dt

-<|x _t|<n

j | f (t) |

Fk (y _ t) dt =

-<|x _t|<n

= C (J 1 (x, k) +J 2 (x, k) +J 3 (x, k)

(29)

It is obvious that

J 1( x, k) < f (x).

(30)

x _n

Further,

J 2 (x, k) =

J f (t)ctg ^ dt + J f (t)

. x - y sin--

- < |x-t| < n k

x-t y-t 1 „ . ^ sin-sin-

k <|x-t|<n 2 2

-dt

Taking into account (26), we have

|J2 (x,k)|<C

J* C

f (x) + h J |f (t)|-

dt

— < |x-t|<n

k

(x - ty

Here [4]

J |f(t)|—— dt = J|f (x +1)|-2 dt <

1 (x -1)2 1

—<| x—t |<n -^t^n

k k

S k

< Ck Tl-y J | f (x +1) | dt < Ckf \ x)

j=1 (2j 1) 2 j-1 2 j

2J

■< t < — k k

if a positive integer S chosen from the condition

2S-1 2s

<n <k k

Hence

| J 2 (x, k) |< Cf J * (x) + ^f *( x) ^ < C (~ * (x) + f * (x))

Finally, in view of (26) and (31)

J 3( X, k) < Cf (x).

Now, according to (29), (30), (32), (33), the estimate (24) is valid at all k < m. Consider now the case of k > m. By (29) we have

n J f (t )Fk (y -1) dt < C J|f (t)kdt +

-n | x—1| < 1/m

J f (t) ctg

y -1

dt

— < | x—1| < n m

(31)

(32)

(33)

1

J| f(t) |

<|x-t|<n

Fk (y -1) dt

= CI —J1 (x, m) +J2 (x, m) + 1 (x, k, m) 1 (34) m J. (34)

According to (30) and (32) we obtain

>¡4 i J'jb jjj \

J 1(x,m) < f (x), |J2 (x, m) | < C(f (x) + f (x)). Further, in view of (28) and (31)

m

I(x,k,m)) < C-f (x) < C f (x). k

+

2

+

It follows now from (34) that

n

j f (t)Fk (y -1) dt

< C (1 + -)(~*(x) + f *(x))

for all k > m.

Thus, the estimate (24) is valid for all k = 1, 2, ..., and lemma is proved.

Proof of theorem 1

We shall prove (21), the proof of (20) is similar to (or even easier, [5]). Applying (9), Abel transform twice [1, vol. 1, p. 15], the obvious estimate | DN (t) |< N, N =1, 2,..., (cf. (22)), and (27), we obtain

\U( f, y; X, h)| =

N

X

Ik=1

lim 1 f f (t)J£x- (h)sin k(y -t)

-1) \dt

1 i " n - lim k n (h) ff(t) 5n (y -1) dt + NAX n-i(h) f f(t) Fn-iC^ -1) dt + n N ^+<»1 J J

I -n -n

N-2

+ X(k+ k=1

V —z

X (k +1) A% (h) f f (t )F- (y -1 ) dt

-n

n N-2

< C lim

N

(n | X N (h)|+ N2 | AX N (h) |) ;

k=1

j| f (t) | dt + Y(k +1)| A2Xk(h) | j f (t)Fk(y -1) dt

- n k=1 -n

Due (18) it follows that

ro n

Uf, y; X, h)| < C Y (k +1)|A2Xk (h)| j f (t)Fk (y -1) dt

According to (24), we obtain

f, y; X, h)\ < C (*(x) + f *(x))Y|A2Xk (h)| k + mm

and, because of the condition (19), we have the assertion (21).

Estimates of the weak and strong type Theorem 2. Under the conditions of Theorem 1 the estimates of weak type

M(x e Q|(Tf )( x) > ç > 0} < C _,x

||fH, p , 1

and strong type

|| Tf ||p < cp,x|| f ||p, p > 1; || Tf || < Cx1 f (ln+ | f | )); HTfHp<CP,X||f ||, 0<p< 1

n

hold if Tf are any of the operators (15) - (17).

The assertion of theorem for the operators (16), (17) follows from Theorem 1 and the corresponding estimates for (12), (13) cited in paragraph 2. The assertion for the operator (15) now follows from (11).

Non-tangential summability

Theorem 3. If the sequence (5) satisfies the conditions (18), (19) and

lim Xk(h) = 1, k = 0,1,..., (35)

h^0

then the relations

lim U (f, y; X, h) = f (x), (36)

(y,h)^( x,0)

(y,h)erd(x)

lim U( f, y; X, h) = f (x) (37)

(y, h) >(x,0)

(y,h)erd (x)

hold almost everywhere for each f e L(Q). Under the same conditions on the sequence (5) the equality

lim 9(9, y; X, h) = 9(exp(ix)) (38)

(y,h) >(x,0)

(y,h)erd(x)

is valid for each 9 e H and almost all x .

The relations (36), (37) follow from the weak type estimates (Theorem 2) and condition (35) by the standard method [1, vol. 2, p. 464-465]. The relation (38) follows from (11), (36) and (37), when you consider that the limit of the left side (38) almost everywhere is

f (x) + i f (x), which is equal 9(exp(ix)) by (2) h (11).

Piecewise convex summation methods

It was noted in [3] that under the conditions (18) every piecewise-convex sequence (5) satisfies the condition

ro

X k Ak (h)| < Cx .

k=1

By virtue of piecewise convex sequence (5) the second finite differences A Xk (h) retain the sign. Suppose for definiteness, it will be a plus sign at all sufficiently large k (depending, generally speaking, from h), namely k > t (m), where t = t (m) -some positive integer,

t = t (m) = T(m, X) < m. (39)

The sum (19) does not exceed

Cx

fro ro 2

r k a2X k (h) + X k;7 |A2Xk (h

Vk=1 k=m

(40)

In the second sum in are positive by (39); applying twice Abel

transform, we have

•JU 1 1 1

(h)| — = - Xk2A2Xk(h) = - m2A2Xm (h) + £(2k - 1)AXk(h)

,1 I m m, m , .

k=m k=m V k=m+1 y

= mA2Xm (h) + AXm+i(h) + — YXk (h) •

m mk =m+2

Thus, under conditions (18) and

1 x

- k (h)| < Cx, (41)

m ^^

k=m+2

the assertions of Theorems 2 and 3 are valid for each piecewise-convex sequence (5). Exponential summation methods

Summation methods

X0(h) = 1, Xk(h) = X(x,h) |x=k , k = 1,2,..., where X(x,h) = exp(-hip(x)),

were studied in [3] in the case of "radial" convergence; in particular, it was given the condition of piecewise convexity of sequence {X k (h)}. In this paper we consider

X(x,h) = exp(- hxa), a> 1.

(42)

This function is a piecewise-convex, since

X'(x, h) = -ah xa-1, X''(x, h) = ah xa-2 (ha xa - (a -1)) (43)

and the second derivative (43) changes its sign once. Consequently, the sequence

Xk(h) = exp(- hka), a > 1, (44)

is a piecewise-convex too, and the second finite differences have constant sign for all

1 1

l 1

2d (a-1) |a a

k ^MO-1^ ma , so that t= 1 +

at the same time t < m for sufficiently large m.

It is obvious that condition (35) is valid for the sequence (44), and is also easy to verify the conditions (18), since (by Lagrange's theorem)

AXk (h) = ah(k + 9)a-1 exp(-h(k + 9)a), where 9 = 9(k) e (0,1).

We verify the satisfiability of condition (41). Because of decrease of the function (42) we have a sum in (41) not exceeding

1 ^ 1 ~ / \ 1 - -œ 1-1 1-- ( 1

- Y | Xk (h)| < - f exp(- hxa )dx =-h a f/a exp(-/)d/ < Cah ar| -

mm J a m J Va y

k=2 0 0

(45)

where r = r^—j - Euler gamma function. For a > 1 the right side of (45) does not

exceed a constant that depends only on a. Thus, Theorems 2 and 3 are valid for exponential summation methods (44).

References

1. Zigmund A. Trigonometricheskie ryady : per. s angl [Trigonometric series: per. from English], Moscow: Mir, 1965, vol. 1, 615 p., vol. 2, 537 p.

2. Bari N.K. Trigonometricheskie ryady [Trigonometric series], Moscow: Fizmatlit, 1961, 936 p. (In Russ.)

3. Nakhman A.D. [Summation of Power Series of Functions of Classes НVp on Boundary of the Convergence Circle], Transactions of Tambov State Technical University, 2014, vol. 20, no. 3, pp. 530-538. (In Russ., abstract in Eng.)

4. Nakhman A.D., Osilenker B.P. [Еxponential Methods of Summation of the Fourier Series], Transactions of Tambov State Technical University, 2014, vol. 20, no. 1, pp. 101-109. (In Russ., abstract in Eng.)

5. Nakhman A.D., Osilenker B.P. [Non-Tangential Convergence of the Generalized Poisson Integral], Transactions of Tambov State Technical University, 2015, vol. 21, no. 4, pp. 660-668, doi: 10.17277/vestnik.2015.04.pp.660-668 (In Russ., abstract in Eng.)

Некасательная суммируемость cтепенных разложений функций классов Харди

А. Д. Нахман

Кафедра «Техническая механика и детали машин», ФГБОУВО «ТГТУ», г. Тамбов, Россия; [email protected]

Ключевые слова: максимальные операторы; некасательная суммируемость; оценки слабого и сильного типов.

Aннотация: Построен класс Н£ обобщенных X -средних степенных рядов аналитических функций ф е Нр. Изучено поведение X -средних при стремлении произвольной точки круга (у, г) к точке (х, 1) по некасательным направлениям. Получены оценки соответствующих максимальных операторов и теоремы о суммируемости почти всюду. В основе результатов лежит представление

Н£ = Ь£ © в котором ¿X и Ьр соответственно, классы X -средних рядов Фурье и сопряженных рядов функций / = Яе ф .

Список литературы

1. Зигмунд, А. Тригонометрические ряды : пер. с англ. : в 2 т. / А. Зигмунд. -М. : Мир, 1965. - Т. 1. - 615 с., Т. 2 - 537 с.

2. Бари, Н. К. Тригонометрические ряды / Н. К. Бари. - М. : ФИЗМАТЛИТ, 1961. - 936 с.

3. Нахман, А. Д. Суммирование степенных рядов функций классов

на границе круга сходимости / А. Д. Нахман // Вестн. Тамб. гос. техн. ун-та. -2014. - Т. 20, № 3. - С. 530 - 538.

4. Нахман, А. Д. Экспоненциальные методы суммирования рядов Фурье / А. Д. Нахман, Б. П. Осиленкер // Вестн. Тамб. гос. техн. ун-та. - 2014. - Т. 20, № 1. - С. 101 - 109.

5. Нахман, А. Д. Нетангенциальная сходимость обобщенного интеграла Пуассона / А. Д. Нахман, Б. П. Осиленкер // Вестн. Тамб. гос. техн. ун-та. - 2015. -Т. 21, № 4. - С. 660 - 668.

Nichttangentiale Summierung der Potenzzerlegungen der Funktionen der Hardy-Klasse

Zusammenfassung: Es ist die Klasse Hp der verallgemeinerten X -mittleren Potenzreihen der analytischen Funktionen 9 e Hp aufgebaut. Es ist das Verhalten von P -mittleren beim Streben eines willkürlichen Punktes des Kreises (y, r) zum Punkt (x, 1) nach den nicht Tangentialrichtungen erlernt. Es sind die Einschätzungen der entsprechenden maximalen Operatoren und des Theorems über der Summierung fast überall erhalten. Zugrunde der Ergebnisse liegt die Beibringung Hp = Lp © iLp, in der

Lp h Lp , entsprechend, die Klassen der P -mittleren Reihen von Foirier und der verknüpften Reihen der Funktionen f = Re^.

Sommation non-tangentiel des extensions de puissance des fonctions de classe Hardy

Résumé: Est construite la classe H f des séries moyennes de puissance des fonctions analytiques . Est étudié le comportement des moyennes P lors de la quête de l'arbitraire d'un point d'un cercle (y, r) à un point (x, 1) par les directions non-tangentielles. Sont obtenues les évaluations des opérateurs maximums correspondants et du théorème de la sommation presque partout. A la base des résultats se trouve la

représentation Hp = Lp ®iLp, dans laquelle Lp h Lf sont respectivement les classes des séries moyennes de Fourier et des séries de fonctions f = Re 9.

Автор: Нахман Александр Давидович - кандидат физико-математических наук, доцент кафедры «Техническая механика и детали машин», ФГБОУ ВО «ТГТУ», г. Тамбов, Россия.

Рецензент: Куликов Геннадий Михайлович - доктор физико-математических наук, профессор, заведующий научно-исследовательской лабораторией «Механика интеллектуальных материалов и конструкций», ФГБОУ ВО «ТГТУ», г. Тамбов, Россия.