SOME TAUBERIAN THEOREMS FOR WEIGHTED MEANS OF DOUBLE INTEGRALS

Let p(x) and q(y) be nondecreasing continuous functions on [0,∞) such that p(0) = q(0) = 0 and p(x), q(y) → ∞ as x, y → ∞. For a locally integrable function f(x, y) on R+ = [0,∞) × [0,∞), we denote its double integral by F (x, y) = ∫ x 0 ∫ y 0 f(t, s)dtds and its weighted mean of type (α, β) by tα,β(x, y) = ∫ x 0 ∫ y 0 ( 1− p(t) p(x) )α ( 1− q(s) q(y) )β f(t, s)dtds where α > −1 and β > −1. We say that ∫∞ 0 ∫∞ 0 f(t, s)dtds is integrable to L by the weighted mean method of type (α, β) determined by the functions p(x) and q(x) if limx,y→∞ tα,β(x, y) = L exists. We prove that if limx,y→∞ tα,β(x, y) = L exists and tα,β(x, y) is bounded on R+ for some α > −1 and β > −1, then limx,y→∞ tα+h,β+k(x, y) = L exists for all h > 0 and k > 0. Finally, we prove that if ∫∞ 0 ∫∞ 0 f(t, s)dtds is integrable to L by the weighted mean method of type (1, 1) determined by the functions p(x) and q(x) and conditions p(x) p′(x) ∫ y 0 f(x, s)ds = O(1) and q(y) q′(y) ∫ x 0 f(t, y)dt = O(1) hold, then limx,y→∞ F (x, y) = L exists.

R y 0 f (t; s)dtds and its weighted mean of type ( ; ) by where > 1 and > 1. We say that R 1 0 R 1 0 f (t; s)dtds is integrable to L by the weighted mean method of type ( ; ) determined by the functions p(x) and q(x) if limx;y!1 t ; (x; y) = L exists. We prove that if limx;y!1 t ; (x; y) = L exists and t ; (x; y) is bounded on R 2 + for some > 1 and > 1, then limx;y!1 t +h; +k (x; y) = L exists for all h > 0 and k > 0. Finally, we prove that if R 1 0 R 1 0 f (t; s)dtds is integrable to L by the weighted mean method of type (1; 1) determined by the functions p(x) and q(x) and conditions p(x) p 0 (x) Z y 0 f (x; s)ds = O(1) and q(y) q 0 (y) hold, then limx;y!1 F (x; y) = L exists.

Introduction
Let p(x) and q(y) be nondecreasing continuous functions on [0; 1) such that p(0) = 0; q(0) = 0 and p(x); q(y) ! 1 as x; y ! 1. For a locally integrable function f (x; y) on R 2 + = [0; 1) [0; 1), we denote its double integral on R 2 + by F (x; y) = R x 0 R y 0 f (t; s)dtds and its weighted mean of type ( ; ) determined by the functions p(x) and q(y) by where > 1 and > 1. An improper double integral is said to be integrable to L by the weighted mean method of type ( ; ) determined by the functions p(x) and q(y) if We use the notion of convergence in Pringsheim's sense, that is, both x and y tend to 1 independently of each other in (3). If we take p(x) = x and q(y) = y in (1), we have the de…nition of (C; ; ) integrability of f (x; y) on [0; 1) [0; 1) given by [3]. The (C; 0; 0) integrability of f (x; y) is convergence of the improper double integral (2).
It is clear that if lim x;y!1 F (x; y) = L exists and F (x; y) is bounded on R 2 + , then the limit (3) also exists for > 1 and > 1. The converse of this implication is not true in general. The converse of this implication may be true only by adding some suitable condition which is called a Tauberian condition. Any theorem which states that convergence of (2) follows from the integrability of f (x; y) by the weighted mean method of type ( ; ) determined by the function p(x) and q(y) and a Tauberian condition is said to be a Tauberian theorem.
In recent years, there has been an increasing interest on summability methods for functions of one and two variables. First, Laforgia [6] obtained a su¢ cient condition under which convergence of the improper integral follows from (C; 1) integrability. Later, Çanak and Totur [1] extended the main results of Laforgia [6] to the (C; ) integrability of functions by weighted mean methods where > 1. Following these works, Totur and Çanak [10] obtained some Tauberian theorems in terms of the concept of the general control modulo of non-integer order for functions of one-variable. Recently, Özsaraç and Çanak [8] obtained Tauberian theorems for the iterations of weighted mean summable integrals. Totur et al. [9] introduced some new Tauberian conditions in terms of the weighted general control modulo for the weighted mean method of integrals. For some interesting Tauberian theorems for Cesàro and weighted integrability in quantum calculus, we refer the readers to Çanak et al. [2], Fitouhi and Brahim [5] and Totur et al. [11], etc. In [7], Móricz obtained one-sided Tauberian conditions which are necessary and su¢ cient in order that convergence follow from summability (C; 1; 1) of (2). More generally, Çanak and Totur [3] obtained a su¢ cient condition under which convergence of (2) follows from (C; ; ) integrability of (2) where > 1 and > 1.
In this paper we prove that if (3) exists and t ; (x; y) is bounded on R 2 + for some > 1 and > 1 , then lim x;y!1 t +h; +k (x; y) = L exists for all h > 0 and k > 0. As a corollary to this result, we show that if (2) is convergent to L and the function F (x; y) is bounded on R 2 + , then lim x;y!1 t 1;1 (x; y) = L. But, the converse of this implication may true under some conditions imposed on p, q and f . Moreover, we give a Tauberian condition under which convergence of improper double integrals follows from the existence of lim

Main Results
Theorem 1. If (3) exists and t ; (x; y) is bounded on R 2 + for some > 1 and > 1, then lim x;y!1 t +h; +k (x; y) = L exists for all h > 0 and k > 0. where where B denotes the Beta function de…ned by '(t; s; x; y)dtds = 1: We …rst prove that Since lim by the hypothesis, there exist numbers x " and y " for any given " > 0 such that jt ; (x; y) Lj < "; x x " ; y y " : It follows from (5) that To prove (6), we need to show that provided that x and y are large enough. We realize that by the hypothesis, the function t ; (x; y) is bounded on R 2 + . Therefore, there exists a constant K such that jt ; (x; y) Lj < K; for 0 x; y < 1: Using (5) and (8), we obtain, by (9), which tends to zero when x; y ! 1 for any …xed x " and y " . Thus, there exist some c '(t; s; x; y)dtds < "; x c x 1 " ; y b y 1 " : which tends to zero when x; y ! 1 for any …xed x " and y " (Note that Similarly, the integral Z x x Z y 0 '(t; s; x; y)dtds tends to to zero when x; y ! 1 for any …xed x " and y " (Note that p(x")=p(x) u (1 u) h 1 du tends to 1 as x ! 1 ). Thus, there exist some c '(t; s; x; y)dtds < "; x c x 3 " ; y b y 3 " : Hence, we have (10) for y " 3 g and this proves (6). We obtain Here, we write I(u; v; x; y) as where (p(t) p(u)) dt; and (q(s) q(v)) ds: Substituting p(t) = p(x) (p(x) p(u))x in I 1 (u; x), we have and similarly we have This completes the proof of Theorem 1.
Proof. Take = = 0 and h = k = 1 in Theorem 1. (2) is integrable to L by the weighted mean method of type (1; 0) determined by the function p(x) and
Theorem 4. If (2) is integrable to L by the weighted mean method of type (0; 1) determined by the function q(y) and q(y) q 0 (y) holds, then (2) converges to L.
Since the proofs of Theorem 3 and Theorem 4 can be obtained by the similar techniques and steps as in the proof of Theorem 2.3 in [1], we omit them.
By the next theorem, we recover convergence of the improper double integral (2) from its weighted mean method of type (1; 1) determined by the functions p(x) and q(y) under conditions (13) and (14).
Theorem 5. If (2) is integrable to L by the weighted mean method of type (1; 1) determined by the functions p(x), q(y) and and q(y) q 0 (y) then (2) converges to L.

SO M E TAUBERIAN THEOREM S FOR W EIG HTED M EANS O F DO UBLE INTEG RALS1459
Proof. Assume that (2) is integrable to L by the weighted mean method of type (1; 1) determined by the functions p(x) and q(y), that is, We rewrite G(x; y) as where It follows from (15), (16) and (17) that @ @y G 1 (x; y) is integrable to L by the weighted mean method of type (0; 1) determined by the functions q(y).

Conclusion
In this paper we have extended Tauberian theorems given for (C; ; ) integrability method to the weighted mean method of type ( ; ) determined by the functions p(x) and q(y). In a forthcoming work, we plan to obtain analogous results for the weighted mean method for functions of three or more variables.