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7URAL MATHEMATICAL JOURNAL, Vol. 3, No. 2, 2017
ON A-CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE TRIGONOMETRIC FOURIER SERIES1
N.N. Nikolay Yu. Antonov
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences; Ekaterinburg, Russia Nikolai.Antonov@imm.uran.ru
Abstract: We consider one type of convergence of multiple trigonometric Fourier series intermediate between the convergence over cubes and the A-convergence for A > 1. The well-known result on the almost everywhere convergence over cubes of Fourier series of functions from the class L(ln+ L)d ln+ ln+ ln+ L([0, 2n)d) has been generalized to the case of the A-convergence for some sequences A.
Key words: Trigonometric Fourier series, Rectangular partial sums, Convergence almost everywhere.
Suppose that d is a natural number, Td = [—n, n)d is a d-dimensional torus, and <p: [0, ^ [0, is a nondecreasing function. Let ^>(L)(Td) be the set of all Lebesgue measurable real-valued functions f on the torus Td such that
J <p(\f (t)|)dt <
Td
Let f € L( Td), k = (k1, k2,..., kd) € Zd, x = (x1, x2,..., xd) € Rd, and kx = kV + k2x2 + ... + kdxd. Denote by
Ck = (w//(t,<!"'kl<it
Td
the kth Fourier coefficient of the function f and by
E ckeikx (1)
kezd
the multiple trigonometric Fourier series of the function f.
Let n = (n1,n2,... ,nd) be a vector with nonnegative integer coordinates, and let Sn(f, x) be the nth rectangular partial sum of series (1):
Sn(f, x) = E Ckeikx.
k=(fc1,...,fcd): |kj|<nj, 1<j<d
Denote by mesE the Lebesgue measure of a set E and let ln+ u = ln(u + e), u > 0.
In 1915, in the case d = 1, N.N. Luzin (see [1]) suggested that the trigonometric Fourier series of any function from L2(T) converges almost everywhere. A.N. Kolmogorov [2] constructed an example of a function F € L(T) whose trigonometric series diverges almost everywhere and, later on [3], of a function from L(T) with the Fourier series divergent everywhere on T. L. Carleson [4] proved that Luzin's conjecture is true: if f € L2(T), then the Fourier series of the function f converges almost
1This work was supported by the Russian Science Foundation (project no. 14-11-00702).
everywhere. R. Hunt [5] generalized the statement about the almost everywhere convergence of the Fourier series to the class L(ln+ L)2(T), particularly, to Lp(T) with p > 1. P. Sjolin [6] generalized it to the wider class L(ln+ L)(ln+ ln+ L)(T). In [7], the author showed that the condition f € L(ln+ L)(ln+ ln+ ln+ L)(T) is also sufficient for the almost everywhere convergence of the Fourier series of the function /. At present, the best negative result in this direction belongs to S.V. Konyagin [8]: if a function <p(u) satisfies the condition <p(u) = o(v,\JIn«/ In In«) as u —> +oo, then, in the class <^(L)(T), there exists a function with the Fourier series divergent everywhere on T.
Let us now consider the case d > 2, i.e., the case of multiple Fourier series. Let A > 1. A multiple Fourier series of a function f is called A-convergent at a point x € Td if there exists a limit
lim Sn(f, x)
min{nj :1<j<d}^+<^
considered only for vectors n = (n1, n2,..., nd) such that 1/A < < A, 1 < < d. The
A-convergence is called the convergence over cubes (the convergence over squares for d = 2) in the case A = 1 and the Pringsheim convergence in the case A = i. e., in the case without any restrictions on the relation between coordinates of vectors n.
N.R. Tevzadze [9] proved that, if f € L2(T2), then the Fourier series of the function f converges over cubes almost everywhere. Ch. Fefferman [10] generalized this result to functions from Lp(Td), p > 1, d > 2. P. Sjolin [11] showed that, if a function f is from the class L(ln+ L)d(ln+ ln+ L)(Td), d > 2, then its Fourier series converges over cubes almost everywhere. The author [12] (see also [13]) proved the almost everywhere convergence over cubes of Fourier series of functions from the class L(ln+ L)d(ln+ ln+ ln+ L)(Td). The best current result concerning the divergence over cubes on a set of positive measure of multiple Fourier series of functions from ^>(L)(Td), d > 2, belongs to S.V. Konyagin [14]: for any function <^(u) = o(u(ln u)d-1 lnln u) as u ^ there exists a function F € ^(L)(Td) with the Fourier series divergent over cubes everywhere.
On the other hand, Ch. Fefferman [15] constructed an example of a continuous function of two variables, i. e., a function from C(T2) whose Fourier series diverges in the Pringsheim sense everywhere on T2. M. Bakhbukh and E.M. Nikishin [16] proved that there exists F € C(T2) such that its modulus of continuity satisfies the condition w(F, ¿) = O (ln-1 (1/^)) as 5 ^ +0 and its Fourier series diverges in the Pringsheim sense almost everywhere. A.N. Bakhvalov [17] established that, for m € N and any A > 1, there is a function F € C(T2m) such that the Fourier series of F is A-divergent everywhere and the modulus of continuity of F satisfies the condition
w(F, 5) = O (ln-m(1/5)) , 5 ^ +0. (2)
Later on, Bakhvalov [18] proved the existence of a function F € C(T2m) satisfying condition (2) and such that its Fourier series is A-divergent for all A > 1 simultaneously.
Let A = {Av}^L1 be a nonincreasing sequence of positive numbers. Assume that
( 1 ni
Qa = \ n = (nl,n2,.. .,nd) € Nd :-— < — < 1 + \nj, 1 < i,j < d
\ i + xni ~ ni ~ nJ'
We will say that a multiple Fourier series of a function f € L(Td) is A-convergent at a point x € Td if there exists a limit
lim Sn(f, x).
nSHA, min{nj:1<j<d
Let us note that, if Av = A — 1 for some A > 1, then the condition of A-convergence turns into the condition of A-convergence defined above. And if Av ^ 0 as v ^ ro, then the condition of A-convergence is weaker than the condition of A-convergence for any A > 1.
The author proved [19] that, if a sequence A = {\v}^=1 satisfies the condition ln2 \v = o(ln v) as v —y to, then there exists a function F € C(T2) such that its Fourier series is A-divergent almost everywhere on T2.
In the present paper, we obtain the following statement that strengthens the result of [12].
Theorem 1. Assume that a nonincreasing sequence of positive numbers A = {Av }^=1 satisfies the condition
A„ = o(I) (3)
and a function <p: [0, +to) — [0, +to) is convex on [0, +to) and such that ^>(0) = 0, ^>(u)u-1 increases on [u0, +to), and (p(u)u-1-6 decreases on [u0, +to) for some u0 > 0 and any 5 > 0. Assume that the trigonometric Fourier series of any function g € f(L)(T) converges almost everywhere on T. Then, for any d > 2, the Fourier series of any function f from the class ^>(L)(ln+ L)d-1(Td) is A-convergent almost everywhere on Td.
Theorem 1 and the result of paper [7] imply the following statement.
Theorem 2. Let a nonincreasing sequence of positive numbers A = {Av}^=1 satisfy condition (3), d > 2. Then the Fourier series of any function f from the class
L(ln+ L)d(ln+ ln+ ln+ L)(Td)
is A-convergent almost everywhere on Td.
Proof of Theorem 1. Let a sequence A = {Av}^=1 and a function tp satisfy the conditions of the theorem. Let ^>d(u) = ^>(u)(ln+ u)d-1 for short. Without loss of generality, we can consider only functions (fid such that the functions (pd,(s/u,) are concave on [0,+oo). Otherwise, we can consider the functions ^>d(u + ad) — bd (with appropriate constants ad and bd) instead of pd. The corresponding class ^>d(L)(Td) will be the same in this case.
Denote by Sn(f, x) the nth cubic partial sum of the Fourier series of the function f:
Sn(f, x) = Sn(f, x), where n = (n, ...,n).
Suppose that
M(f, x) = sup\Sn(f, x)\, neN
MA(f,x) = sup \Sn(f,x)\. nenA
Under the conditions of the theorem (see [12, formula (3.1) and Lemma 3]), there are constants Kd > 0 and yd > 0 such that
mes{x€Td:M(/,x)>y}<^( y^d(|/(x)|)dx + l), y > yd, f £ <pd(L)( Td). (4)
Td
Using (4), we will prove that, for every y > yd and f € pd(L)(Td),
mes{x€Td:MA(/,x)>y}<^ j ^d(|/(x)|) dx + l) (5)
Td
and, for every f € ^>d+1(L)(Td),
J MA(/, x)dx < Bd( J (x)|) dx + 1^ , (6)
where Ad is independent of / and y; Bd is independent of /.
The proof is by induction on d. Consider the base case, i. e., d =1: statement (5) immediately follows from (4) because M(/, x) = Ma(/, x) in the one-dimensional case. Similarly, (6) is a consequence of [5, Theorem 2].
Let d > 2. Suppose that statements (5) and (6) hold for d — 1 and let us show that the same is true for d.
First, let us prove the validity of (5). Let n = (n1,«2,... , nd) € Qa. According to (3), there is an absolute constant C > 0 such that Avv < C for all natural numbers v. Combining this with the definition of Qa, we obtain that, for all i, j € {1,2,..., d},
|n - nj |< C. (7)
Recall that, if n = (n1,«2,... , nd), then the following representation holds for the nth rectangular partial sum of the Fourier series of the function /:
1 f d
>',,(/• x) 37 /II i/?)-/'i-r' ' ......' /'î'//' ••• dtd, (8)
Td j = 1
where Dn(t) = sin((n + 1/2)t)/(2sin(t/2)) is the one-dimensional Dirichlet kernel of order n. Let
d
us add to and subtract from the d-dimensional Dirichlet kernel H Dnj (tj) of order n the sum
j=i
d / k d E ll Dm (tj ) II Dnj (tj )
k=2 ^ j=1 j=k+i
(here and in what follows, we suppose that all products n with an upper index less than a lower one are equal to 1). Rearranging the terms, we obtain
d d—1 / k d k+1 d ■> d
riDnj (tj ) = E( IlDni (tj ) U Dn> (tj ) -n D„1 (tj ) [] D„j (tj )]+[] D„1 (tj ) = j=1 k=1 ^ j=1 j=k+1 j=1 j=k+2 ' j=1 d ✓ k— 1 d ■> d
= E ( II Dni(tj) Ü Dnj (tj) (Dnk(tk) - Dni(tk)) ) + Ü Dni(tj). k=2 ^ j=1 j=k+1 ' j=1
From this and (8), it follows that
d / k—1 d Sn(f^) = Y,-d / (II Dn^j) II Dn](tn{Dnk(tk)-D.,Atk)) ix k=2 ^ Td j=1 j=k+1
d
:f{xl + i1,..., xd + td) dt1... dtd + y fj £)„i (i^) /(x1 + t1,...,xd + td)dt1... dtd
Td j=1
d 1 f
Ep J (Dn* (tk) - Dni(tk) ]x
k—1 d >. / r[Dni (tj) n Dnj (tj )f (x1 +i1,...,xd+id) dt1 ...dik—1dtk+1 ...didJ dtk+Sni (f, x). Td-i j=1 j=k+1
(9)
Note that the latter term on the right hand side of (9) is the n1th cubic partial sum of the Fourier series of the function f. By (7), for all k € {2,3,..., d} and t € T, we have \Dnk(t) — Dni(t)\ < C. Combining this with (9), we obtain
d
k—1 d
k=2 t Td-i j^1 j=k+1
xf (x1 +11,..., xk—1 + tk—1, tk, xk+1 + tk+1,..., xd + td) dt1... dtk—1dtk+1... dtd
dtk + \Sni (f, x)\.
Applying the definitions of MA(f, x) and M(f, x), from the latter estimate, we obtain
d a j A k 1 d
Ma(/,x)<M(/,x) + -E / sup — / n^^') II
n uo-J n= (ni ,n2, ••• ,nd)eHA n J „•_ i.,1
-i j=1 j=k+1
dtk = (10)
T Td-i
xf (x1 +11,..., xk—1 + tk—1, tk, xk+1 + tk+1,..., xd + td) dt1... dtk—1dtk+1... dtd
Cd
= M(/,x) + -VMfc(/,x), n z—' k=2
where Mk(f, x) denotes the kth term of the sum on the left hand side of the equality in (10). Let k € {2,3,... ,d}. Consider Mk(f, x). Denote by gk,tk the function of d — 1 variables that can be obtained from the function f by fixing the kth variable tk:
gk,tk(t1,...,tk—1,tk+1,...,td) = f(t1,...,tk—1,tk,tk+1,...,td), (t1,...,tk—1,tk+1,...,td) € Td—1.
Define QA as the set of mk = (m1,..., mk—1, mk+1,..., md) € Nd—1 such that m = (m1,..., md) € Qa. Note that, in view of the invariance of Qa with respect to a rearrangement of variables, the set QA is independent of k. Suppose that n'k = (n1,..., n1, nk+1,..., nd) € Nd—1. Then
k—1 d t / IlA^) n n<> i/?)x
nd—
Td-i
-i j=1 j=k+1
xf (x1 +11,..., xk—1 + tk—1, tk, xk+1 + tk+1,..., xd + td) dt1... dtk—1dtk+1. . . dtd =
and
dxk.
= Snk (gMk, (x1 ,...,xfc-1,xfc+1,...,xd))
Mfc (/, x)=/ sup Snk (gKxk, (x1 ,...,xfc-1,xfc+1,...,xd))
J nken' k v '
Further,
me^ x € Td : Mk (/, x) > y} = 2n me^(x1,...,xk-1,xfc+1,...,xd) € Td-1 : Mk (/, x) > y} <
2n f
< — / Mk(f, x) dx1... dxk~1dxk+1... dxd =
1
Sn'k (gfc,xk , (x , • • • , x , x + , . . . , x ^
pd-1
2n f
- / sup
y J n', en'
Td
/ sup ^(fiffe^fc ^a;1,...,^ \xfc+1,...,xd))
T Td-1
dx =
dx ... dx : 1dxk+1... dxd | dxk
(11)
From this, applying the induction hypothesis (more precisely, statement (6) for the dimension d — 1) to the inner integral on the right hand part of (11), we obtain
mes jx g Td : .1//,(/. x) > y} < y j i j <£d(|/(x)|) dx1... dx^dx^1... dxd + l) dxfc <
pd-1
<
(27r)2Bd_1
y
Pd(|/(x)|) dx + 1
Td
(12)
According to (10),
{x g Td : MA(/,x) > y} c {x g Td : M(/,x) > |} |J ( |J {x g Td : .\/,i/.x) > ^J^}) •
k=2 (13)
Combining (13), (4) and (12), we obtain (5) with the constant Ad = 2Kd + 8n(d — 1)2Bd-^.
Now, we only need to prove the validity of statement (6). To this end, let us use statement (5) proved above.
From (5), it follows that the majorant Ma(/, x) is finite almost everywhere on Td for all / € ^>d(L)(Td), in particular, for all / € L2(Td). Applying Stein's theorem on limits of sequences of operators [20, Theorem 1], we see that the operator Ma(/, ■) is of weak type (2,2), i.e., there is a constant Ad > 0 such that, for all y > 0 and / € L2(Td),
mes jx g Td : M\(f, x) > y j < ^ J |/(x)|2dx
Td
(14)
Similarly, from [20, Theorem 3], we can obtain the following refinement of statement (5): there is a constant Ad > 0 such that, for all y > yd/2 = Ad and / € ^>d(L)(Td),
mes {x g Td : MA(/,x) > y} < J <pd (^y^j < ^ J ^(|/(x)|) dx. (15)
Further, let f € ^>d(L)(Td) and y > 0. Suppose that
if(x), \f(x)\ >y,
g(x) = gy(x) = s h(x) = (x) = f (x) — g(x).
\f(x)\ < y;
Define A/(y) = mes {x € Td : MA(f, x) > y}. Then A/(y) < me^x € Td : MA(g,x) > y/2} + me^x € Td : MA(h,x) > y/2} = Ag(y/2) + Afc(y/2). From this, using the equality
MA(f, x) dx = —J ydA/(y) = J A/ (y) dy
Td 0 0
(see, for example, [21, Chapter 1, § 13, formula (13.6)]), we obtain
oo oo o
I MA(/,x) dx<yd(2vr)d + J Xf(y)dy<yd(^)d + f Afl(|) dy + J A„ (|) dy. (16)
Td Sd Sd Sd
Taking into account that g € ^>d(L)(Td) and h € Lo(Td) c L2(Td) and applying estimate (15) to Ag(y/2) and estimate (14) to Ah(y/2), from (16), we obtain
J M\(f, x) dx<yd(27r)d + 2Ad yYi J <pd(\g(t)\)dt\dy + 4AlJ J \h(t)\2 dt) dy =
Td yd ^ Td Sd ^ Td
= yd(27r)d + 2Adj J M\m\)dt\dy + AAlJ J |/(t)|2dtW
Sd ^ {teTd |/(t)|>y} ' Sd ^ {teTd |/(t)|<y} '
(17)
Applying Fubibi's theorem to the integrals on the right hand side of (17), we conclude that
, 1/(t)| V
J M\(f,x) dx < 2Ad J <Ai(|/(t)|)l J j)dt+
Td {teTd |/(t)|>Sd} Sd
+ ^2dJ\m\2( j ||]dt+yd(2vr)d,
Td |/(t)|
hence, statement (6) follows easily.
Finally, the A-convergence of the Fourier series of an arbitrary function from the class ^d(L)(Td) can be obtained from (5) by means of standard arguments (see, for example, [12, Lemma 3]). Theorem 1 is proved. □
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