On weighted Hardy inequality with two-dimensional rectangular operator -- extension of the E. Sawyer theorem
aa r X i v : . [ m a t h . F A ] S e p On weighted Hardy inequality with two-dimensional rectangularoperator – extension of the E. Sawyer theorem
V. D. Stepanov and E. P. Ushakova Abstract:
A characterization is obtained for those pairs of weights v and w on R , for whichthe two–dimensional rectangular integration operator is bounded from a weighted Lebesgue space L pv ( R ) to L qw ( R ) for < p = q < ∞ , which is an essential complement to E. Sawyer’s result [14]given for < p ≤ q < ∞ . Besides, we declare that the E. Sawyer theorem is actual if p = q only,for p < q the criterion is less complicated. The case q < p is new. Key words:
Rectangular integration operator; Hardy inequality; weighted Lebesgue space.
MSC:
Let n ∈ N . For Lebesgue measurable functions f ( y , . . . , y n ) on R n + := (0 , ∞ ) n the n –dimensional rectangular integration operator I n is given by the formula I n f ( x , . . . , x n ) : = Z x . . . Z x n f ( y , . . . , y n ) dy . . . dy n ( x , . . . , x n > . The dual transformation I ∗ n has the form I ∗ n f ( x , . . . , x n ) : = Z ∞ x . . . Z ∞ x n f ( y , . . . , y n ) dy . . . dy n ( x , . . . , x n > . Let < p, q < ∞ and v, w ≥ be weight functions on R n + . Consider Hardy’s inequality (cid:18)Z R n + (cid:0) I n f (cid:1) q w (cid:19) q ≤ C n (cid:18)Z R n + f p v (cid:19) p ( f ≥ (1)on the cone of non–negative functions in weighted Lebesgue space L pv ( R n + ) . The constant C n > in (1) is assumed to be the least possible and independent of f . For a fixed weight v and a parameter p > the space L pv ( R n + ) consists of all measurable on R n + functions f such that R R n + | f | p v < ∞ .The problem of characterizing the inequality (1) is well known and has been consideredby many authors (see [1, 4, 8, 12, 14, 16, 17] and references therein). The one–dimensionalcase of this inequality has been completely studied (see [5–7, 13]). However, for n > difficulties arise, preventing characterizing (1) without additional restrictions on v and w .Nevertheless, E. Sawyer’s result is well known for arbitrary v, w in the case < p ≤ q < ∞ .To formulate it we denote p ′ := p/ ( p − and σ := v − p ′ . Theorem [14, Theorem 1A]
Let n = 2 and < p ≤ q < ∞ . The inequality (1) holds for all measurable non-negative functions f on R if and only if A := A ( p, q ) := sup ( t ,t ) ∈ R (cid:2) I ∗ w ( t , t ) (cid:3) q (cid:2) I σ ( t , t ) (cid:3) p ′ < ∞ , (2) Computing Center of FEB RAS, Khabarovsk 680000, Russia; E-mail: [email protected]. V. A. Trapeznikov Institute of Control Sciences of RAS, Moscow 117997, Russia; E-mail:[email protected]. := A ( p, q ) := sup ( t ,t ) ∈ R (cid:18)Z t Z t (cid:0) I σ (cid:1) q w (cid:19) q (cid:2) I σ ( t , t ) (cid:3) − p < ∞ , (3) A := A ( p, q ) := sup ( t ,t ) ∈ R (cid:18)Z ∞ t Z ∞ t (cid:0) I ∗ w (cid:1) p ′ σ (cid:19) p ′ (cid:2) I ∗ w ( t , t ) (cid:3) − q ′ < ∞ , (4) and C ≈ A + A + A with equivalence constants depending on parameters p and q . Note that in one–dimensional case the analogs of the conditions (2)–(4) are equivalentto each other [2]. For n = 2 this, generally speaking, is not true. Moreover, as shownin [14, § 4] for p = q = 2 , no two of the conditions (2)–(4) guarantee (1). However, theconstruction of the second counterexample in [14, § 4] is unexpandable to the case p < q. The purpose of this paper is to obtain new criteria for the fulfillment of Hardy’sinequality (1) for n = 2 and < p = q < ∞ . The solution to this problem is contained inTheorem 2.1 (see § 2). In § 3 an alternative sufficient condition is found for v and w , when(1) is true for n = 2 and < q < p < ∞ . Recall that the criterion for (1) when n = 2 and < p ≤ q < ∞ , established in [14], is that the sum of three independent functionalsis bounded (see Theorem 1.1). It is proven in Theorem 2.1 that for < p = q < ∞ theinequality (1) is characterized by only one functional.Analogs of Theorem 2.1 are also valid for the dual operator I ∗ and mixed Hardyoperators (see [14, Remark 1] for details).In § 4, for completeness, we present known results about the operator I n for arbitrary n , provided that at least one of the two weight functions in (1) is factorizable, that is, canbe represented as a product of n one–dimensional functions.Since A ≤ C , we may and shall assume that I σ ( x, y ) < ∞ and I ∗ w ( x, y ) < ∞ forany ( x, y ) ∈ R . In particular, σ, w ∈ L loc ( R ) . Throughout the work, the notation of the form Φ . Ψ means that the relation Φ ≤ c Ψ holds with some constant c > , independent of Φ and Ψ . We write Φ ≈ Ψ in the case of Φ . Ψ . Φ . The symbols Z and N are used for denoting the sets of integers and naturalnumbers, respectively. The characteristic function of the subset E ⊂ R n + is denoted by χ E . Symbols := and =: are used to define new values. Denote α ( p, q ) := p ( q − q − p , p < q ; β ( p, q ) := 2 q +1 qr − · ( qp − qr , rp ≥ , , rp < , q < p, where /r := 1 /q − /p ; A := A ,B := B := B ( p, q ) := (cid:18)Z R d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:19) r = (cid:18)Z R (cid:2) I σ ( x, y ) (cid:3) rp ′ d x d y (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:19) r = (cid:18)Z R (cid:2) I ∗ w ( x, y ) (cid:3) rq d x d y (cid:2) I σ ( x, y ) (cid:3) rp ′ (cid:19) r , B := B ( p, q ) := Z R (cid:2) I σ ( x, y ) (cid:3) − rp d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq ! r ,B := B ( p, q ) := Z R (cid:2) I ∗ w ( x, y ) (cid:3) − rq ′ d x d y (cid:18)Z ∞ x Z ∞ y ( I ∗ w ) p ′ σ (cid:19) rp ′ ! r . Notice that lim q ↑ p B i ( p, q ) = A i ( p, p ) , i = 1 , , . (5)Let us recall the result we need in what follows from the work [3]. Proposition [3, Proposition 2.1]
Let < γ < ∞ and let { a k } k ∈ Z , { ρ k } k ∈ Z , { τ k } k ∈ Z be non-negative sequences. (a) If ρ := inf k ∈ Z ρ k +1 /ρ k > then X k ∈ Z (cid:16)X m ≥ k a m (cid:17) γ ρ γk ≤ X m ∈ Z a γm ρ γm · ( ρ γ ρ γ − , < γ ≤ , ρ γ ( ρ γ ′− − γ − ( ρ γ − − , γ > . (b) If τ := sup k ∈ Z τ k +1 /τ k < then X k ∈ Z (cid:16)X m ≤ k a m (cid:17) γ τ γk ≤ X m ∈ Z a γm τ γm · ( τ − γ τ − γ − , < γ ≤ , τ − γ ( τ − γ ′ − γ − ( τ − γ − , γ > . We start with some auxiliary technical statements.
Lemma
Let ≤ a < b < ∞ and ≤ c < d < ∞ . If < p < q < ∞ then V ( a,b ) × ( c,d ) := Z ba Z dc w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) q dy dx ≤ α ( p, q ) (cid:16)Z ba Z dc σ (cid:17) qp A q . For < q < p < ∞ the following inequality holds: V ( a,b ) × ( c,d ) ≤ β ( p, q ) (cid:16)Z ba Z dc σ (cid:17) qp × (cid:20)Z ba Z dc χ supp w ( x, y ) d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:21) qr . Proof.
Assume < p < q < ∞ and write V ( a,b ) × ( c,d ) = Z ba Z dc (cid:16)Z xa Z yc σ (cid:17) q d y (cid:20) − Z dy w ( x, t ) dt (cid:21) dx = q Z ba Z dc (cid:16)Z xa Z yc σ (cid:17) q − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z dy w ( x, t ) dt (cid:17) dy dx = q Z dc Z ba (cid:16)Z xa Z yc σ (cid:17) q − (cid:16)Z xa σ ( s, y ) ds (cid:17) d x h − Z bx Z dy w i dy = q Z ba Z dc (cid:26) ( q − (cid:16)Z xa Z yc σ (cid:17) q − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + (cid:16)Z xa Z yc σ (cid:17) q − σ ( x, y ) (cid:27)(cid:16)Z bx Z dy w (cid:17) dx dy. V ( a,b ) × ( c,d ) ≤ qA q Z ba Z dc (cid:26) ( q − (cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy. The assertion of the lemma for the case p < q follows from the chain of inequalities: q Z ba Z dc (cid:26) ( q − (cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy = p Z ba Z dc (cid:26) qp (cid:16) qp − qp ′ (cid:17)(cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + qp (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy ≤ p Z ba Z dc (cid:26) qp (cid:16) qp − (cid:17)(cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + qp (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy + p q p ′ q ( q − p ) Z ba Z dc (cid:26) qp (cid:16) qp − (cid:17)(cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + qp (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy = (cid:20) p + pq ( p − q − p (cid:21) Z ba Z dc (cid:26) qp (cid:16) qp − (cid:17)(cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + qp (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy = α ( p, q ) Z ba Z dc (cid:26) qp (cid:16) qp − (cid:17)(cid:16)Z xa Z yc σ (cid:17) qp − (cid:16)Z xa σ ( s, y ) ds (cid:17)(cid:16)Z yc σ ( x, t ) dt (cid:17) + qp (cid:16)Z xa Z yc σ (cid:17) qp − σ ( x, y ) (cid:27) dx dy = α ( p, q ) (cid:16)Z ba Z dc σ (cid:17) qp . Now suppose that q < p . By analogy with the proof of [14, Theorem 1A] we define thedomains ω k := n ( x, y ) ∈ ( a, b ) × ( c, d ) : Z xa Z yc σ > k o , −∞ < k ≤ K σ . The restriction K σ < ∞ follows from the condition [14, (1.6)], which is necessary for anyrelations between p and q . Then V ( a,b ) × ( c,d ) = X k ≤ K σ Z ω k \ ω k +1 w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) q dy dx ≤ q X k ≤ K σ kq (cid:12)(cid:12) ω k \ ω k +1 (cid:12)(cid:12) w ≤ q X k ≤ K σ kq (cid:12)(cid:12) ω k (cid:12)(cid:12) w , (cid:12)(cid:12) ω k (cid:12)(cid:12) w := R ω k w . Denote α k := inf (cid:8) x : a ≤ x : wχ ω k ( x, y ) > (cid:9) , β k := inf (cid:8) y : c ≤ y : wχ ω k ( x, y ) > (cid:9) . Observe that α k > a and β k > c and write (cid:12)(cid:12) ω k (cid:12)(cid:12) w = Z bα k Z dβ k wχ ω k = (cid:18)Z dβ k d y (cid:20) − (cid:16)Z dy Z bα k wχ ω k (cid:17) rq (cid:21)(cid:19) qr = (cid:18)Z dβ k d y (cid:26) − Z bα k d x (cid:20) − (cid:16)Z bx Z dy wχ ω k (cid:17) rq (cid:21)(cid:27)(cid:19) qr . Since h − (cid:16)R bx R dy wχ ω k (cid:17) r/q i ′ x = 0 out of the set ω k ∩ supp w for each fixed y ≥ β k and,analogously, h − (cid:16)R bx R dy wχ ω k (cid:17) r/q i ′ y = 0 outside ω k ∩ supp w for all x ≥ α k , then (cid:12)(cid:12) ω k (cid:12)(cid:12) w = (cid:18)Z dβ k Z bα k d x d y (cid:16)Z bx Z dy wχ ω k (cid:17) rq (cid:19) qr = (cid:18)Z ω k χ supp w ( x, y ) d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr . Due to the choice of ω k , kq (cid:12)(cid:12) ω k (cid:12)(cid:12) w = 2 kq (cid:18)Z ω k χ supp w ( x, y ) d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr ≤ kqr (cid:18)Z ω k χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) r − d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr . It follows from Proposition 2.1(a) with ρ = 2 and γ = q/r < that X k ≤ K σ kqr (cid:18)Z ω k χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) r − d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr = X k ≤ K σ kqr (cid:18)X m ≥ k Z ω m \ ω m +1 χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) r − d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr ≤ qr qr − X k ≤ K σ kqr (cid:18)Z ω k \ ω k +1 χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) r − d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr ≤ β ( p, q )2 q + qp X k ≤ K σ kqp (cid:18)Z ω k \ ω k +1 χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) rp ′ d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr . Using H¨older’s inequality with exponents r/q and p/q , we obtain X k ≤ K σ kqp (cid:18)Z ω k \ ω k +1 χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) rp ′ d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr ≤ qp qKσp (cid:18) X k ≤ K σ Z ω k \ ω k +1 χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) rp ′ d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr ≤ qp (cid:16)Z ba Z dc σ (cid:17) qp (cid:18)Z ba Z dc χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) rp ′ d x d y (cid:16)Z bx Z dy w (cid:17) rq (cid:19) qr . r/q > and r/p ′ > , then integrating by parts over the variable y yields Z ba Z dc χ supp w ( x, y ) (cid:16)Z xa Z yc σ (cid:17) rp ′ d y d x (cid:16)Z bx Z dy w (cid:17) rq = Z ba Z dc χ supp w ( x, y ) d y (cid:16)Z xa Z yc σ (cid:17) rp ′ d x (cid:20) − (cid:16)Z bx Z dy w (cid:17) rq (cid:21) ≤ Z ba Z dc χ supp w ( x, y ) d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17) . A similar statement holds with the (inner) integral of w . Lemma
Let ≤ a < b < ∞ and ≤ c < d < ∞ . If < p < q < ∞ then W ( a,b ) × ( c,d ) := Z ba Z dc σ ( x, y ) (cid:16)Z bx Z dy w (cid:17) p ′ dy dx ≤ α ( q ′ , p ′ ) (cid:16)Z ba Z dc w (cid:17) p ′ q ′ A p ′ . In the case < q < p < ∞ W ( a,b ) × ( c,d ) ≤ β ( q ′ , p ′ ) (cid:16)Z ba Z dc w (cid:17) p ′ q ′ × (cid:20)Z ba Z dc χ supp σ ( x, y ) d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:21) p ′ r . Introduce notations: α := α ( p, q ) , β := β ( p, q ) , α ′ := α ( q ′ , p ′ ) , β ′ := β ( q ′ , p ′ ) , C α,α ′ := 3 q h(cid:16) (cid:17) q max n α, q ( q ′ ) qp ′ o(cid:16) p − p − − (cid:17) qp + 3 p ( α ′ ) p ′ (cid:16) q − q − − (cid:17) q ′ i , C β,β ′ := 3 q h(cid:16) (cid:17) q max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o(cid:16) p − p − − (cid:17) qp + 3( β ′ ) p ′ (cid:16) q − q − − (cid:17) q ′ i . The main result of the work is the following statement.
Theorem
Let < p = q < ∞ . If p < q then the inequality (cid:18)Z R (cid:0) I f (cid:1) q w (cid:19) q ≤ C (cid:18)Z R f p v (cid:19) p ( f ≥ (6) holds if and only if A < ∞ . Besides, A ≤ C ≤ C α,α ′ A. In the case q < p the inequality (6) is true if and only if
B < ∞ . Moreover, − p ′ (cid:16) qr (cid:17) q (cid:16) p ′ r (cid:17) p ′ B ≤ C ≤ C β,β ′ B. Proof. ( Sufficiency ) Similarly to how it was done in E. Sawyer’s paper [14] for the case < p ≤ q < ∞ , we show that the conditions of the theorem are sufficient, limitingourselves to proving the inequality (6) on the subclass M ⊂ L pv ( R ) of all functions6 ≥ bounded on R with compact supports contained in the set { I σ > } . Then theinequality (6) for arbitrary ≤ f ∈ L pv ( R ) follows by the standard arguments.Suppose A < ∞ for p < q (or B < ∞ in the case of q < p ) and fix f ∈ M . By analogywith the proof of [14, Theorem 1A], we define the domains Ω k : = (cid:8) I f > k (cid:9) , k ∈ Z . Then, by our assumptions on f , there exists K ∈ Z such that Ω k = ∅ for k ≤ K, Ω k = ∅ for k > K , S k ∈ Z Ω k = R and k < I f ( x, y ) ≤ k +1 , k ≤ K, ( x, y ) ∈ (Ω k \ Ω k +1 ) . ✻ ✲ x k x k x k y k y k y k e S k e R k T k (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ∂ Ω k +1 ∂ Ω k Fig. 1
We can write down that Z R ( I f ) q w = X k ≤ K − Z Ω k +2 \ Ω k +3 ( I f ) q w ≤ q X k ≤ K − kq | Ω k +2 \ Ω k +3 | w , where | Ω k +2 \ Ω k +3 | w := R Ω k +2 \ Ω k +3 w and Ω K \ Ω K +1 = Ω K , since Ω K +1 is empty.Next, as in the proof of [14, Theorem 1A], we introduce rectangles. For this, we fix k such that Ω k +1 = ∅ , and choose points ( x kj , y kj ) , ≤ j ≤ N = N k , lying on theboundary ∂ Ω k in such a way to have ( x kj , y kj − ) belonging to ∂ Ω k +1 for ≤ j ≤ N and Ω k +1 ⊂ S Nj =1 S kj , where S kj is a rectangle of the form ( x kj , ∞ ) × ( y kj , ∞ ) . We also definerectangles e S kj = ( x kj , x kj +1 ) × ( y kj , y kj − ) for ≤ j ≤ N and R kj = (0 , x kj +1 ) × (0 , y kj ) , e R kj = ( x kj , x kj +1 ) × ( y kj +1 , y kj ) and T kj = ( x kj +1 , ∞ ) × ( y kj , ∞ ) for ≤ j ≤ N − . Put y k = x kN +1 = ∞ (see Figure 1).Now we choose the sets E kj ⊂ T kj so that E kj ∩ E ki = ∅ for j = i and S j E kj =(Ω k +2 \ Ω k +3 ) ∩ (cid:16)S j T kj (cid:17) . Since Ω k +2 \ Ω k +3 ⊂ Ω k +1 ⊂ (cid:16)S j T kj (cid:17) ∪ (cid:16)S j e S kj (cid:17) , then − q Z R ( I f ) q w ≤ X k,j kq (cid:12)(cid:12) E kj (cid:12)(cid:12) w + X k,j kq (cid:12)(cid:12) e S kj ∩ (Ω k +2 − Ω k +3 ) (cid:12)(cid:12) w =: I + II. (7)7o estimate II we denote D kj := e S kj \ Ω k +3 and turn to the reasoning of E. Sawyer onpage 6 in [14], from which it follows that I ( χ D kj f )( x, y ) > k if ( x, y ) ∈ e S kj ∩ (Ω k +2 \ Ω k +3 ) . Further, according to [14, p. 6], (cid:12)(cid:12) e S kj ∩ (Ω k +2 \ Ω k +3 ) (cid:12)(cid:12) w ≤ − k Z e S kj ∩ (Ω k +2 \ Ω k +3 ) I ( χ D kj f )( x, y ) w ( x, y ) dxdy ≤ − k Z D kj (cid:16)Z xx kj Z yy kj f (cid:17) w ( x, y ) dxdy = 3 − k Z D kj f ( s, t ) (cid:16)Z ∞ s Z ∞ t wχ D kj (cid:17) dsdt ≤ − k (cid:18)Z D kj f p v (cid:19) p (cid:18)Z D kj σ ( s, t ) (cid:16)Z ∞ s Z ∞ t wχ D kj (cid:17) p ′ dsdt (cid:19) p ′ . (8)By applying Lemma 2.2 to ( a, b ) × ( c, d ) = e S kj , we obtain for p < q that W e S kj = Z D kj σ ( s, t ) (cid:16)Z ∞ s Z ∞ t wχ D kj (cid:17) p ′ dsdt ≤ α ′ A p ′ (cid:12)(cid:12) e S kj (cid:12)(cid:12) p ′ q ′ w , (9)and in the case q < p W e S kj ≤ β ′ (cid:12)(cid:12) e S kj (cid:12)(cid:12) p ′ q ′ w (cid:18)Z D kj d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:19) p ′ r . For q < p , from this and H¨older’s inequality with q and q ′ , ( β ′ ) − p ′ · II ≤ X k,j k ( q − (cid:18)Z D kj f p v (cid:19) p (cid:18)Z D kj d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:19) r (cid:12)(cid:12) S kj (cid:12)(cid:12) q ′ w ≤ (cid:18)X k,j kq (cid:12)(cid:12) S kj (cid:12)(cid:12) w (cid:19) q ′ "X k,j (cid:18)Z D kj f p v (cid:19) qp Z D kj d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)! qr q . On the strength of [14, (2.6)] N k X j =1 χ S kj ≤ − k χ Ω k I f for all k. Then X k,j kq (cid:12)(cid:12) S kj (cid:12)(cid:12) w = X k kq N k X j =1 Z R χ S kj w = X k kq Z R (cid:16) N k X j =1 χ S kj (cid:17) w ≤ X k k ( q − Z R χ Ω k ( I f ) w = X k k ( q − X m ≥ k Z R χ Ω m \ Ω m +1 ( I f ) w = X m m ( q − Z R χ Ω m \ Ω m +1 ( I f ) w X m ≥ k ( k − m )( q − ≤ q − q − − X m m ( q − Z R χ Ω m \ Ω m +1 ( I f ) w X k,j kq (cid:12)(cid:12) S kj (cid:12)(cid:12) w ≤ q − q − − X m Z Ω m \ Ω m +1 (cid:0) I f (cid:1) q w = 3 q − q − − Z R (cid:0) I f (cid:1) q w. Further, H¨older’s inequality with p/q , r/q and the estimate P k,j χ D kj ≤ P k χ Ω k \ Ω k +3 ≤ entail X k,j (cid:18)Z D kj f p v (cid:19) qp (cid:18)Z D kj d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:19) qr ≤ (cid:18)X k,j Z D kj f p v (cid:19) qp X k,j Z D kj d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:19) qr ≤ (cid:18)Z R f p v (cid:19) qp Z R d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)! qr . Thus, for q < p , II ≤ β ′ ) p ′ B (cid:16) q − q − − (cid:17) q ′ (cid:18)Z R f p v (cid:19) p (cid:18)Z R (cid:0) I f (cid:1) q w (cid:19) q ′ . (10)In the case p < q a similar estimate of the form II ≤ p ( α ′ ) p ′ A (cid:16) q − q − − (cid:17) q ′ (cid:18)Z R f p v (cid:19) p (cid:18)Z R (cid:0) I f (cid:1) q w (cid:19) q ′ (11)follows from (8), (9) and the reasoning on pages 6–7 in [14].To estimate I in (7), in full accordance with the proof of [14, Theorem 1A, pp. 8–9],we put gσ := f and write: q I = X k,j ( k +1) q (cid:12)(cid:12) E kj (cid:12)(cid:12) w = X k,j (cid:12)(cid:12) E kj (cid:12)(cid:12) w (cid:18)Z R kj f (cid:19) q = X k,j (cid:12)(cid:12) E kj (cid:12)(cid:12) w (cid:12)(cid:12) R kj (cid:12)(cid:12) qσ (cid:18) (cid:12)(cid:12) R kj (cid:12)(cid:12) σ Z R kj gσ (cid:19) q . (12)For an integer l , by Γ l we denote the set of pairs ( k, j ) such that (cid:12)(cid:12) E kj (cid:12)(cid:12) w > and l < (cid:12)(cid:12) R kj (cid:12)(cid:12) σ Z R kj gσ ≤ l +1 , ( k, j ) ∈ Γ l and observe that Γ l ′ ∩ Γ l ′′ = ∅ , l ′ = l ′′ . For fixed l the family { U li } i ( l ) i =1 consists of maximal rectangles from the collection { R kj } ( k,j ) ∈ Γ l , that is, each R kj with ( k, j ) ∈ Γ l is contained in some U li (or coincides withit). In [14, p. 8] it is shown that e U li are disjoint for fixed l , where we denote e U li = e R li if U li = R li .Let χ li be the characteristic function of the union of the sets E kj over all ( k, j ) ∈ Γ l such that R kj ⊂ U li . Further, following [14, (2.13)], we arrive to X ( k,j ) ∈ Γ l (cid:12)(cid:12) E kj (cid:12)(cid:12) w (cid:12)(cid:12) R kj (cid:12)(cid:12) qσ = i ( l ) X i =1 X ( k,j ): R kj ⊂ U li Z E kj w (cid:2) I ( χ U li σ )( x kj +1 , y kj ) (cid:3) q ≤ i ( l ) X i =1 Z R χ li w (cid:2) I ( χ U li σ ) (cid:3) q . (13)9y analogy with [14, (2.8)], let us first show the validity of the estimate Z R χ li w (cid:2) I ( χ U li σ ) (cid:3) q ≤ max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o(cid:0) B li (cid:1) q (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ (14)for U li = (0 , a ) × (0 , b ) in the case q < p , where (cid:0) B li (cid:1) r = Z R χ li ( x, y ) d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17) . On (0 , a ) × (0 , b ) = U li , in view of Lemma 2.1, V U li = Z U li χ li w (cid:0) I σ (cid:1) q ≤ β (cid:18)Z U li χ li ( x, y ) d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17)(cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ ≤ β (cid:0) B li (cid:1) q (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ . On the rectangle ( a, ∞ ) × ( b, ∞ ) we obtain the estimate: Z ( a, ∞ ) × ( b, ∞ ) χ li w (cid:12)(cid:12) U li (cid:12)(cid:12) qσ = Z ( a, ∞ ) × ( b, ∞ ) χ li ( x, y ) d x d y (cid:2) I ∗ wχ li ( x, y ) (cid:3) rq ! qr (cid:12)(cid:12) U li (cid:12)(cid:12) qσ ≤ (cid:18)Z ( a, ∞ ) × ( b, ∞ ) χ li ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rp ′ d x d y (cid:2) I ∗ wχ li ( x, y ) (cid:3) rq (cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ ≤ (cid:18)Z R χ li ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rp ′ d x d y (cid:2) I ∗ wχ li ( x, y ) (cid:3) rq (cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ , whence by integration by parts Z R χ li ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rp ′ d x d y (cid:2) I ∗ wχ li ( x, y ) (cid:3) rq = (cid:0) B li (cid:1) r . In the first of the two mixed cases — (0 , a ) × ( b, ∞ ) and ( a, ∞ ) × (0 , b ) — we obtain,using the criteria for the fulfillment of the one–dimensional weighted Hardy inequality for f p ( x ) = R b σ ( x, y ) dy (see [7, § 1.3.2]): Z (0 ,a ) × ( b, ∞ ) χ li ( x, y ) w ( x, y ) (cid:18)Z x Z b σ (cid:19) q dxdy = Z a (cid:18)Z ∞ b χ li ( x, y ) w ( x, y ) dy (cid:19)(cid:18)Z x (cid:18)Z b σ ( s, t ) dt (cid:19)(cid:19) q dx ≤ q ( p ′ ) q − (cid:18)Z a (cid:18)Z ∞ s Z ∞ b χ li w (cid:19) rp (cid:18)Z s Z b σ (cid:19) rp ′ (cid:18)Z ∞ b χ li ( s, t ) w ( s, t ) dt (cid:19) ds (cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ = q ( p ′ ) q − (cid:16) qr (cid:17) qr (cid:18)Z a (cid:18)Z s Z b σ (cid:19) rp ′ d s (cid:20) − (cid:18)Z ∞ s Z ∞ b χ li w (cid:19) rq (cid:21)(cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ = q ( p ′ ) q − (cid:16) qr (cid:17) qr (cid:18)Z a (cid:18)Z s Z b σ (cid:19) rp ′ d s (cid:20) − Z ∞ b d t (cid:20) − (cid:18)Z ∞ s Z ∞ t χ li w (cid:19) rq (cid:21)(cid:21)(cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ = q ( p ′ ) q − (cid:16) qr (cid:17) qr (cid:18)Z a Z ∞ b χ li ( s, t ) (cid:18)Z s Z b σ (cid:19) rp ′ d s d t (cid:18)Z ∞ s Z ∞ t χ li w (cid:19) rq (cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ ( p ′ ) q − (cid:16) qr (cid:17) qr (cid:18)Z a Z ∞ b χ li ( s, t ) (cid:2) I σ ( s, t ) (cid:3) rp ′ d s d t (cid:18)Z ∞ s Z ∞ t χ li w (cid:19) rq (cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ ≤ q ( p ′ ) q − (cid:16) qr (cid:17) qr (cid:18)Z R χ li ( s, t ) (cid:2) I σ ( s, t ) (cid:3) rp ′ d s d t (cid:18)Z ∞ s Z ∞ t χ li w (cid:19) rq (cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ ≤ q ( p ′ ) q − (cid:16) qr (cid:17) qr (cid:18)Z R χ li ( s, t ) d t (cid:2) I σ ( s, t ) (cid:3) rp ′ d s (cid:16) − (cid:2) I ∗ w ( s, t ) (cid:3) rq (cid:17)(cid:19) qr (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ . (15)The second mixed case is estimated in a similar way. So, (14) is proven. Continuing (13),we obtain, using [14, (2.11)] and H¨older’s inequality with r/q , p/q : X ( k,j ) ∈ Γ l (cid:12)(cid:12) E kj (cid:12)(cid:12) w (cid:12)(cid:12) R kj (cid:12)(cid:12) qσ ≤ max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o X i (cid:0) B li (cid:1) q (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ ≤ max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o X i (cid:0) B li (cid:1) q (cid:16) − l Z e U li ∩{ g> l − } gσ (cid:17) qp ≤ max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o(cid:18)X i (cid:0) B li (cid:1) r (cid:19) qr (cid:18)X i − l Z e U li ∩{ g> l − } gσ (cid:19) qp ≤ max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o − lq/p (cid:0) B l (cid:1) q (cid:18)Z { g> l − } gσ (cid:19) qp . The last estimate is valid with (cid:0) B l (cid:1) r := Z R χ {∪ ( k,j ) ∈ Γ l E kj } ( x, y ) d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x (cid:16) − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:17) due to the fact that for fixed l the rectangles e U li do not intersect (see [14, p. 8]). Combiningit with (12), we obtain, taking into account the relation X l l ( p − χ { g> l − } ≤ p − p − − g p − for p > , H¨older’s inequality with r/q and p/q and the fact that all E kj are disjoint: I ≤ (cid:16) (cid:17) q X l lq X ( k,j ) ∈ Γ l (cid:12)(cid:12) E kj (cid:12)(cid:12) w (cid:12)(cid:12) R kj (cid:12)(cid:12) qσ ≤ (cid:16) (cid:17) q max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o X l lq (cid:0) B l (cid:1) q (cid:18) − l Z { g> l − } gσ (cid:19) qp ≤ (cid:16) (cid:17) q max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o(cid:16)X l (cid:0) B l (cid:1) r (cid:17) qr (cid:18)X l l ( p − Z { g> l − } gσ (cid:19) qp ≤ (cid:16) (cid:17) q max n β, q ( p ′ ) q − (cid:16) qr (cid:17) qr o(cid:16) p − p − − (cid:17) qp B q (cid:18)Z R f p v (cid:19) qp . (16)Combining (16) with (10) we arrive at the required upper bound for q < p .For p < q , the term I is estimated identically to the case p ≤ q in [14, p. 9], i.e. I ≤ (cid:16) (cid:17) q max n α, q ( q ′ ) qp ′ o(cid:16) p − p − − (cid:17) qp A q (cid:18)Z R f p v (cid:19) qp , (17)11elying on an analog of the inequality (14) of the form Z R χ li w (cid:2) I ( χ U li σ ) (cid:3) q ≤ max n α, q ( q ′ ) qp ′ o A q (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ for U li = (0 , a ) × (0 , b ) . Note that in this case, unlike [14, (2.8)], to perform the estimate on the rectangle (0 , a ) × (0 , b ) = U li one should apply the statement of Lemma 2.1, from which it follows that V U li ≤ α (cid:12)(cid:12) U li (cid:12)(cid:12) qp σ A q . The final upper estimate Z R ( I f ) q w ≤ C (cid:18)Z R f p v (cid:19) p (cid:18)Z R ( I f ) q w (cid:19) q ′ + C q (cid:18)Z R f p v (cid:19) qp follows from (7) combined with (11) and (17) for p < q (or (10) and (16) if q < p ) with C = A · C α,α ′ in case p < q and C = B · C β,β ′ for q < p .( Necessity ) The validity of A ≤ C follows by substituting f = χ (0 ,s ) × (0 ,t ) into theinitial inequality (6). To establish B . C in the case q < p , we apply the test function f ( s, y ) = σ ( s, y ) (cid:20)Z ∞ s (cid:2) I σ ( x, y ) (cid:3) rq ′ (cid:2) I ∗ w ( x, y ) (cid:3) rp (cid:16)Z ∞ y w ( x, t ) dt (cid:17) dx (cid:21) p =: σ ( s, y ) J ( s, y ) into (6). Then Z R f p v = Z R σ ( s, y ) (cid:2) J ( s, y ) (cid:3) p dsdy = Z R (cid:2) I σ ( x, y ) (cid:3) rq ′ (cid:2) I ∗ w ( x, y ) (cid:3) rp (cid:16)Z ∞ y w ( x, t ) dt (cid:17)(cid:16)Z x σ ( s, y ) ds (cid:17) dxdy = p ′ qr Z R d y (cid:2) I σ ( x, y ) (cid:3) rp ′ d x h − (cid:2) I ∗ w ( x, y ) (cid:3) rq i = p ′ qr B r . (18)To estimate the left–hand side of the inequality (6), we write (cid:2) J ( s, y ) (cid:3) p = qr (cid:2) I σ ( s, y ) (cid:3) rq ′ (cid:2) I ∗ w ( s, y ) (cid:3) rq + qq ′ Z ∞ s (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:2) I ∗ w ( x, y ) (cid:3) rq (cid:16)Z y σ ( x, t ) dt (cid:17) dx = : qr (cid:2) J ( s, y ) (cid:3) p + qq ′ (cid:2) J ( s, y ) (cid:3) p . (19)Then, for our chosen f , F ( u, z ) := Z u Z z f = Z u Z z σ ( s, y ) J ( s, y ) dyds ≥ − p ′ (cid:18)(cid:16) qr (cid:17) p Z u Z z σ ( s, y ) J ( s, y ) dyds + (cid:16) qq ′ (cid:17) p Z u Z z σ ( s, y ) J ( s, y ) dyds (cid:19) = : 2 − p ′ (cid:0) F + F (cid:1) .
12o estimate F , we observe that (cid:16) q ′ q (cid:17) p F = Z u Z z σ ( s, y ) J ( s, y ) dyds ≥ (cid:2) I ∗ w ( u, z ) (cid:3) rqp Z u Z z σ ( s, y ) (cid:20)Z us (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z y σ ( x, t ) dt (cid:17) dx (cid:21) p dyds. Since Z us (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z y σ ( x, t ) dt (cid:17) dx ≤ q ′ r (cid:2) I σ ( u, y ) (cid:3) rq ′ , (20)then Z u Z z σ ( s, y ) (cid:20)Z us (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z y σ ( x, t ) dt (cid:17) dx (cid:21) − p ′ dyds ≥ (cid:16) q ′ r (cid:17) − p ′ Z u Z z σ ( s, y ) (cid:2) I σ ( u, y ) (cid:3) − rq ′ p ′ (cid:20)Z us (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z y σ ( x, t ) dt (cid:17) dx (cid:21) dyds ≥ (cid:16) q ′ r (cid:17) − p ′ (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ Z u Z z σ ( s, y ) (cid:20)Z us (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z y σ ( x, t ) dt (cid:17) dx (cid:21) dyds = (cid:16) q ′ r (cid:17) − p ′ (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ Z u Z z (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z x σ ( s, y ) ds (cid:17)(cid:16)Z y σ ( x, t ) dt (cid:17) dydx and, therefore, F ≥ (cid:16) qq ′ (cid:17) p (cid:16) rq ′ (cid:17) p ′ (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, z ) (cid:3) rqp × Z u Z z (cid:2) I σ ( x, y ) (cid:3) rq ′ − (cid:16)Z x σ ( s, y ) ds (cid:17)(cid:16)Z y σ ( x, t ) dt (cid:17) dxdy = : (cid:16) qr (cid:17) p rq ′ (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, z ) (cid:3) rqp J ( u, z ) . For F we obtain: F = (cid:16) qr (cid:17) p Z u Z z σ ( s, y ) (cid:2) I σ ( s, y ) (cid:3) rq ′ p (cid:2) I ∗ w ( s, y ) (cid:3) rqp dyds ≥ (cid:16) qr (cid:17) p (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, z ) (cid:3) rqp Z u Z z σ ( s, y ) (cid:2) I σ ( s, y ) (cid:3) rq ′ dyds = : (cid:16) qr (cid:17) p (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, z ) (cid:3) rqp J ( u, z ) . It holds that F ( u, z ) ≥ − p ′ (cid:16) qr (cid:17) p (cid:2) I σ ( u, z ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, z ) (cid:3) rqp (cid:0) J ( u, z ) + rq ′ J ( u, z ) (cid:1) . Integrating by parts we find: J ( u, z ) = q ′ r Z u dx Z z (cid:16)Z y σ ( x, t ) dt (cid:17) d y (cid:2) I σ ( x, y ) (cid:3) rq ′ = q ′ r Z u (cid:16)Z z σ ( x, t ) dt (cid:17)(cid:2) I σ ( x, z ) (cid:3) rq ′ dx − q ′ r J ( u, z )= q ′ p ′ r (cid:2) I σ ( u, z ) (cid:3) rp ′ − q ′ r J ( u, z ) . F ( u, z ) ≥ − p ′ (cid:16) qr (cid:17) p p ′ r (cid:2) I σ ( u, z ) (cid:3) rqp ′ (cid:2) I ∗ w ( u, z ) (cid:3) rqp . (21)We write making use of (19): Z R ( I f ) q w = Z R f ( x, y ) (cid:18)Z ∞ x Z ∞ y w ( u, z ) (cid:2) F ( u, z ) (cid:3) q − dzdu (cid:19) dxdy ≥ − p ′ Z R σ ( x, y ) (cid:18)Z ∞ x Z ∞ y wF q − (cid:19)(cid:26)(cid:16) qr (cid:17) p (cid:2) I σ ( x, y ) (cid:3) rq ′ p (cid:2) I ∗ w ( x, y ) (cid:3) rqp + (cid:16) qq ′ (cid:17) p (cid:20)Z ∞ x (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:2) I ∗ w ( s, y ) (cid:3) rq (cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) p (cid:27) dxdy = : 2 − p ′ (cid:0) G + G (cid:1) . (22) G is evaluated with (21) as follows: G = (cid:16) qr (cid:17) p Z R σ ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rq ′ p (cid:2) I ∗ w ( x, y ) (cid:3) rqp (cid:18)Z ∞ x Z ∞ y w (cid:19)(cid:2) F ( x, y ) (cid:3) q − dxdy ≥ − q − p ′ (cid:16) qr (cid:17) qp (cid:16) p ′ r (cid:17) q − Z R σ ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rq ′ (cid:2) I ∗ w ( x, y ) (cid:3) rq dxdy. (23)It is true for G : (cid:16) q ′ q (cid:17) p G = Z R σ ( x, y ) (cid:20)Z ∞ x (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:2) I ∗ w ( s, y ) (cid:3) rq (cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) p × (cid:18)Z ∞ x Z ∞ y w ( u, z ) (cid:2) F ( u, z ) (cid:3) q − dzdu (cid:19) dxdy = Z R Z u σ ( x, y ) (cid:20)Z ∞ x (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:2) I ∗ w ( s, y ) (cid:3) rq (cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) p dx × (cid:18)Z ∞ y w ( u, z ) (cid:2) F ( u, z ) (cid:3) q − dz (cid:19) dudy ≥ Z R Z u σ ( x, y ) (cid:20)Z ux (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:2) I ∗ w ( s, y ) (cid:3) rq (cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) p dx × (cid:18)Z ∞ y w ( u, z ) (cid:2) F ( u, z ) (cid:3) q − dz (cid:19) dudy ≥ Z R (cid:2) I ∗ w ( u, y ) (cid:3) rpq Z u σ ( x, y ) (cid:20)Z ux (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) − p ′ dx × (cid:18)Z ∞ y w ( u, z ) (cid:2) F ( u, z ) (cid:3) q − dz (cid:19) dudy (20) ≥ (cid:16) rq ′ (cid:17) p ′ Z R (cid:2) I σ ( u, y ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, y ) (cid:3) rqp × Z u σ ( x, y ) (cid:20)Z ux (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) dx (cid:18)Z ∞ y w ( u, z ) (cid:2) F ( u, z ) (cid:3) q − dz (cid:19) dudy ≥ (cid:16) rq ′ (cid:17) p ′ Z R (cid:2) I σ ( u, y ) (cid:3) − rq ′ p ′ (cid:2) I ∗ w ( u, y ) (cid:3) rqp (cid:18)Z ∞ y w ( u, z ) dz (cid:19)(cid:2) F ( u, y ) (cid:3) q − × (cid:20)Z u (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:16)Z s σ ( x, y ) dx (cid:17)(cid:16)Z y σ ( s, t ) dt (cid:17) ds (cid:21) dudy. Integrating by parts we find Z u (cid:2) I σ ( s, y ) (cid:3) rq ′ − (cid:16)Z s σ ( x, y ) dx (cid:17)(cid:16)Z y σ ( s, t ) dt (cid:17) ds = q ′ r (cid:16)Z u σ ( x, y ) dx (cid:17)(cid:2) I σ ( u, y ) (cid:3) rq ′ dx − q ′ r Z u (cid:2) I σ ( s, y ) (cid:3) rq ′ σ ( s, y ) ds. Hence, continuing the reasoning, we obtain for G using (21): (cid:16) q ′ q (cid:17) p G ≥ − q − p ′ (cid:16) q ′ r (cid:17) p (cid:16) qr (cid:17) q − p (cid:16) p ′ r (cid:17) q − Z R (cid:2) I ∗ w ( u, y ) (cid:3) rp (cid:18)Z ∞ y w ( u, z ) dz (cid:19) × (cid:20)(cid:2) I σ ( u, y ) (cid:3) rq ′ Z u σ ( x, y ) dx − Z u (cid:2) I σ ( s, y ) (cid:3) rq ′ σ ( s, y ) ds (cid:21) dudy. (24)Since Z R (cid:2) I ∗ w ( u, y ) (cid:3) rp (cid:18)Z ∞ y w ( u, z ) dz (cid:19)(cid:20)Z u (cid:2) I σ ( s, y ) (cid:3) rq ′ σ ( s, y ) ds (cid:21) dudy = qr Z R (cid:2) I ∗ w ( u, y ) (cid:3) rq (cid:2) I σ ( u, y ) (cid:3) rq ′ σ ( u, y ) dudy then from (22) we obtain, applying (23) and (24), qp ′ Z R ( I f ) q w ≥ (cid:16) qr (cid:17) qp (cid:16) p ′ r (cid:17) q − Z R σ ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rq ′ (cid:2) I ∗ w ( x, y ) (cid:3) rq dxdy + (cid:16) qr (cid:17) qp (cid:16) p ′ r (cid:17) qq ′ Z R (cid:2) I ∗ w ( u, y ) (cid:3) rp (cid:18)Z ∞ y w ( u, z ) dz (cid:19)(cid:2) I σ ( u, y ) (cid:3) rq ′ (cid:18)Z u σ ( x, y ) dx (cid:19) dudy − (cid:16) qr (cid:17) qp (cid:16) p ′ r (cid:17) q − qr Z R σ ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rq ′ (cid:2) I ∗ w ( x, y ) (cid:3) rq dxdy = (cid:16) qr (cid:17) qp (cid:16) p ′ r (cid:17) q − qp Z R σ ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rq ′ (cid:2) I ∗ w ( x, y ) (cid:3) rq dxdy + (cid:16) qr (cid:17) qp +1 (cid:16) p ′ r (cid:17) q Z R d u (cid:16) − (cid:2) I ∗ w ( u, y ) (cid:3) rq (cid:17) d y (cid:2) I σ ( u, y ) (cid:3) rp ′ ≥ (cid:16) qr (cid:17) qp +1 (cid:16) p ′ r (cid:17) q B r . In view of (18), the required lower bound for C in the case q < p is proven.Recall that in the case p ≤ q the best constant C of the two–dimensional inequality(6) is equivalent to P i =1 A i (see Theorem 1.1). However, by virtue of the statements ofLemmas 2.1 and 2.2, for p < q the following inequalities take place: A ≤ C ≤ C , (cid:2) A + A + A (cid:3) ≤ C , (cid:2) α ( p, q ) q + α ( q ′ , p ′ ) p ′ (cid:3) A . (25)15oreover, lim p ↑ q (cid:2) α ( p, q ) + α ( q ′ , p ′ ) (cid:3) = ∞ . Thus, the last estimate in (25) and the upper bound in the main theorem have blow-upfor p ↑ q .Estimates similar to (25) hold also in the case q < p if conditions r/p ≥ and r/q ′ ≥ are simultaneously satisfied, namely, (cid:16) qr (cid:17) q (cid:16) p ′ r (cid:17) p ′ B ≤ C ≤ C , (cid:2) B + B + B (cid:3) ≤ C , (cid:2) β ( p, q ) + β ( q ′ , p ′ ) (cid:3) B , (26)where β ( p, q ) = 2 /q +1 (2 ( r − q ) /p − /r (2 q/r − /p . Observe that lim q ↑ p (cid:2) β ( p, q ) + β ( q ′ , p ′ ) (cid:3) = ∞ . In the rest cases, the following inequalities take place for q < p : (cid:16) qr (cid:17) q (cid:16) p ′ r (cid:17) p ′ B ≤ C ≤ C ,β ′ (cid:2) B + B (cid:3) ≤ C ,β ′ (cid:2) β ( p, q ) (cid:3) B , rp ≥ rq ′ < , C β, (cid:2) B + B (cid:3) ≤ C β, (cid:2) β ( q ′ , p ′ ) (cid:3) B , rp < rq ′ ≥ , C β,β ′ B rp < rq ′ < . (27)On the strength of the restrictions on the parameters p and q , all coefficients in (27) arefinite. In the first zone r → ∞ only if p, q → ∞ ; similarly, in the second zone r → ∞ onlyif p, q → ; and in the third zone r cannot approach ∞ . In addition, C , in (26) does notdiverge for q ↑ p , and, therefore, the second inequality gives an upper bound in Sawyer’stheorem for p = q , since lim q ↑ p B i = A i , i = 1 , , (see (5)).The upper estimates in (26)–(27) can be proven similarly to the upper bound for C in the case q < p in the main theorem. The only difference is that for r/p ≥ , instead ofLemma 2.1, one should use the inequality V ( a,b ) × ( c,d ) ≤ (cid:2) I σ ( b, d ) (cid:3) qp (cid:20)Z ba Z dc χ supp w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) rp d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq (cid:21) qr . Similarly, for r/q ′ ≥ , instead of Lemma 2.2, the following estimate should be applied: W ( a,b ) × ( c,d ) ≤ (cid:2) I ∗ w ( a, c ) (cid:3) p ′ q ′ (cid:20)Z ba Z dc χ supp σ ( x, y ) (cid:2) I ∗ w ( x, y ) (cid:3) rq ′ d x d y (cid:18)Z ∞ x Z ∞ y ( I ∗ w ) p ′ σ (cid:19) rp ′ (cid:21) p ′ r . To establish B ≤ β ( p, q ) B we split R into domains ω k (as in Lemma 2.1). Then Z R (cid:2) I σ ( x, y ) (cid:3) − rp d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq = X k ≤ K σ Z ω k \ ω k +1 (cid:2) I σ ( x, y ) (cid:3) − rp d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq X k ≤ K σ − kr/p Z ω k \ ω k +1 d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq ≤ X k ≤ K σ − kr/p Z R \ ω k +1 d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq . Since Z R \ ω k +1 d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq = Z R χ R \ ω k +1 ( x, y ) d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq = Z R d x d y (cid:18)Z x Z y χ R \ ω k +1 ( I σ ) q w (cid:19) rq = (cid:18)Z R \ ω k +1 ( I σ ) q w (cid:19) rq , then we have X k ≤ K σ − kr/p Z R \ ω k +1 d x d y (cid:18)Z x Z y ( I σ ) q w (cid:19) rq = X k ≤ K σ − kr/p (cid:18)Z R \ ω k +1 ( I σ ) q w (cid:19) rq . From Proposition 2.1(b) with τ = 2 qp and γ = r/q X k ≤ K σ − kr/p (cid:18)Z R \ ω k +1 ( I σ ) q w (cid:19) rq = X k ≤ K σ − kr/p (cid:18)X m ≤ k Z ω m \ ω m +1 ( I σ ) q w (cid:19) rq ≤ r/p (2 ( r − q ) /p − q/r − r/p X k ≤ K σ − kr/p (cid:18)Z ω k \ ω k +1 ( I σ ) q w (cid:19) rq ≤ r/p + r (2 ( r − q ) /p − q/r − r/p X k ≤ K σ kr/p ′ (cid:18)Z ω k \ ω k +1 w (cid:19) rq . By analogy with the proof of Lemma 2.1, we can write | ω k \ ω k +1 | rq w ≤ | ω k | rq w = Z R d x d y (cid:2) I ∗ ( χ ω k w )( x, y ) (cid:3) rq = Z ω k d x d y (cid:2) I ∗ w ( x, y ) (cid:3) rq . Hence (see Proposition 2.1(a)), X k ≤ K σ kr/p ′ (cid:18)Z ω k \ ω k +1 w (cid:19) rq ≤ X k ≤ K σ kr/p ′ Z ω k d x d y (cid:2) I ∗ w ( x, y ) (cid:3) rq ≤ X k ≤ K σ k Z ω k (cid:2) I σ ( x, y ) (cid:3) rq ′ d x d y (cid:2) I ∗ w ( x, y ) (cid:3) rq ≤ X k ≤ K σ Z ω k \ ω k +1 (cid:2) I σ ( x, y ) (cid:3) rp ′ d x d y (cid:2) I ∗ w ( x, y ) (cid:3) rq = B r . Similarly, one can show that B ≤ β ( q ′ , p ′ ) B . Thus, (26) and (27) are valid.17 Sufficient condition
The one–dimensional analog of the condition (2) is the boundedness of the Muckenhouptconstant [9], of the condition (3) — the boundedness of the Tomaselli functional [15,definition (11)], and the analogs of the constants B , B are the Maz’ya–Rozin [7, § 1.3.2]and Persson–Stepanov [10, Theorem 3] functionals, respectively. The constants have beengeneralized to the scales of equivalent conditions in [11] (see also [2] for the case p ≤ q ). Inthe following theorem we find a sufficient condition for the inequality (6) to hold, havingthe form (28), where B v is a two–dimensional analog of the constant B (1) MR (1 /r ) from [11]in the one–dimensional case. Theorem
Let < q < p < ∞ . The inequality (6) holds if B v := (cid:18)Z R σ ( u, z ) (cid:18)Z ∞ u Z ∞ z ( I σ ) q − w (cid:19) rq du dz (cid:19) r < ∞ , (28) where C . B v .Proof. We apply Sawyer’s scheme of partitioning R into rectangles from the proof ofthe sufficiency in Theorem 2.1. Compared to Figure 1, Figure 2 below has a rectangle Q kj = (0 , x kj ) × (0 , y kj ) added. ✻ ✲ x k x k x k y k y k y k e S k e R k T k Q k (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ∂ Ω k +1 ∂ Ω k Fig. 2
Denote e E kj := E kj ∪ (cid:0) e S kj ∩ (Ω k +2 − Ω k +3 ) (cid:1) . Then (see (7)) Z R ( I f ) q w ≈ X k,j kq (cid:12)(cid:12) e E kj (cid:12)(cid:12) w . (29)Put gσ := f and write X k,j kq (cid:12)(cid:12) e E kj (cid:12)(cid:12) w = X k,j (cid:12)(cid:12) e E kj (cid:12)(cid:12) w (cid:18)Z Z Q kj f (cid:19) q = X k,j (cid:12)(cid:12) e E kj (cid:12)(cid:12) w (cid:12)(cid:12) Q kj (cid:12)(cid:12) qσ (cid:18) (cid:12)(cid:12) Q kj (cid:12)(cid:12) σ Z Z Q kj gσ (cid:19) q . (30)18or an integer l by Γ l we denote the set of pairs ( k, j ) such that (cid:12)(cid:12) e E kj (cid:12)(cid:12) w > and l < (cid:12)(cid:12) Q kj (cid:12)(cid:12) σ Z Z Q kj gσ ≤ l +1 , ( k, j ) ∈ Γ l . By analogy with how it was done in the proof of [14, Theorem 1A], we show that l − < (cid:12)(cid:12) Q kj (cid:12)(cid:12) σ Z Z Q kj gσχ { g> l − } , for all j, k. Indeed, this follows from the fact that l < (cid:12)(cid:12) Q kj (cid:12)(cid:12) σ Z Z Q kj gσ = 1 (cid:12)(cid:12) Q kj (cid:12)(cid:12) σ (cid:20)Z Z Q kj ∩{ g> l − } gσ + Z Z Q kj ∩{ g ≤ l − } gσ (cid:21) ≤ (cid:12)(cid:12) Q kj (cid:12)(cid:12) σ Z Z Q kj ∩{ g> l − } gσ + 2 l − . Further, we write for fixed l : X ( k,j ) ∈ Γ l (cid:12)(cid:12) e E kj (cid:12)(cid:12) w (cid:12)(cid:12) Q kj (cid:12)(cid:12) qσ (31) . − l X ( k,j ) ∈ Γ l (cid:12)(cid:12) e E kj (cid:12)(cid:12) w (cid:12)(cid:12) Q kj (cid:12)(cid:12) q − σ Z Z Q kj gσχ { g> l − } ≤ − l X ( k,j ) ∈ Γ l Z e E kj w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) q − (cid:18)Z x Z y gσχ { g> l − } (cid:19) dx dy. Combining the last estimate and (30), we obtain X k,j kq (cid:12)(cid:12) e E kj (cid:12)(cid:12) w . X l lq X ( k,j ) ∈ Γ l (cid:12)(cid:12) e E kj (cid:12)(cid:12) w (cid:12)(cid:12) Q kj (cid:12)(cid:12) qσ ≤ X l l ( q − X ( k,j ) ∈ Γ l Z e E kj w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) q − (cid:18)Z x Z y gσχ { g> l − } (cid:19) dx dy = X k,j Z e E kj w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) q − (cid:18)Z x Z y gσ hX l l ( q − χ { g> l − } i(cid:19) dx dy. Since l − < g ( s, t ) ≤ l almost everywhere for fixed ( s, t ) then g ( s, t ) > l − for l ≤ l and, therefore, X l l ( q − χ { g> l − } = X l ≤ l l ( q − = 2 l ( q − X l ≤ l ( l − l )( q − ≈ l ( q − . From this and H¨older’s inequalities with exponents p/q and r/q , we find that X k,j kq (cid:12)(cid:12) e E kj (cid:12)(cid:12) w . X k,j Z E kj w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) q − (cid:18)Z x Z y g q ( s, t ) σ ( s, t ) ds dt (cid:19) dx dy = Z R w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) q − (cid:18)Z x Z y g q ( s, t ) σ ( s, t ) ds dt (cid:19) dx dy = Z R g q ( s, t ) σ ( s, t ) (cid:18)Z ∞ s Z ∞ t w ( x, y ) (cid:2) I σ ( x, y ) (cid:3) q − dx dy (cid:19) ds dt (cid:18)Z R g p σ (cid:19) qp (cid:18)Z R σ ( s, t ) (cid:18)Z ∞ s Z ∞ t ( I σ ) q − w (cid:19) rq ds dt (cid:19) qr = B qv (cid:18)Z R g p σ (cid:19) qp , (31)since the sets e E kj are disjoint and g p σ = f p v . The estimates (29) and (31) imply thevalidity of (6) for all f from the subclass M. There is also a dual statement of the last theorem with the functional B w := (cid:18)Z R w ( u, z ) (cid:18)Z u Z z ( I ∗ w ) p ′ − σ (cid:19) rp ′ du dz (cid:19) r instead of B v . The proof of this fact is similar and can be carried out through the operator I ∗ f .If the weights v and w are factorizable, then the condition B v < ∞ (or B w < ∞ ) isnecessary and sufficient for the (6) to be true in the case of < q < p < ∞ , moreover C ≈ B v ≈ B w . It was established by A. Wedestig in [16] (see also [17]) for the case n = 2 that if theweight function v in (1) is factorizable, that is, v ( x , x ) = v ( x ) v ( x ) , then it is possibleto characterize the inequality (1) by only one functional for all < p ≤ q < ∞ . Theorem [17, Theorem 1.1]
Let n = 2 , < p ≤ q < ∞ , s , s ∈ (1 , p ) and v ( x , x ) = v ( x ) v ( x ) . Then the inequality (1) holds for all f ≥ if and only if A W ( s , s ) : = sup ( t ,t ) ∈ R (cid:2) I σ ( t ) (cid:3) s − p (cid:2) I σ ( t ) (cid:3) s − p × (cid:18)Z ∞ t Z ∞ t (cid:0) I σ (cid:1) q ( p − s p (cid:0) I σ (cid:1) q ( p − s p w (cid:19) q < ∞ , where σ i := v − p ′ i , i = 1 , . Moreover, C ≈ A W ( s , s ) with equivalence constantsdependent on parameters p , q and s , s only. The result of this theorem can be generalized to n > . A number of statements similar to [17, Theorem 1.1] were obtained in [12] under thecondition that weight functions v or w satisfy v ( y , . . . , y n ) = v ( y ) . . . v n ( y n ) (32)or w ( x , . . . , x n ) = w ( x ) . . . w n ( x n ) . (33) Theorem [12, Theorems 2.1, 2.2]
Let < p ≤ q < ∞ and the weight function v satisfy the condition (32) . Then the inequality (1) holds for all f ≥ (i) if and only if A M n < ∞ , where A M n : = sup ( t ,...,t n ) ∈ R n + (cid:2) I ∗ n w ( t , . . . , t n ) (cid:3) q (cid:2) I σ ( t ) (cid:3) p ′ . . . (cid:2) I σ n ( t n ) (cid:3) p ′ ; if and only if A T n < ∞ , where A T n = sup ( t ,...,t n ) ∈ R n + (cid:2) I σ ( t ) (cid:3) − p . . . (cid:2) I σ n ( t n ) (cid:3) − p (cid:18)Z t . . . Z t n (cid:0) I σ (cid:1) q . . . (cid:0) I σ n (cid:1) q w (cid:19) q . Besides, C n ≈ A M n ≈ A T n with equivalence constants depending on p, q and n. Theorem [12, Theorems 2.4, 2.5]
Let < p ≤ q < ∞ and the weight w satisfythe condition (33) . Then the inequality (1) is true (i) if and only if A ∗ M n < ∞ , where with σ := v − p ′ A ∗ M n : = sup ( t ,...,t n ) ∈ R n + (cid:2) I n σ ( t , . . . , t n ) (cid:3) p ′ (cid:2) I ∗ w ( t ) (cid:3) q . . . (cid:2) I ∗ w n ( t n ) (cid:3) q ; (ii) if and only if A ∗ T n < ∞ , where A ∗ T n = sup ( t ,...,t n ) ∈ R n + (cid:2) I ∗ w ( t ) (cid:3) − q ′ . . . (cid:2) I ∗ w n ( t n ) (cid:3) − q ′ (cid:18)Z ∞ t . . . Z ∞ t n (cid:0) I ∗ w (cid:1) p ′ . . . (cid:0) I ∗ w n (cid:1) p ′ σ (cid:19) p ′ . Besides, C n ≈ A ∗ M n ≈ A ∗ T n with equivalence constants depending on p, q and n. Theorem [12, Theorems 3.1, 3.2]
Let < q < p < ∞ . Suppose that the weightfunction v in (1) satisfies the condition (32) and I σ ( ∞ ) = . . . = I σ n ( ∞ ) = ∞ . Then (1) is valid for all f ≥ on R n + with C n < ∞ independent of functions f (i) if and only if B MR n < ∞ , where B MR n := (cid:18)Z R n + (cid:2) I ∗ n w ( t , . . . , t n ) (cid:3) rq (cid:2) I σ ( t ) (cid:3) rq ′ σ ( t ) . . . (cid:2) I σ n ( t n ) (cid:3) rq ′ σ n ( t n ) dt . . . dt n (cid:19) r ; (ii) if and only if B P S n < ∞ , where B P S n := (cid:18)Z R n + (cid:18)Z t . . . Z t n (cid:2) I σ ( t ) (cid:3) q . . . (cid:2) I σ n ( t n ) (cid:3) q w ( x , . . . , x n ) dx . . . dx n (cid:19) rq × (cid:2) I σ ( t ) (cid:3) − rq σ ( t ) . . . (cid:2) I σ n ( t n ) (cid:3) − rq σ n ( t n ) dt . . . dt n (cid:19) r . Moreover, C n ≈ B MR n ≈ B P S n with equivalence constants dependent on p, q and n. Theorem [12, Theorems 3.3, 3.4]
Let < q < p < ∞ . Assume that w in (1) satisfies (33) and I ∗ w (0) = . . . = I ∗ w n (0) = ∞ . Then (1) is valid for all f ≥ on R n + with C n < ∞ independent of functions f (i) if and only if B ∗ MR n < ∞ , where B ∗ MR n := (cid:18)Z R n + (cid:2) I n σ ( t , . . . , t n ) (cid:3) rp ′ (cid:2) I ∗ w ( t ) (cid:3) rp w ( t ) . . . (cid:2) I ∗ w n ( t n ) (cid:3) rp w n ( t n ) dt . . . dt n (cid:19) r ; (ii) if and only if B ∗ P S n < ∞ , where B ∗ P S n := (cid:18)Z R n + (cid:18)Z ∞ t . . . Z ∞ t n (cid:0) I ∗ w (cid:1) p ′ . . . (cid:0) I ∗ w n (cid:1) p ′ σ (cid:19) rp ′ × (cid:2) I ∗ w ( t ) (cid:3) − rp ′ w ( t ) . . . (cid:2) I ∗ w n ( t n ) (cid:3) − rp ′ w n ( t n ) dt . . . dt n (cid:19) r . Moreover, C n ≈ B ∗ MR n ≈ B ∗ P S n with equivalence constants dependent on p, q and n. ibliographyibliography