(1. ¥µ ËºÐµÐ ÐÐ µÐ« Ç ½ Ô , ¥µ 100875; 2. À ¶Ð µÐÇ, À ¶ 323000) Ð ´ Ù ¸ Ü°ØAE


ºÔ A ¨: ´ 37 ¾ ½¥ ± ( ¼³), 1 < |λ 1 | • • • |λ n |. ÄØ n × n » P , ∆ = {x ∈ R n : |P x| < 1} Ø ß. ¸ Ø ß ∆ Ü r > 1, ∆ ⊂ r∆ ⊂ A∆, Ü ∆ Lebesgue Â |∆| = 1. AE B k := A k ∆, k ∈ Z, B k ⊂ rB k ⊂ B k+1 , |B k | = b k , Ô¹ b = | det A| > 1. AE B ½¥Ñ § A ¼ß ©Þ, « B = {x + B k : x ∈ R n , k ∈ Z}. ω Î 2B 0 ⊂ A ω B 0 = B ω ³. Ø ¢» § » A Ö ¶ Ê³³ ρ A : R n → [0, ∞) Î ρ A (x) > 0, Ä x = 0, ρ A (Ax) = | det A|ρ A (x), Ä x ∈ R n , ρ A (x + y) C A (ρ A (x) + ρ A (y)), Ä x, y ∈ R n , Ô¹ C A 1 Ø ³. ½¥¢» § » A Ö ¶ Ê³À ( ÂÍ [6, 2.4]). Þ R n AE¤ ¼ A ² Ö ¶ Ê³ ρ ρ(x) =    b j , ¦Õ x ∈ B j+1 \B j , 0, ¦Õ x = 0. Ä¢Ý x, y ∈ R n , ¥ ρ(x + y) b ω (ρ(x) + ρ(y)). λ -, λ + Î 1 < λ -< |λ 1 | |λ n | < λ +
¢ÝÄ ³. ÛÉ ÂÍ [6] ¹Ú : ¸ Ø ³ c > 0, c -1 ρ(x) ln λ-/ ln b |x| cρ(x) ln λ+/ ln b , 
Ä ρ(x) 1, (1.1) 
c -1 ρ(x) ln λ+/ ln b |x| cρ(x) ln λ-/ ln b , Ä ρ(x) 1. 
(1.2) 
Ø Þ R n AE C ∞ ±Ø³ ϕ ± § Schwartz S, ¦Õ½¥ Å¼³AE α Ü m ∈ N ¥ ||ϕ|| α,m := sup x∈R n ρ(x) m |∂ α ϕ(x)| < ∞. ÄØ N ∈ N, AE S N = {ϕ ∈ S : ||ϕ|| α,m 1 Ä |α| N, m N }. S Ä , «¤ S AE ¥°Î ÌØ , R n AE ¢ÒÞØ³ , AE S ′ . Ä ϕ ∈ S, k ∈ Z, Þ ϕ ±Â k ¼ ϕ k (x) = b -k ϕ(A -k x). , ¦Õ A = 2Id, Ô¹ Id ¯ ¾Á» , Ï¢» § A ¼º Ö Ç´ ¼. f ∈ S ′ . f Ñ § ϕ Û ¦¹Ø³ Þ M ϕ f (x) := sup{|f * ϕ k (y)| : x -y ∈ B k , k ∈ Z}. f Ñ § ϕ ¸ ¦¹Ø³ Þ M 0 ϕ f (x) := sup k∈Z |f * ϕ k (x)|. Ä N ∈ N, f Û Grand ¦¹Ø³ Þ M N f (x) := sup ϕ∈SN M ϕ f (x). f ¸ Grand ¦¹Ø³ Þ M 0 N f (x) := sup ϕ∈SN M 0 ϕ f (x).
¡ : ª Hardy ª ² 1405 
•ÂÍ [6, 3.10] 
°, Û Grand ¦¹Ø³Ü¸ Grand ¦¹Ø³ , « Ä¢Ý ³ N 0 ¸ Ø ³ C = C(N ), Ä½¥ f ∈ S ′ , Â M 0 N f (x) M N f (x) CM 0 N f (x), Ä x ∈ R n .
(1.3) 
Ä 0 < p < ∞ AE N p :=    [(1/p -1) ln b/ ln λ -] + 2, 0 < p 1, 2, p > 1, Ô¹ [t] ¯ t ³ . Ä¢Ý N N p , Þ¢» § A ß Hardy H p (R n ) = H p A (R n ) := {f ∈ S ′ : M N f ∈ L p (R n )}, Ê³ ||f || H p = ||M N f || p . ´ÖÎ N N p , H p Þ Ù § N ±. Ø «AE (p, q, s) , ¦Õ 0 < p 1, 1 q ∞, p < q, s ∈ N, Ü s [(1/p -1) ln b/ ln λ -]. (p, q, s) . Ø ¢» § ¼ A (p, q, s)-¯ ³ ÕØ Ø³ a, ¿Î : (1) supp a ⊂ x 0 + B j , Ä x 0 ∈ R n Ü j ∈ Z; (2) ||a|| q |B j | 1/q-1/p ; (3) R n a(x)P (x) dx = 0 Ä½¥ P (x) ∈ P s , Ô¹ P s ¯ ½¥ n ³ Ö s Å . Bownik [6] ÂAE H p (R n ) ¯ Í. Ø f ∈ H p (R n ) Ü ³ ÝÞÉ Â f = ∞ j=1 µ j a j , Ô¹ a j (p, q, s)-¯ , Ü ∞ j=1 |µ j | p < ∞. ´Ø , ||f || p H p © inf ∞ j=1 |µ j | p , Ô¹ inf f ½¥ Õ . Ä Ì ² ß © Hardy Þ. 1.1 Ä ¼ A, 0 < p < ∞ Ü N N p , Þ¢» § A ß © Hardy H p,∞ = H p,∞ A (R n ) := {f ∈ S ′ : M N f ∈ L p,∞ } , Ê³ ||f || H p,∞ = ||M N f || p,∞ . × 1.1 © Ö Ý ØÕ, ß © Hardy H p,∞ ´¥ 0 < p 1 ± Ñ¹. µ ³², º H p,∞ AE Ê³Ù § N , ¿•ÛÉ Í 2.1 °, Ö N N p , H p,∞ Þ Ù § N ±. ÛÉ ©Â ¾Ö Õ. Ê 1.1 f ∈ H p,∞ , Ô¹ 0 < p 1, ¸ ØÇ¥°Ø³ {f k } ∞ k=-∞ Î ¦É ¸: (a) f = ∞ k=-∞ f k ÝÞÉ Â, ÜÄ½¥ k ∈ Z, ||f k || ∞ C2 k . (b) f k Û¯ f k = i∈N β k i , ÝÞÉ Â, Ô¹Ø³ β k i Î : (i) supp β k i ⊂ Bk i , Ü i∈N | Bk i | C 1 2 -kp , Ô¹ Bk i ∈ B; (ii) ||β k i || ∞ C2 k ; (iii) Ä½¥ k ∈ Z, i ∈ N Ü P (x) ∈ P s , Â β k i (x)P (x) dx = 0, Ô¹ ℄ ³ s [(1/p -1) ln b/ ln λ -]. ÉÖ, ¦Õ f ∈ S ′ , Î (a) Ü (b), Ï f ∈ H p,∞ , Ü ||f || p H p,∞ ∼ C 1 . 2 ÖAEÂ ÒÏ 1.1 ÚÈ Ù §ÂÍ [6, 1 , 6 ] Ø Í, ¿ × , Ä ¥Ç² Í. È ¥, Þ ÂÍ [6]. É , Ä Ö¦É © ¯ ºÔ A ¨: ´ 37 ¾
, ¿ÁÂ ¨Stein [12] ª Kalton [13] . 
Ê 2.1 g m ØÇ Ø³, 0 < q < 1. |{x : |g m (x)| > λ}| C/λ q , Ô¹ C Ù § m Ü λ, Ä½¥ {c m } ∈ l q Â x : m c m g m (x) > λ 2 -q 1 -q • C λ q m |c m | q . Ê 1.1 ÑÎ f ∈ H p,∞ . ϕ ∈ S, ϕ = 1. Ä l ∈ N, AE f (l) = f * ϕ -l , ¤ÂÍ [6, 3.8], l → ∞ , f (l) → f S ′ ¹ Â. ¤ÂÍ [6, 6.6] °, M N +2 f (l) (x) C ϕ M N f (x), Ä¢Ý x ∈ R n Ü l ∈ N, (2.1) Ô¹ ³ C ϕ ´Ù § ϕ Ù § N , µ f (l) ∈ H p,∞ . ÊÁ£©ÂÍ [6, 6.4] £©Ï ³ ÍÈ f (l) . Ä¢Ý k ∈ Z Ü N N p , AE Ω k = {x : M N f (x) > 2 k }. {x k i + B l k i } i∈N Ω k Whitney , Ω k = i∈N (x k i + B l k i ), (2.2) 
(x k i + B l k i ) ∩ (x k j + B l k j ) = ∅, i = j, (2.3) 
(x k i + B l k i +4ω ) ∩ Ω c k = ∅, (x k i + B l k i +4ω+1 ) ∩ Ω c k = ∅, i ∈ N, (2.4) 
(x k i + B l k i +2ω ) ∩ (x k j + B l k j +2ω ) = ∅ ⇒ l k i -l k j ω, (2.5 
) 
♯{j ∈ N : (x k i + B l k i +2ω ) ∩ (x k j + B l k j +2ω ) = ∅} L, i ∈ N, (2.6 
) 
Ô¹ Ù ♯ ¯ £³AE L Ø Ù § Ω k ³ ( ÂÍ [6, p. 23]). Ð θ ∈ C ∞ (R n ), supp θ ⊂ B ω , Ü B 0 AE θ ≡ 1. Ä¢Ý k ∈ Z Ü i ∈ N Þ θ k i (x) = θ(A -l k i (x -x k i )), Ü ζ k i (x) =      θ k i (x)/ j θ k j (x), x ∈ Ω k , 0, x / ∈ Ω k . ³ s [(1/p -1)] ln b/ ln λ -]. P s Ê³ Þ ||P || = 1 ζ k i R n |P (x)| 2 ζ k i (x) dx 1/2 , Ä P ∈ P s . Ø³ f l (x) (× Û ) ² P s AE¦É Þ Ø ÌØ Q → 1 ζ k i f l , Qζ k i , Ä Q ∈ P s . ¤ Riesz µÌØ ¤ Ø¸ Å P l,k i ∈ P s ¯ , «Ä½¥ Q ∈ P s , 1 ζ k i f l , Qζ k i = 1 ζ k i P l,k , Qζ k i = 1 ζ k i R n P l,k i (x)Q(x)ζ k i (x) dx. b l,k = f (l) χ Ω k - i∈N P l,k i ζ k i = i∈N (f (l) -P l,k i )ζ k i ¡ : ª Hardy ª ² 1407 Ü g l,k = f (l) χ Ω c k + i∈N P l,k i ζ k i , f (l) = b l,k + g l,k . Ä¢Ý x ∈ R n , Î ζ k i (x) = 0 ³ i ³ Ö L, Ô¹ L (2.6) ¹ ³. µ, ¦Õ ÂÍ [6, 5.2 Ü 5.3] Ö°¹£ N + 2 N Á £ (2.1) , Ü Ä½¥ x ∈ R n ¥ | i∈N P l,k i ζ k i |(x) C2 k . Ë , |f (l) (x)χ Ω c k (x)| C(M N f )(x)χ Ω c k (x) C2 k . ½ÛÄ½¥ x ∈ R n , |g l,k (x)| C2 k . µ ØµÝÞÉ Â lim k→-∞ b l,k (x) = f (l) (x). (2.7) ÉØÍ , Ý supp b l,k ⊂ Ω k ª |Ω k | = O(2 -kp ) → 0, k → ∞, lim k→∞ b l,k (x) = 0, a.e.
(2.8) 
¤ (2.7) Ü (2.8) , f (l) f (l) (x) = ∞ k=-∞ {b l,k (x) -b l,k+1 (x)} a.e. ¦ b l,k -b l,k+1 = i∈N (f (l) -P l,k i )ζ k i - j∈N (f (l) -P l,k+1 j )ζ k+1 j . Ý i ζ k i = χ Ω k Ü supp ζ k+1 j ⊂ Ω k+1 ⊂ Ω k . ÛAE b l,k -b l,k+1 = i∈N (f (l) -P l,k i )ζ k i - j∈N (f -P l,k+1 j )ζ k i ζ k+1 j . Á£ÂÍ [6, 6.3], ÛAE b l,k -b l,k+1 = i∈N (f (l) -P l,k i )ζ k i - j∈N ((f (l) -P l,k+1 j )ζ k i -P l,k+1 ij )ζ k+1 j := i∈N β l,k i , Ô¹½¥¬³ Ã S ′ ÝÞÉ Â, Ü β l,k i = (f (l) -P l,k i )ζ k i - j∈N ((f (l) -P l,k+1 j )ζ k i -P l,k+1 ij )ζ k+1 j . •AE, f (l) ¯ f (l) (x) = k∈Z f l k = k∈Z i∈N β l,k i (x) a.e., Ô¹ f l k = b l,k -b l,k+1 . b l,k -b l,k+1 = g l,k+1 -g l,k , ½Û ||f l k || ∞ C2 k . Ý P l,k+1 ij = 0 ¤× supp ζ k+1 j ∩ supp ζ k i = ∅, ½Û¤ÂÍ [6, 6.1] °supp ζ k+1 j ⊂ x k+1 j + B l k+1 j +ω ⊂ 1408 ºÔ A ¨: ´ 37 ¾ x k i +B l k i +4ω , µ supp β l,k i ⊂ x k i +B l k i +4ω . AE Bk i = x k i +B l k i +4ω . Ë , i | Bk i | = b 5ω i |x k i + B l k i -ω | b 5ω |Ω k | C||f || p H p,∞ 2 -kp . Ü, ¤ P l,k i Ü P l,k+1 ij , ¥ β l,k i (x)P (x) dx = 0, ∀ P (x) ∈ P s . β l,k i , Ô β l,k i = f (l) χ Ω c k+1 ζ k i -P l,k i ζ k i + j∈N P l,k+1 j ζ k+1 j + j∈N P l,k+1 ij ζ k+1 j . Ý |f (l) (x)χ Ω c k+1 (x)ζ k i (x)| C2 k , Ü ¼ÂÍ [6, 5.3 Ü 6.1 ª 6.2] ( 5.3 Ü 6.2 £ N + 2 N ÜÁ£ (2.1) ) ¥ ||β l,k i || ∞ C2 k . Ì |k| N1 f l k → 0 (N 1 → ∞) (2.9) S ′ ¹Ä l ∈ N Øµ Â. Ä»Ý Ì. 0 < p < 1 , ´ |k| N1 f l k → 0 (N 1 → ∞) H p0 +H 1 Ê³ÝÞÉÄ l ∈ N Øµ Â, Ô¹ 0 < p 0 < p Î [(1/p 0 -1) ln b/ ln λ -] s. µ, ´ lim N2→-∞ N2 k=-∞ f l k H 1 = 0 ØµÄ l ∈ N (2.10) Ü lim N3→∞ ∞ k=N3 f l k H p 0 = 0 ØµÄ l ∈ N. (2.11) AE, (C2 k b l k i +4ω ) -1 β l,k i Ø (1, ∞, s)-¯ . µ, ¼ÂÍ [6, 4.2], N2 k=-∞ f l k H 1 N2 k=-∞ i β l,k i H 1 C N2 k=-∞ i 2 k b l k i +4ω C N2 k=-∞ 2 k |Ω k | C N2 k=-∞ 2 k(1-p) ||f || p H p,∞ , ½Û (2.10) Â. ¸ , (C2 k b (l k i +4ω)/p0 ) -1 β l,k i Ø (p 0 , ∞, s)-¯ . s [(1/p 0 - 1) ln b/ ln λ -], ¤ÂÍ [6, 4.2] ∞ k=N3 f l k p0 H p 0 ∞ k=N3 f l k p0 H p 0 ∞ k=N3 i β l,k i p0 H p 0 C ∞ k=N3 i (2 k b (l k i +4ω)/p0 ) p0 C ∞ k=N3 2 kp0 |Ω k | C ∞ k=N3 2 k(p0-p) ||f || p H p,∞ , ½Û (2.11) Â. ¡ : ª Hardy ª ² 1409 Ä p = 1 Ý , H p0 +H 1 H p0 +L 2 , Ô¹ 0 < p 0 < 1 Î s [(1/p 0 -1) ln b/ ln λ -]. (2.9) , ´ Ý N2 k=-∞ f l k 2 N2 k=-∞ f l k 2 C N2 k=-∞ 2 k |Ω k | 1/2 C||f || 1/2 H p,∞ N2 k=-∞ 2 k/2 . Á£ Banach-Alaoglu Ü Ä ÍÈ Í, Û Ø Ç {l v } ∞ v=1 ⊂ N, Ä¢Ý k ∈ Z Ü i ∈ N, v → ∞ , β lv ,k i → β k i L ∞ ¹© * Ã. Ë , supp β k i ⊂ Bk i , ||β k i || ∞ C2 k , ÜÄ½¥ P (x) ∈ P s ¥ β k i (x)P (x) dx = 0. Ê f k = i∈N β k i . Ä ÃÒ v → ∞ S ′ ¹ f lv k → f k . AE, ¢ ψ ∈ S, ¤ • Ã (¥ Ü¬³ ) f lv k , ψ = i∈N β lv ,k i , ψ = i∈N β lv ,k i , ψ → i∈N β k i , ψ = i∈N β k i , ψ = f k , ψ , v → ∞. Õ S ′ AE |k| N1 f k → 0 (N 1 → ∞). Ø£ , ´ S ′ AE f = ∞ k=-∞ f k . µ , ¢ ψ ∈ S Ü ǫ > 0. •ÛAE °, ¸ Ø Ù l v ³ N 1 , | |k|>N1 f lv k , ψ | < ǫ/3 Ü | |k|>N1 f k , ψ | < ǫ/3. Äµ N 1 , ¸ Ø ³ v 1 , v v 1 , f lv k -f k , ψ < ǫ 6N 1 , Ä k = 0, ±1, ±2, . . . , ±N 1 . µ, v v 1 ¥ ∞ k=-∞ f lv k , ψ - ∞ k=-∞ f k , ψ |k|>N1 f lv k , ψ + |k|>N1 f k , ψ + k=N1 k=-N1 f lv k -f k , ψ < ǫ/3 + ǫ/3 + ǫ/3 = ǫ. µÝÀ lim v→∞ ∞ k=-∞ f lv k , ψ = ∞ k=-∞ f k , ψ , µ f, ψ = lim v→∞ f lv , ψ = lim v→∞ ∞ k=-∞ f lv k , ψ = ∞ k=-∞ f k , ψ , º AE S ′ ¹ f = ∞ k=-∞ f k . Ì ÉÖ . f ∈ S ′ , ÜÎ (a) Ü (b). ´ Ì s = [(1/p -1) ln b/ ln λ -] Ý . ¼ (1.3) , ´ Ä N N p , ¥ |{x : M 0 N (f )(x) > α}| Cα -p . ¤ (a) °, ||M 0 N (f k )|| C2 k . Ä¢Ý α > 0, k 0 , 2 k0 α < 2 k0+1 , f = k0-1 k=-∞ f k + ∞ k=k0 f k := F 1 + F 2 , M 0 N (F 1 )(x) k0-1 k=-∞ M 0 N (f k )(x) C k0-1 k=-∞ 2 k C 0 α. ºÔ A ¨: ´ 37 ¾ µ |{x : M 0 N (f )(x) > (C 0 + 1)α}| |{x : M 0 N (F 2 )(x) > α}|. ½Û ÉÖ , ´ |{x : M 0 N (F 2 )(x) > α}| Cα -p . Bk i = x k i + B lk i , A k0 = ∪ ∞ k=k0 ∪ i {x k i + B lk i +ω }, |A k0 | ∞ k=k0 i |x k i + B lk i +ω | CC 1 ∞ k=k0 2 -kp CC 1 α -p . µ´ |{x / ∈ A k0 : M 0 N (F 2 )(x) > α}| Cα -p . x / ∈ A k0 , ϕ ∈ S N , l ∈ Z. P ³ Ö s ½ Å , |(β k i * ϕ l )(x)| = b -l R n β k i (y)ϕ(A -l (x -y)) dy = b -l x k i +B lk i β k i (y) ϕ(A -l (x -y)) -P (A -l (x -y)) dy Cb -l 2 k x k i +B lk i ϕ(A -l (x -y)) -P (A -l (x -y)) dy C2 k A -l (x-x k i )+B lk i -l |ϕ(y) -P (y)| dy C2 k b lk i -l sup y∈A -l (x-x k i )+B lk i -l |ϕ(y) -P (y)| .
(2.12) 
x ∈ x k i + B lk i +ω+m+1 \B lk i +ω+m , Ô¹ ³ m 0, A -l (x -x k i ) + B lk i -l ⊂ A -l (B lk i +ω+m+1 \B lk i +ω+m ) + B lk i -l = A lk i -l ((B ω+m+1 \B ω+m ) + B 0 ) ⊂ A lk i -l (B m ) c = (B lk i -l+m ) c . ÌÄ»Ý . ¨l k i l, P = 0, µ sup y∈A -l (x-x k i )+B lk i -l |ϕ(y) -P (y)| sup y∈A -l (x-x k i )+B lk i -l min(1, ρ(y) -N ) b -N ( lk i -l+m) .
(2.13) 
¨l k i < l, Å P ϕ A -l (x -x k i ) ³ s Taylor . ¼ Taylor ¨ Ü (1.2) , ¥ sup y∈A -l (x-x k i )+B lk i -l |ϕ(y) -P (y)| C sup z∈B lk i -l sup 0 θ 1 sup |α|=s+1 ∂ α ϕ(A -l (x -x k i ) + θz) |z| s+1 Cλ (s+1)( lk i -l) - sup y∈A -l (x-x k i )+B lk i -l min(1, ρ(y) -N ) Cλ (s+1)( lk i -l) - min(1, b -N ( lk i -l+m) ). (2.14) Þ (2.12)-(2.14) , Ä x ∈ x k i + B lk i +ω+m+1 \B lk i +ω+m , m 0, ¥ M 0 N β k i (x) = sup ϕ∈SN sup l∈Z |(β k i * ϕ l )(x)| C2 k max sup l∈Z,l lk i b ( lk i -l) b -N ( lk i -l+m) + C sup l∈Z,l> lk i b ( lk i -l) λ (s+1)( lk i -l) - min(1, b -N ( lk i -l+m) ) . Ý l lk i AE ° l = lk i ³ . N N p = s + 2, ½Û bλ s+1 - b N , Ü l > lk i AE ° lk i -l + m = 0 ³ . AE, ´ lk i < l k 0 + m Ü l lk i + m Ä Ó « . µ, ¤ bλ s+1 -b -N 1 Ü s = [(1/p -1) ln b/ ln λ -] M 0 N β k i (x) C2 k max b -N m , C(bλ s+1 -) -m C2 k (bλ s+1 -) -m C2 k b lk i {(s+1) ln λ-/ ln b+1} b ( lk i +ω+m){-(s+1) ln λ-/ ln b-1} . µÄ½¥ x / ∈ A k0 , M 0 N β k i (x) C2 k | Bk i | (s+1) ln λ-/ ln b+1 ρ(x -x k i ) -((s+1) ln λ-/ ln b+1) . Ê g k,i (x) = ρ(x -x k i ) -((s+1) ln λ-/ ln b+1) , C k,i = C2 k | Bk i | (s+1) ln λ-/ ln b+1 , q = 1 (s+1) ln λ-/ ln b+1 , |{x / ∈ A k0 : M 0 ψ F 2 (x) > α}| x / ∈ A k0 : ∞ k=k0 i M 0 ψ β k i (x) > α x : ∞ k=k0 i C k,i g k,i (x) > α . Ý |{x : g k,i (x) > α}| = |{x : ρ(x -x k i ) -1/q > α}| Cα -q . Á£ 2.1, ¥ |{x / ∈ A k0 : M 0 ψ F 2 (x) > α}| Cα -q ∞ k=k0 i |C k,i | q Cα -q ∞ k=k0 i 2 kq | Bk i | CC 1 α -q ∞ k=k0 2 k(q-p) = CC 1 α -q 2 k0(q-p) CC 1 α -p , q -p = 1 (s+1) ln λ-/ ln b+1 -p < 0. ª. Ì 2.1 0 < p 1, N > N p , H p,∞ (N ) (R n ) = H p,∞ (Np) (R n ), Ô¹ÉAEÄ § Þ 1.1 ¹ Grand ¦¹Ø³¹ ³, Ü Ê³ || • || H p.∞ (N ) Ü || • || H p,∞ (Np ) . ×º •, ¸ Ù § N ³ C, Ä½¥ f ∈ H p,∞ Â ||f || H p,∞ (N ) ||f || H p,∞ (Np ) C||f || H p,∞ (N ) , ∀ f ∈ H p,∞ . (2.15) ÑÎ Ë ¥ H p,∞ (Np) (R n ) ⊂ H p,∞ (N ) (R n ) Ü ||f || H p,∞ (N ) ||f || H p,∞ (Np ) . ÉÖ , f ∈ H p,∞ (N ) (R n ), f 1.1 , Ô¹ ³ C 1 ∼ ||f || H p,∞ (N ) . ÉØÍ , • 1.1 É ºÔ A ¨: ´ 37 ¾ Ö °¸ Ø Ù § N ³ C ′ , ||M Np f || p,∞ C ′ C 1 , µ (2.15) Â. Í ª. 3 © ÊÁ£ Bownik [6] Â ß Hardy Í Ä ± , Ô Î ÂÍ [3], ´ ¦• AE ¬ . ©Â 1.1 £, Ä ÛAEÄ ± Ò ß © Hardy AE. ©Â ß Ñ¹ Õß¥ ¹ ß © H 1 ¥° , 1.1 ÉØ £. X 1 Ü X 2 Ù § Ñ Å Ä Î Ê , X 1 + X 2 AE Peetre K- ÌØ Þ K(t, f ) = inf {||f 1 || X1 + t||X 2 || X2 : f 1 ∈ X 1 , f 2 ∈ X 2 , f = f 1 + f 2 } (t > 0), AE ± [X 1 , X 2 ] θ,q (0 < θ < 1, 0 < q ∞) Þ [X 1 , X 2 ] θ,q = f ∈ X 1 + X 2 : ||f || θ,q = ∞ 0 (t -θ K(t, f )) q dt/t 1/q < ∞ . Ø Ñ § ± ¶ Õ [L p0 , L p1 ] θ,p = L p , Ô¹ 1 p 0 , p 1 ∞, 1/p = (1 -θ)/p 0 + θ/p 1 , 0 < θ < 1 ( ÂÍ [14]). © Ê µ Õ Ò ß Hardy AE. Ê 3.1 ¦Õ 0 < p 0 < p 1 ∞, 0 < θ < 1, 1/p = (1 -θ)/p 0 + θ/p 1 , Ï [H p0 , H p1 ] θ,p = H p . ÑÎ Ä [ , ] θ,p £É ( ÂÍ [14, 3.11.5]), ´ p 1 = ∞. 0 < p 0 1, Ô¾Ý « ¶ Ý , µ 0 < p 0 1, 0 < θ < 1, p = p 0 /(1 -θ). N ∈ N ¹. Ê ¶ ¹ M N H p0 ¥° § L p0 , L ∞ ¥° § L ∞ , ½Û [H p0 , L ∞ ] θ,p ¥° § [L p0 , L ∞ ] θ,p . ¿ [L p0 , L ∞ ] θ,p = L p , ½Û [H p0 , L ∞ ] θ,p ⊂ H p . ÉÖ , ¤ÂÍ [6, Í 5.11] °, ´ H p ∩ L 1 ⊂ [H p0 , L ∞ ] θ,p . ¦Õ f ∈ H p ∩ L 1 , F M N f (0, ∞) AE ¼ , « F (y) = inf{σ : |{x : |M N f (x)| > σ}| y}, y ∈ (0, ∞), ÜÄ t > 0, f = g t + b t i f Ñ § M N ³ s Â α = F (t p0 ) Calderón-Zygmund , Ô¹ N > s [ln b/(p ln λ -)]. ¤ÂÍ [6, 5.7] , M N i b t i p0 p0 C {x∈R n :MN f (x)>α} M N f (x) p0 dx = C t p 0 0 F (y) p0 dy. ¶Ö¬Å t p0 → t Á£ Hardy ( ÂÍ [15, p.272]), ∞ 0 t -θ M N b t i p0 p dt/t C ∞ 0 t -θp t p 0 0 F (y) p0 dy p/p0 dt/t = Cp -1 0 ∞ 0 t -θp/p0 t 0 F (y) p0 dy p/p0 dt/t Cp -1 0 θ -p/p0 ∞ 0 t -θp/p0 (tF (t) p0 ) p/p0 dt/t ¡ : ª Hardy ª ² 1413 = C ∞ 0 F (t) p dt = C||M N f || p p . (3.1) 
ÉØÍ , ¤ÂÍ [6, 5.10] , ∞ 0 (t (1-θ) ||g t || ∞ ) p dt/t C ∞ 0 t (1-θ)p F (t p0 ) p dt/t = Cp -1 0 ∞ 0 t (1-θ)p/p0 F (t) p dt/t = C ∞ 0 F (t) p dt = C||M N f || p p . (3.2) 
¿Ë ¥ 
K(t, f ) M N b t i p0 + t||g t || ∞ , ½Û, p 1 (3.1) Ü (3.2) Ï , AE p > 1 Á£ Minkowski , ¥ ∞ 0 (t -θ K(t, f )) p dt/t 1/p C||M N f || p , •AE f ∈ [H p0 , L ∞ ] θ,p . ª. Ê 3.2 0 < p < 1 < q ∞, T L 1 Ø³ ¶Î ¹ ÜÎ ||T f || p p C||f || p H p , f ∈ L 1 ∩ H p , ||T f || q C||f || q , f ∈ L 1 ∩ L q , T © (1, 1) . ÑÎ Ä f ∈ L 1 Ü α > 0, f = g + b i f Ñ § M N ³ s Â α/2 Calderón-Zygmund , Ô¹ N > s [(1/p -1) ln b/ ln λ -]. £ÂÍ [6, 1 , 5 ] AEÙ, b i ß x i + B li+ω AE, Ü {x i + B li } Ω = {x : M N f (x) > α/2} Whitney . λ i = ||b i || 1 |B li+ω | 1/p-
λ p i = ||b i || p 1 |B li+ω | 1-p ||b i || 1 p |B li+ω | 1-p b 2ω(1-p) ||b i || 1 p |{x : M N f (x) > α/2}| 1-p C||f || p 1 (2||f || 1 /α) 1-p Cα p-1 ||f || 1 . ½Û b i ∈ H p Ü || b i || p H p Cα p-1 ||f || 1 . ´Ø , ¤ÂÍ [6, 5.8], b i ∈ L 1 , || b i || 1 C||f || 1 , AE ||g|| 1 C||f || 1 . ÉØÍ , ¤ÂÍ [6, 5.8 Ü 5.10] , g ∈ L 1 ∩ L ∞ , Ü ||g|| ∞ Cα. µ, ||g|| q ||g|| q-1 ∞ |g| 1/q Cα 1-1/q ||f || 1/q 1 . T f |T g| + |T b i |, ½Û |{x : |T f (x)| > α}| |{x : |T g(x)| > α/2}| + x : T b i (x) > α/2 .
µ, ¤Ø² , 
|{x : |T f (x)| > α}| (2||T g|| q /α) q + 2 T b i p /α p C||g|| q q /α q + C b i p p /α p ºÔ A ¨: ´ 37 ¾ C α q-1 ||f || 1 /α q + α p-1 ||f || 1 /α p = C||f || 1 /α, µ• T © (1, 1) . ª. Ê 3.3 ¦Õ 0 < p 0 < p 1, 0 < θ < 1, 1/p = (1 -θ)/p 0 , Ï [H p0 , L ∞ ] θ,∞ = H p,∞ . ÑÎ Ø£ ¸ § 3.1. ³ N N p . M N H p0 ¥° § L p0 , L ∞ ¥° § L ∞ , µ [H p0 , L ∞ ] θ,∞ ¥° § [L p0 , L ∞ ] θ,∞ . ¿ [L p0 , L ∞ ] θ,∞ = L p,∞ , µ [H p0 , L ∞ ] θ,∞ ⊂ H p,∞ . ÉÖ , ¤ 1.1, Ä¢Ý f ∈ H p,∞ , f = ∞ k=-∞ i β k i , Ô¹ β k i Î : supp β k i ⊂ Bk i ∈ B, i∈N | Bk i | C||f || p H p,∞ 2 -kp , ||β k i || ∞ C2 k , ÜÄ½¥ P (x) ∈ P s Â β k i (x)P (x) dx = 0. b l (x) = ∞ k=l i β k i (x), λ i,k = ||β k i || ∞ | Bk i | 1/p0 , b l = ∞ k=l i λ i,k a k i , Ô¹ a k i (p 0 , ∞, s)-¯ . ∞ k=l i λ p0 i,k C ∞ k=l i 2 kp0 | Bk i | C ∞ k=l 2 k(p0-p) ||f || p H p,∞ C2 l(p0-p) ||f || p H p,∞ , ½Û b l ∈ H p0 , ||b|| p0 H p 0 C2 l(p0-p)||f || p H p,∞ . µ, ||b|| H p 0 C2 -lθ/(1-θ) ||f || 1/(1-θ) H p,∞ . g = l-1 k=-∞ i β k i , ||g l || ∞ C l-1 k=-∞ 2 k C2 l . Ä t > 0, l ∈ Z, 2 -l t 1-θ /||f || H p,∞ < 2 -l+1 , µ t -θ ||b l || H p 0 C||f || H p,∞ , t 1-θ ||g|| ∞ C||f || H p,∞ , •AE sup t>0 t -θ K(t, f ) C||f || H p,∞ , º AE ÉÖ . ª. Ê 3.4 0 < q < p 1 < p 0 < ∞. T ¶Î ¹ Ü (L p0 , L po ) Ü (H q , H q ) , T H p,∞ AE¥°. ÑÎ ³ N N p . Ä f ∈ H p,∞ Ü λ > 0, ´ |{x ∈ R n : M N (T f )(x) > λ}| Cλ -p ||f || p H p,∞ , Ô¹ ³ C Ù § f Ü λ. k 0 ∈ Z, 2 k0 λ < 2 k0+1 . ¼ H p,∞ , f f = k0 k=-∞ f k + ∞ k=k0+1 f k := F 1 + F 2 . Ý p 0 > 1, ¥ ||F 1 || p0 C k0 k=-∞ ||f k || p0 C k0 k=-∞ 2 k i | Bk i | 1/p0 C k0 k=-∞ 2 k(1-p/p0) ||f || p/p0 H p,∞ C2 k0(1-p/p0) ||f || p/p0 H p,∞ . ½Û |{x ∈ R n : |M N (T F 1 )(x)| > λ}| ||M N (T F 1 )|| p0 p0 λ -p0 C||T F 1 || p0 p0 λ -p0 ||F 1 || p0 p0 λ -p0 C2 k0(p0-p) ||f || p H p,∞ λ -p0 Cλ p0-p ||f || p H p,∞ λ -p0 = Cλ -p ||f || p H p,∞ . µ, ´ |{x ∈ R n : |M N (T F 2 )(x)| > λ}| Cλ -p ||f || p H p,∞ . ¡ : ª Hardy ª ² 1415 Ý M N (T F 2 )(x) ∞ k=k0 M N (T f k )(x), Ô¹ f k = i β k i . Ü°C -1 2 -k | Bk i | -1/q β k i (q, ∞, s)-¯ , µ f k ∈ H q , Ü ||f k || q H q C i 2 kq | Bk i | C2 k(q-p) ||f || p H p,∞ . T H q AE¥°, ½Û |{x ∈ R n : M N (T f k )(x) > λ}| Cλ -q ||T f k || q H q Cλ -q ||f k || q H q . µ |{x ∈ R n : M N (T (f k /||f k || H q ))(x) > λ}| λ -q . Á£ 2.1, |{x ∈ R n : M N (T F 2 )(x) > λ}| x ∈ R n : ∞ k=k0+1 ||f k || H q M N (T (T f k /||f k || H q ))(x) > λ 2 -q 1 -q 1 λ q ∞ k=k0+1 ||f k || q H q C||f || p H p,∞ λ q ∞ k=k0 2 k(q-p)
C2 k0(q-p) ||f || p H p,∞ /λ q C||f || p H p,∞ /λ p . 
ª. ¦É ÂÍ [9] ¹Ø Õ ß © Hardy AE Ò. Ê 3.5 T f (x) = p.v. R n K(xy)f (y) dy ½¥ Õß¥ ¹ , Ü L 2 (R n ) AE ¥°. Û K Î ¦É Dini : ÑÎ Ä α > 0, k 0 ∈ Z, 
2 k0 α < 2 k0+1 . f = ∞ k=-∞ f k = ∞ k=-∞ i∈N β k i f ¤ 1.1 ¿ ¯ , Ô¹ supp β k i ⊂ Bk i = x k i + B l k i +4ω . f = k0 k=-∞ f k + ∞ k=k0+1 f k := F 1 + F 2 , F 1 ∈ L 2 (R n ), Ü ||F 1 || 2 C k0 k=-∞ 2 k i | Bk i | 1/2 C||f || 1/2 H 1,∞ k0 k=-∞ 2 k/2 C||f || 1/2 H 1,∞ α 1/2 . µ, ¤ § T L 2 (R n ) AE¥°, |{x ∈ R n : |T F 1 (x)| > α}| ||T F 1 || 2 2 α 2 ||T || 2 2 ||F 1 || 2 2 α 2 C||T || 2 2 ||f || H 1,∞ /α. ºÔ A ¨: ´ 37 ¾ Ä k > k 0 , m k ∈ N, b m k -1 ( 3 2 ) k-k0 < b m k . A k0 = ∪ ∞ k=k0 ∪ i {x k i + B l k i +4ω+m k }, |A k0 | ∞ k=k0 i b m k | Bk i | C ∞ k=k0 3 2 k-k0 2 -k ||f || H 1,∞ C ∞ k=k0 3 4 k-k0 ||f || H 1,∞ /α C||f || H 1,∞ /α.
´ 
A c k 0 |T F 2 (x)| dx C||f || H 1,∞ . ¤ Fubini Ü℄ » , ¥ A c k 0 |T F 2 (x)| dx C ∞ k=k0+1 2 k i Bk i (x k i +B l k i +4ω+m k ) c |K(x -y) -K(x -x k i )| dx dy = C ∞ k=k0+1 2 k i Bk i (B l k i +4ω+m k ) c |K(x ′ -(y -x k i )) -K(x ′ )| dx ′ dy. Ý ¦Õ x ′ ∈ (B l k i +4ω+m k ) c Ü y ∈ Bk i , Ï ρ(x ′ ) > ( 3 
2 ) k-k0 ρ(yx k i ). 
µ 
A c k 0 |T F 2 (x)| dx C ∞ k=k0+1 2 k i | Bk i |Γ 2 3 k-k0 C||f || H 1,∞ ∞ k=k0+1 Γ 2 3 k-k0 C 1 0 Γ(δ) δ dδ • ||f || H 1,∞ .
ª. 
Ô ¼¡ ¢ •±ª². Ç 

1 0

 1 (Γ(δ)/δ) dδ < ∞, Ô¹ Γ(δ) = sup y =0 ρ(x)>δ -1 ρ(y) |K(x + y) -K(x)| dx. ÏÄ½¥ f ∈ H 1,∞ , |{x ∈ R n : |T f (x)| > α}| C||f || H 1,∞ /α.
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